Improved Arrangement and Combination of Apparatus to be Employed in Harmonic Analysis.
Year: 1894

page_text: Date of Application, 3rd Aug., 1894-Accepted, 22nd Sept., 1894
Improved Arrangement and Combination of Apparatus to be
Employed in Harmonic Analysis.
I, GEORGE UDNY YULE of 40 Argyll Road, Kensington, in the County of
London, Demonstrator of Applied Mathematics, University College, London, do
hereby declare the nature of this invention and in what manner the same is
to be performed to be particularly described and ascertained in and by the
following statement :-
My invention is illustrated by the accompanying drawings in which
Figure 1 is a plan of the combined apparatus in a position it might assume
in practice.
Figure 2 is a cross section of the rolling parallel ruler showing a rolling disc in
use therewith and
Figures 3 and 4 and Figures 5 and 6 are respectively underside views and cross
sections of other forms of rolling discs.
The theorem upon which my invention is based is that known as Fourier's by
which it is known that if y be a function of x, y may be expanded in terms of sines
and cosines of multiples of .r, so that if x range from + to
y = A + A, sin x + A, sin 2x + A, sin 3x +
0
1
3
+ B, cos x + B₂ cos 2r + B, cos 3.r +
.
we may write
[Price Su.]
area of OQPO ± nr π B₁.

page_text: The sign+or
-
to be taken depends on the initial position of A or A' to the
right or left, above or below the centre of the disc.
may conveniently be made a multiple of
TT
The area of O.Q.P.O, the original curve will have been already determined in
order to obtain A. Consequently if we can determine the areas of the curves
described by the points A A' of the disc we shall have the coefficients A,
and B.
The following is a method of carrying the invention into effect. L L' is a
bar of sheet brass or other metal with a fine rack cut along its fore edge, and
attached to the front of a rolling parallel ruler. D is a flat toothed wheel gearing
into the rack its centre may be given by an inserted disc d of glass marked on the
underside with two lines at right angles, the continuations of the marked lines
passing through or perpendicularly under the points A A'. As many such wheels
or discs will be required as terms in the Fourier series, the numbers of teeth
in them being in the ratios 1,,, etc. and the largest wheel having as many
teeth as the length O.P. of the rack.
To obtain the area of the curve described by A or A' slight depressions may be
formed at these points on the upper side of the wheel or disc. If now the tracer X¹
of a planimeter X be allowed to rest in one of these depressions A A' while the
centre of the disc is carried completely round the curve O.Q.P the difference
between the initial and final planimeter readings will be proportioned to the area
of OQPO + A, (or B, as the case may be).
As the tracer X of the planimeter will be raised above the surface of the paper
on which the curve is drawn by this proceeding, it may be made adjustable
vertically within its socket.
It may be desirable, especially in obtaining the coefficients of the higher terms
of the series, to bring the points A A' outside the toothed wheel D. In this case a
metal disc D is attached above the wheel D, as shown at Figures 3 to 6 so as to
sweep over the rack bar L L' when the wheel D gears into it, and the depressions
at A A¹ may be formed in the upper disc D¹.
I would here remark that the form of planimeter indicated in the drawing is that
known as "Amsler's." I do not however confine myself to the use of such form of
planimeter as other forms may also be employed.
Having now particularly described and ascertained the nature of my said inven-
tion and in what manner the same is to be performed I declare that what I
claim is :-
1. The improved apparatus to be employed in harmonic analysis consisting in the
arrangement and combination of a parallel moving rack, a set of toothed discs,
having centrally inserted glass disc marked with lines crossing each other at right
angles to indicate the centre and having depressions in the continuations of such
cross lines to receive the tracer of the planimeter and a planimeter substantially as
described.
2. The combination with a planimeter and a parallel ruler having a rack on its
fore edge, of a set of toothed discs having centrally inserted glass discs marked
with lines crossing each other at right angles to indicate the centre and having
depressions in the continuations of such cross lines to receive the tracer of the
planimeter substantially as herein shown and described and for the purpose stated.
Dated August 3rd 1894.
G. U. YULE,
By Harris & Mills,
London, Agents.