Method for providing desired balance of bowling balls

A method of using an article to facilitate determination of the location of the balance hole necessary to change the side, finger/thumb and top/bottom weights of a bowling ball from known starting weights to desired ending weights, or to determine a set of ending weights necessary to place the center of gravity of the ball at a desired location. The article may also be used to determine the amount of weight to be removed by the balance hole in order to provide the desired ending weights. The article includes a base sheet, having a set of X-Y coordinates and an arcuate scale printed on one surface, and a linear reference line movable with respect to the base sheet about an axcis through the intersection of the X-Y coordinates. Indicia scales adjacent the X and Y coordinates are marked in uniform increments. The arcuate scale intersects the X and Y coordinates at points appreciably spaced from the intersection thereof and having a radius centered at the intersection. The arcuate scale is marked in equal increments, from O at the Y coordinate to the value equal to one-quarter of the ball circumference (e.g., 63/4") at the X coordinate. The reference line is preferably printed on a transparent sheet pivotally attached to the base sheet, and an additional indicia scale, in uniform increments equal to those of the X and Y coordinate indicia scale, is preferably printed along the reference line.

BACKGROUND OF THE INVENTION 
The present invention relates to a method, and an article useful in the 
practice of such method, for altering the balance of a bowling ball to 
conform to a desired set of specifications. More specifically, the 
invention relates to an improved and simplified method of determining the 
location and size of a hole to be drilled in a bowling ball having 
predrilled grip holes and known balance characteristics in order to 
optimize such balance characteristics for a particular bowler. 
Bowling balls are normally composed of a machineable, thermosetting 
plastic, comprising an inner core and a shell with a smooth, spherical 
surface. Although the plastic materials forming the core and shell are 
essentially homogeneous, a weight block having a density greater than that 
of the ball material is normally incorporated in the core, thereby placing 
the center of gravity at some point other than the center of the ball. 
Furthermore, removal of material by drilling the grip holes, i.e., two 
finger holes and a thumb hole, shifts the center of gravity to a location 
different from that of the undrilled ball, as well as changing the balance 
characteristics between pairs of adjacent ball hemispheres. 
The action of a bowling ball as it travels down the lane is dependent upon 
the bowler's release, as well as the location of the weight block, and the 
location and amount of weight at the center of gravity. When a bowling 
ball is released in the usual manner, it tends to slide along the alley in 
its initial travel, then changing at some point, depending upon the 
previously noted factors, from a slide to a roll. The path of the ball is 
not usually the same during the slide and roll portions of the ball's 
travel. The change of direction in the transition from slide to roll is 
known as the "break" or "snap." The point in the ball's travel where the 
break begins, as well as the relative abruptness (i.e., whether a 
relatively gradual or a sharp break), also depend largely on the factors 
noted above. Of course, the coefficient of friction between the ball and 
lane also has a significant effect; thus, it is not uncommon for a bowler 
to have more than one ball, each having characteristics suited to the 
bowler's preference for the particular lane. 
In order to achieve maximum striking power, it is necessary that the bowler 
use a consistent delivery or release, and that the ball have balance 
characteristics and a center of gravity located to provide an optimum 
break point for that particular release. By "balance characteristics" is 
meant the relative balance or imbalance of two adjacent hemispheres of the 
ball taken on each of three, mutually perpendicular, imaginary planes. 
These planes are in a specified relation to the position of the grip 
holes, as is well known to ball drillers and other advanced bowlers, and 
as explained in greater detail hereinafter. 
Once a bowler has acquired sufficient experience and technique to develop a 
reasonably consistent release, the balance characteristics and center of 
gravity location of the ball must be established to optimize ball action, 
and thus score, for that particular release on a lane of known 
characteristics. The bowler's release, basically a combination of 
velocity, lift and spin imparted by the bowler, as well as the position of 
the weight block, will result in a specific "track," i.e., the circular 
area on the ball surface which contacts the lane during ball travel. The 
track concentrically surrounds the axis of rotation which, in virtually 
all cases, is disposed at an oblique angle with respect to the horizontal 
lane surface. The farther the weight block is removed from the track, the 
longer the ball will slide and the sharper it will break. Conversely, the 
closer the weight block is to the track, the earlier in its travel the 
ball will begin to roll and the more gradual the break. 
The grip holes are drilled in the ball at a position relative to the weight 
block and the bowler's normal track to provide the general characteristics 
of slide, roll and break desired for the particular bowler's delivery or 
release. In order to optimize balance characteristics, it is then 
necessary to drill an additional hole, removing sufficient material from 
the ball to produce the previously mentioned relative imbalances between 
the three sets of ball hemispheres. The present invention is concerned 
with methods and means of determining the position on the surface of the 
ball for drilling the balancing hole and the amount of material to be 
removed to provide optimum ball action for the particular bowler using the 
ball. 
For many years it has been the practice to weigh a bowling ball drilled 
with finger and thumb holes in each of three specified orientations on a 
device known as a dodo scale to determine the relative imbalance between 
three pairs of hemispheres. The location of, and amount of weight to be 
removed by, the balancing hole in order to change the known, i.e., the 
measured or "starting" weights, to the desired ending weights is then 
calculated. This operation has traditionally been performed by manual 
calculation involving comparison of each set of starting and ending 
weights with one another, and with the weights of other sets, and marking 
off calculated distances in specified directions from a reference point. 
The calculations require considerable expertise and know-how in ball 
geometry and balance characteristics on the part of the ball driller. In 
order to simplify the process, and to render it more accurate, the 
mathematics of locating and sizing a balance hole, as well as other 
parameters of ball weight and drilling calculations, have been reduced to 
formulae and programmed into a pocket computer, as disclosed in U.S. Pat. 
No. 4,742,620, issued May 10, 1988 to R. C. Manker. Performance of the 
calculations in this manner, while considerably faster and possibly more 
accurate than the usual, manual method, requires access to and familiarity 
with operation of relatively expensive and technically sophisticated 
computer equipment. 
It is an object of the present invention to provide an article of 
manufacture useful in implementing a novel and improved method of 
determining the location of a balance hole in a bowling ball having 
pre-drilled finger and thumb holes to optimize balance characteristics for 
a particular bowler. 
An additional object is to provide an article of manufacture useful in 
implementing a novel and improved method of determining the amount of 
weight to be removed by a balance hole in a bowling ball in order to 
provide desired balance characteristics. 
Another object is to provide an article of manufacture useful in 
implementing a method of determining side finger and top weights in a 
previously drilled bowling ball necessary to place the center of gravity 
in a desired location, which method may be performed easily and quickly, 
without the requirement of relatively expensive equipment such as computer 
hardware and custom software. 
A further object is to provide a relatively inexpensive and simple article 
of manufacture which may be used with minimal training and experience to 
aid in determining values concerning the balance characteristics of 
bowling balls. Other objects will in part be obvious and will in part 
appear hereinafter. 
SUMMARY OF THE INVENTION 
In a first aspect the invention is practiced in connection with a bowling 
ball which has been pre-drilled with two finger holes and a thumb hole, 
having known (starting) side, finger, and top weights. The difference 
between each set of weights is determined, including an indication of 
whether each of the ending (desired) weights is more or less than the 
starting (known) weight. The value of the difference in side weight is 
compared to the difference in finger weight to determine a first distance 
value. The difference in side weight is compared to the difference in top 
weight to determine a second distance value, each comparison being made 
and the distance values determined with the aid of the article of 
manufacture of the invention. 
A first reference point is established on the surface of the ball at a 
position 63/4" (1/4 of the circumference of a standard bowling ball) from 
a point midway between the finger and thumb (grip) holes, i.e., the 
"center-of-grip" point, the along a first arc perpendicular to a line 
between the grip hole centerlines, in a direction depending upon whether 
the ball is to be used by a right-handed or a left-handed bowler and 
whether side weight is to be added or subtracted from the starting side 
weight to arrive at the desired side weight. The first distance is 
measured on the surface of the ball from the first reference point, along 
a second arc perpendicular to the first arc, in a direction depending upon 
whether finger weight is to be added or subtracted in order to arrive at 
the desired finger weight, to establish a second reference point. The 
second distance is measured on the surface of the ball from the second 
reference point along a third arc perpendicular to the second arc, in a 
direction depending on whether top weight is to be added or subtracted in 
order to arrive at the desired top weight. The point established by the 
second distance measurement defines the position for drilling the balance 
hole. 
The amount of material to be removed in order to provide the desired 
weights is determined by first comparing the difference in starting and 
ending side weights with the difference in starting and ending finger 
weights. A first value is obtained from a predetermined relationship 
between the side and finger weight differences. The difference in starting 
and ending side weights is then compared to the difference in starting and 
ending top weights, and a second value is obtained from a predetermined 
relationship therebetween. The first and second values are then compared 
according to a predetermined relationship to arrive at a figure indicating 
the amount of weight to be removed by the balance hole. 
The invention provides a relatively simple and inexpensive article of 
manufacture for making the aforementioned comparisons of the weight values 
in order to arrive at the first and second distance values, as well as the 
values indicating the weight to be removed. The article includes a base 
sheet bearing a set of X-Y coordinates with associated numerical scales, 
and a transparent cover sheet pivotally attached to the base sheet at the 
intersection of the coordinates. The base sheet also includes an arcuate 
line having a radius centered at the intersection of the X and Y 
coordinates. The cover sheet has imprinted thereon a linear reference 
line, also preferably with an associated numerical scale, extending from 
the point of pivotal attachment of the base and cover sheets and 
intersecting the arcuate line on the base sheet. The cover sheet is moved 
relative to the base sheet about the pivotal attachment to align the 
reference line with coordinate points on the base sheet and obtain values 
from the scale on the arcuate line. 
In another aspect of the invention, the article of manufacture may be used 
to determine a set of side, finger/thumb and top/bottom weights which will 
cause the center of gravity of the ball to be located in a desired 
position. The article may then be used in the manner described above to 
determine the position of and amount of weight to be removed by a balance 
hole in order to provide the determined set of weights. 
The foregoing and other features of the invention will be more readily 
understood and appreciated from the following detailed description, taken 
in conjunction with the accompanying drawings.

DETAILED DESCRIPTION 
The invention will be best understood with reference to bowling ball 10 
shown in FIGS. 1-5 with a series of markings which may be applied to the 
surface during progression of the series of steps involved in locating the 
position for drilling the balance hole. As previously mentioned, the 
method of the invention is performed in conjunction with a ball having 
pre-drilled grip holes, i.e., finger holes 12 and 14, and thumb hole 16. 
The position of the grip holes is selected with respect to the position 
and orientation of the weight block relative to the bowler's track to 
provide generally the type of ball action desired by the bowler, as 
previously described. Optimization of the desired ball action is achieved 
by adjusting the balance characteristics of the drilled ball in accordance 
with the invention. 
As a preliminary step, the ball 10 is weighed in each of three positions on 
a conventional scale, commonly known in the bowling industry as a dodo 
scale, to determine the amount of imbalance, if any, between adjacent 
hemispheres. One pair of hemispheres is that on opposite sides of a plane 
passing through the midpoint between finger holes 12 and 14, and the 
center of thumb hole 16, a portion of the intersection of such plane with 
the surface of ball 10 being represented by line 18. The second pair of 
hemispheres lie on opposite sides of a plane perpendicularly intersecting 
line 18 at a point midway between the center of a line connecting the 
centers of finger holes 12 and 14 and the center of thumb hole 16; a 
portion of the intersection of the plane separting the second set of 
hemispheres with the surface of ball 10 is represented by line 20. The 
third pair of hemispheres is defined by a plane mutually perpendicular to 
the first two planes, line 22 representing a segment of the intersection 
of the third plane with the surface of ball 10. 
The division of bowling balls into three pairs of hemispheres by imaginary 
planes in the positions indicated in FIG. 1, and weighing the balls on a 
dodo scale to determine the amount of imbalance between adjacent 
hemispheres has been standard practice for many years. The amount of 
imbalance between the hemispheres on opposite sides of the plane 
represented by line 18 is commonly termed "side weight" with a + or - sign 
to indicate whether the greater weight is in the right or left side, 
respectively, when facing the side of the ball with the grip holes with 
the finger holes above the thumb hole (for right-handed bowlers). The 
imbalance between the hemispheres on opposite sides of line 20 is termed 
"finger weight" or "thumb weight," depending upon which of the hemispheres 
having the finger holes and the thumb hole is heavier. Likewise, the 
imbalance between the third pair of hemispheres is termed "top weight" if 
the heavier hemisphere is that containing the grip holes, and "bottom 
weight" if the reverse. 
Rules of the American Bowling Congress presently provide that regulation 
balls can have not more than one ounce (+ or -) side weight, one ounce 
finger or thumb weight, and three ounces top or bottom weight. In addition 
to these constraints, the ball may be dynamically balanced about its 
rotational axis. The rotational axis is the ball axis perpendicular to the 
plane through the "truck," i.e., the area of the ball surface which 
contacts the alley surface during travel down the alley. The position of 
the track relative to the grip holes is dependent largely upon the 
particular bowler's style of delivery or release. Assuming the bowler uses 
a reasonably consistent release, the track may be located in any of a 
number of ways, such as noting the position of scuff marks on a ball used 
for some time exclusively by that bowler. An example of the track area on 
ball 10 is indicated by reference numeral 24, the axis of rotation about 
this track being denoted as X--X in FIG. 1. 
After the finger and thumb holes have been drilled, and the ball is weighed 
to determine the side, finger/thumb and top/bottom weights, a balance hole 
may be drilled in order to provide balance characteristics which will 
optimize ball action for the release and delivery techniques of the bowler 
who will be using the ball. The balance hole must have a size (diameter 
and depth) and location on the ball surface such that the side (+ or -), 
finger or thumb and top or bottom weights conform to the desired balance 
characteristics. The American Bowling Congress also limits the maximum 
diameter of the balance hole to 11/4" under the present rules. Bowlers 
with sufficient experience in the game may know the balance 
characteristics which they desire; that is, an experienced bowler may know 
the values of side (+ or -), finger or thumb, and top or bottom weight 
which produce the ball action best suited to his or her style of delivery 
and release. The method and article of the invention may then be employed 
to expedite calculation of the location and size of the balance hole 
necessary to provide the desired balance characteristics. 
In other cases the bowler may know, or it may be determined by a bowling 
professional or experienced ball driller, where the center-of-gravity of 
the ball should be placed to best suit the bowler's needs. In the latter 
case, the method and article of manufacture of the present invention may 
be advantageously used to determine the imbalance weights necessary to 
place the center-of-gravity at the desired location. An example of each 
embodiment of the invention, i.e., locating the balance hole based on 
known balance characteristics or on a known, optimum position of the 
center of gravity, will now be described. 
As a preliminary step in the practice of both embodiments the difference 
between each set of weights is calculated. For example, if the ball has a 
starting side weight of +11/4 oz. and a desired ending side weight of +3/4 
oz., the difference is -1/2 oz., the minus sign indicating that positive 
side weight is to be removed when adjusting from starting to ending side 
weight. Likewise, the ball may have a starting finger weight of 1 oz. and 
a desired ending finger weight of 3/8 oz., for a difference of -5/8 oz., 
indicating that 5/8 oz. of finger weight is to be removed. Also, assuming 
the ball has a starting top/bottom weight of zero, i.e., the top and 
bottom hemispheres are evenly balanced, and an ending top weight of 3/16 
oz., the difference is +3/16 oz., indicating that 3/16 oz. of top weight 
is to be added. Thus, in this example, the starting and ending weights, 
and the difference between the two, may be tabulated as follows: 
TABLE I 
______________________________________ 
Start End Diff. 
______________________________________ 
side +11/4 +3/4 -1/2 
finger 1 3/8 -5/8 
top 0 3/16 +3/16 
______________________________________ 
For many years it has been the normal practise of ball drillers to 
calculate the differences in the three weights, i.e., the amount of change 
in each weight necessary to provide a ball having the desired ending 
weights, and then calculate the size and position of the balance hole 
which would produce the changes from starting to ending weights. Various 
ball drillers have used different methods, most involving a considerable 
degree of estimation, in sizing and locating the balance hole. In any 
case, a certain amount of mathematical calculation was required, combined 
with a comprehensive knowledge of bowling ball geometry and dynamics, in 
order to start from a reference point on the ball and locate the balance 
hole with respect thereto. The reference point most commonly used was the 
center-of-grip (point A in FIG. 1) and coordinates for locating the 
balance hole were calculated from that point. 
The mathematical calculations necessary to locate the balance hole are 
greatly simplified in the present invention by the use of an article of 
manufacture comprising a set of coordinates printed on a base member and a 
transparent sheet pivotally attached to the base member for movement with 
respect thereto of a cooperative reference line. The method of the 
invention involves a sequence of steps involving location of coordinates 
on the base member and, with the aid of the movable reference line, the 
determination of corresponding distance values on an arc printed on the 
base member. Another sequence of steps may be performed with the 
assistance of indicia on the movable reference line to determine the 
amount of weight to be removed by the balance hole. 
Before describing the various steps involved in performing the method(s) of 
the invention, the preferred embodiment of the article of manufacture will 
be described, with reference to FIGS. 2-4, wherein the article is shown in 
top perspective, exploded perspective and plan views, respectively. The 
article comprises base member 26, preferably an opaque sheet of rigid or 
semi-flexible paperboard, plastic, or other suitable material, and cover 
sheet 28 of transparent material pivotally attached to base 26 by 
two-piece rivet 30. Printed on the upper surface of base 26 is a set of 
X-Y coordinates, generally indicated in FIG. 4 by reference numerals 32 
and 34, respectively, and arcuate scale 36. 
The X-Y coordinates are of equal scale and marked in equal increments, 
although not necessarily corresponding to actual measurements in inches or 
other units. The arc of scale 36 has a radius centered at the intersection 
of the X-Y coordinates, at which point the pivotal connection of base 26 
and cover 28 is located. Scale 36 is marked in equal increments from 0 at 
the Y coordinate to 63/4 at the X coordinate. The 63/4 increments of the 
arcuate scale correspond to the number of inches in one-quarter of the 
actual circumference of a standard 27 inch circumference bowling ball. 
Although the increments of scales 32 and 34 must be equal to one another, 
they need not bear any particular relationship to the increments of scale 
36, which may intersect the X and Y scales at any desired location 
consistent with easy reading of the scales. 
The edges of base 26 adjacent X-Y coordinates 32 and 34 are straight and 
the edge outwardly adjacent scale 36 is arcuately concentric therewith, 
cover 28 being of the same size and configuration, although it will be 
understood that the peripheral configuration of the base and cover have no 
effect on operation of the invention. It is preferred, for ease of finding 
the intersection of points on the X and Y coordinates, that base member 26 
be printed with a grid, as indicated in FIG. 4. The grid, as well as the 
numerical scales 32, 34 and 36, are visible through transparent cover 
sheet 28, which bears reference line 38, extending linearly from the point 
of pivotal attachment of the base and cover at the intersection of the X-Y 
coordinates. Scale 40 is provided adjacent reference line 38 for use in 
practicing one aspect of the invention, as explained later. The increments 
of scale 40 are equal to those of scales 32 and 34, beginning at 0 at the 
intersection of the X and Y coordinates. 
Turning now to the steps involved in practicing the invention to locate the 
balance hole after the differences between the beginning and ending side 
(.+-.), finger/thumb, and top/bottom weights have been determined as 
previously explained, the values of the differences in finger and side 
weight are located on the X and Y coordinates, respectively. Continuing 
with the example of the values given in Table I, 5/8 is located on scale 
32 and 1/2 on scale 34; the + or - signs indicating whether the weight is 
to be increased or decreased from starting to ending weight are not taken 
into account at this point in the procedure. Cover sheet 28 is moved as 
necessary to make reference line 38 cross the intersection of X coordinate 
5/8 and Y coordinate 1/2, i.e., in the position shown in FIG. 5, where it 
will be noted that reference line 38 crosses arcuate scale 36 at indicia 
mark 35/8. This is the distance in inches from reference point B, measured 
along line 22, in the upward direction if finger weight is to be removed 
(as in the present example) and in the downward direction if finger weight 
is to be added, for establishing a second reference point. The terms 
"upward" and "downward" are meant to apply to the ball in the orientation 
shown in FIG. 6, i.e., with finger holes 12 and 14 above thumb hole 16, 
where the position of reference point C is indicated. 
As the next step, the value of the differences in top weight and side 
weight are located on the X and Y coordinates, respectively, and cover 
sheet 28 is moved to cause reference line 38 to cross the intersection of 
these coordinates on base 26. Following the example of Table I, 1/2 is 
located on scale 34 and 3/16 on scale 32, again without considering the 
sign of the weight difference value, and cover 28 is positioned so that 
line 38 crosses the intersection of these coordinates, as shown in FIG. 7. 
In this position, line 38 crosses arcuate scale 36 at indicia mark 11/2. 
This distance (11/2 inches) is then measured from point C, perpendicularly 
to line 22, in a direction away from the grip holes if ton weight is to be 
added (as in the present example), or toward the grip holes if top weight 
is to be reduced (subtracted). The point so located is marked on ball 10, 
as indicated at point D in FIG. 8, and the balance hole is drilled at this 
point. 
The amount of weight to be removed to provide the desired ending weights 
may be quickly and easily determined with the assistance of the article of 
manufacture. Cover sheet 28 is moved to make reference line 38 cross the 
intersection of the X-Y coordinates corresponding to the differences in 
finger weight and side weight, respectively. This is the same step 
previously performed to determine the distance from reference point B to 
be measured along line 22, i.e., reference line 38 is positioned as shown 
in FIG. 5. The indicia reading on scale 40, adjacent reference line 38, at 
the intersection of the X-Y coordinates corresponding to the differences 
in finger and side weights provides a first value, which may be noted when 
cover sheet 28 is originally positioned to determine the distance value 
from point B to point C. 
As the next step in determining the amount of weight to be removed, cover 
sheet 28 is positioned to cause reference line 38 to cross the 
intersection of the X and Y coordinates corresponding to the differences 
in the starting and ending top weights and side weights, respectively, 
i.e., the cover sheet is positioned as in FIG. 7. The indicia reading on 
scale 40 at the intersection of the X and Y coordinates at this point 
provides a second value; again, this value may be noted when performing 
the step to determine the distance from point C to point D, as previously 
described. As the final step, cover sheet 28 is positioned to cause 
reference line 38 to cross the intersection of the X and Y coordinates 
corresponding to the first and second values obtained from scale 40. The 
indicia reading on scale 40 adjacent this intersection indicates the 
number of ounces to be removed in drilling the balance hole. The proper 
combination of drill bit size (hole diameter) and depth of the hole to 
remove the required amount of material may be obtained from standard 
charts. 
In the preceding example, it was assumed that the bowler knew from prior 
experience the ending weights necessary to provide the desired ball 
action. However, instead of knowing the desired ending weights, the bowler 
may know the optimum location for the center of gravity relative to the 
bowler's track. That is, since ball action for a particular release 
depends on the location of the center of gravity and its relation to the 
bowler's track, the desired type of ball action can be achieved by proper 
placement of the center of gravity as determined by the location of a 
balance hole and the amount of weight to be removed. The previousy 
described method proceeds from three sets of known starting and ending 
weights, and the difference between the two weights of each set; the 
location of a balance hole and the amount of weight to be removed 
necessary to provide the desired ending weights is determined without 
specific reference to where the center of gravity will be located after 
the balance hole is drilled. When the desired position of the center of 
gravity is known, the steps described in the following paragraphs may be 
performed in order to determine the required ending weights. 
The desired position of the center of gravity (actually, the point on the 
surface of the ball closest to the center of gravity) is marked on the 
surface of the ball, for example, the point indicated at E in FIG. 9. 
Points A and B are located and arc 22 is marked on the ball surface, 
perpendicular to arc 20, as in the previous example, point B being located 
on the side of point A nearest point E. Arc 42 is drawn on the surface of 
the ball extending through points A and E, intersecting line 22 at point 
F. The distance along line 22 between points B and F is measured, and 
reference line 38 is positioned to intersect arcuate scale 36 at the 
indicia marking corresponding to the distance measured, in inches. A set 
of values for ending finger weight and corresponding ending side weight is 
provided by indicia scales 32 and 34, respectively, at all points of 
intersection of the X-Y coordinates with reference line 38. Such values 
may be chosen at any intersection point, keeping in mind the maximum 
allowable side and finger weights. 
After the ending side weight is selected from scale 34 and the 
corresponding ending finger weight (or thumb weight, if point E is below 
line 20) is noted from scale 32, the distance on the surface of the ball 
from point E to point F is measured. Reference line 38 is then located in 
a second position, intersecting arcuate scale 36 at the indicia mark 
corresponding to this distance (in inches). The point of intersection with 
reference line 38 of the value previously chosen from scale 34 to 
represent ending side weight is located; the ending top/bottom weight is 
the value on indicia scale 32 at this point of intersection. After thus 
determining the ending side, finger/thumb and top/bottom weights necessary 
to place the ending center of gravity in the desired location, the 
differences between these ending weights and the starting weights 
determined by weighing the ball on a dodo scale are calculated and the 
location and the amount of weight to be removed are determined according 
to the steps previously described. The article of manufacture may be 
advantagously used to calculate the approximate ending side, finger/thumb 
and top/bottom weights which will place the center of gravity at a desired 
location relative to the grip position whether or not the grip holes have 
been pre-drilled. This method may be performed strictly as a "paper" 
calculation without specific reference to the ball surface by way of 
marking points, axes, or the like, thereon, and will be best understood 
with reference to FIG. 10. 
A center-of-grip point A' is established by perpendicular lines in the 
usual manner, and a first reference point B' is placed at a distance 1/4 
of the ball circumference from the center-of-grip point, along line or 
vector 2D' as in previous examples. Point E' corresponding to the point on 
the surface of the ball which is closest to the ending center of gravity 
is then placed in the desired position relative to the center-of-grip. 
Many bowlers, for example, prefer the "center-of-gravity point to be 
substantially coincident with the axis of rotation, as determined by their 
particular track, which is in a known relation to the grip position. A 
second reference point C' is located on line or vector 22', at the 
intersection thereof with line 43, line 22' being perpendicular to line 
20' and line 43 being perpendicular to line 22' and passing through point 
E'. A first distance d.sub.1 is measured from point B' to point C', and a 
second distance d.sub.2 is measured from point C' to point E'. Reference 
line 38 is positioned to intersect arcuate scale 36 at the indicia marking 
thereon corresponding to the first distance d.sub.1. A set of 
corresponding finger/thumb and side weights is then selected from the X 
and Y coordinate scales, respectively, at any point intersected by 
reference line 38, keeping in mind the maximum allowable imbalance 
weights. 
The reference line is then positioned to intersect arcuate scale 36 at the 
indicia mark corresponding to the second distance d.sub.2. The ending 
top/bottom weight is the value on the X coordinate indicia scale which 
intersects reference line 38 at the position intersected by the side 
weight value selected in the preceding step on the Y coordinate indicia 
scale. 
Once a set of ending side, finger/thumb and top/bottom weights have been 
determined in this manner, the position of and amount of weight to be 
removed by the balance hole may be determined, for a pre-drilled ball 
having known beginning weights, by the previously described methods. It 
should be noted that the last-described method is not as accurate as that 
described in connection with FIG. 9 linear relationships are used, rather 
than the actual, spherical relationship of the ball surface. However, the 
approximations will be adequate for most purposes.