Patent GB-728242-A (GB)
Title: Apparatus for the determination of polynomial functions and the solution of algebraic equations
IPC: G06G7/36
Filed: 1952-10-06
Abstract: 728,242. Electric analogue calculating systems; ohmic resistors. WALKER, W. J., WALKER, R. M., DUGGAN, T. O., and ARCHER, A. I. Oct. 6, 1952 [Oct. 11, 1951], No. 25014/52. Class 37. [Also in Group XIX] A mechanical computing apparatus for calculating, according to mathematical principles set forth, powers and polynomial functions of a variable and the solution of polynomial equations therein, comprises means defining a pair of rectangular axes having common origin and respectively scaled linearly and logarithmically, means for establishing a point co-ordinated with the axes and corresponding to the variable in its location on a straight line intersecting the axes at a slope representing the negative quantitative value of the logarithm of the variable, whereby to establish an intercept on the linearly scaled axis corresponding to the exponent of the required power function, and to establish an intercept on the logarithmically scaled axis corresponding to the point and proportional to the logarithm of the required power function, and means from which the value thereof may be read off directly for an infinite angular variation of the slope of the line; the axisdefining means being a sectional chart divided into rectangular co-ordinates, the point establishing means being a point marking on the chart, the means providing angular variation being a transparent mask having closely spaced parallel lines marked across it and applied against the chart so as to intercept its axes and there being provided a mounting for the mask whereby it may be rotated on an axis perpendicular to the plane of the chart axes to determine a desired slope of the lines with respect to the axes. In an analogous electromechanical computing apparatus, the axes are respectively defined by the straight edge of a resistance element constructed so that its resistance along such edge varies logarithmically, and by the direction of a line perpendicular to the resistance edge; and the straight edge represented by an arm angularly movable in a plane parallel to the resistance edge about a perpendicular axis representing the point marking of the graph corresponds to one of the lines of the rotary mask, while the point of engagement with the resistance edge of an electric contact traversible along the latter edge and the arm edge responsively to angular movement of the arm represents the intersection of the rotary mask line with the logarithmically scaled axis; the resistance intercepted by the contact along the element being connected with a current. source and calibrated indicating galvanometer. A mechanical computing device for evaluating the function a r x<SP>r</SP> comprises (Fig. 2) a sectional chart 4 in rectangular x and y co-ordinates scaled linearly along the axis 6 and logarithmically along the axis 7, carried on a baseboard 5. A disc 1 of transparent plastic material inscribed with parallel lines 2 and mounted within an L-section circular frame 3 is rotatable by knobs 1a about the axis of the chart and locatable in fixed positions by pegs 3b inserted in holes 3a; while an arrow 9 coincident with an inscribed diametrical line of the disc is movable along a quadrantal scale 8 extending from a zero line 8a collinear with the x-axis through the centre of the sectional chart. A point is selected on the sectional chart defined by the coefficient a r  along the logarithmic y axis and the power r along the linear x axis, while the rotary disc is adjusted so that the arrow indicates the value of x on the scale 8 which is calibrated so that the slope in of the incribed parallel lines is given by It is shown that the intercept on the y-axis of the line intersecting the selected point represents arxr and that the polynomial may be constructed and solved for all real values of x when f(x)=0 by using the device to compute successive terms of the polynomial, and determining the value of x for which their algebraic sum is zero. It is also shown that when the appropriate line through the selected point (a r r) intersects the y-axis outside the limits of the chart, the intercept on the parallel axis r=b represents the quantity so that f(x) may be determined by multiplying the algebraic sum of the intercepts by the constant xb and the roots of f(x)=0 may be determined as before. The sectional chart may be graduated with reference to other systems of co-ordinates. Fig. 4 shows an electrical analogue of the above mechanical calculator in which the y or logarithmic axis is represented by the straight vertical edge of one of a plurality of parallelly mounted logarithmic resistances RL1, RL2, RL3 and the x or linear axis by a perpendicular thereto. Contacts B<SP>1</SP>, B<SP>11</SP>, B<SP>111</SP> engaging the edges of the respective resistances are slidably mounted on straight parallel arms A<SP>1</SP>, All, A<SP>111</SP> carried by a common spindle 10 rotatable by a drum 11 calibrated in terms of x so that the slope of the arms is given by m=log 10 x. Each resistance corresponds to a term a r x of the polynomial f(x) and is mechanically adjustable in space along the x and y axes with reference to scales (not shown) to define the appropriate co-efficient a r  and power r, while the several resistances are energized from a current source 13 through respective reversing switches SR1, SR2, SR3 determining the sign of each term. The slider of each resistance is connected to the beginning of each succeeding resistance; the beginning of the first and the slider of the last resistance being interconnected through a galvanometer G, whose reading is proportional to the value of the polynomial f(x) for the preset parameters a r , x and r. The real roots of the polynomial are obtainable by adjusting the input value of x to zeroize the galvanometer reading. Fig. 5 shows a modified electric analogue computer for extending the range of r over several decades in which the plural logarithm resistances (such as RL1, RL2) are A.C. energized from transformer T over sign determing reversing switches SR1, SR2, current equalizing resistors R1<SP>1</SP>, R1<SP>2</SP> and RC1, RC2 and connections xx, yy, and the input dial ID mechanically traverses the resistances over successive pairs of contacts B10 B11, B11 B12 ... B13 B14 and B20 B21; B22 B23 . : . B23 B24 which are spaced apart by precisely one decade of the slightly longer logarithmic resistors so that the potential tapped off each contact is 10 times that of the previous contact of the same pair. Currents derived from the several contacts are connected through ganged switches SC1, SC2 adapted to translate the contact connections over one complete decade to corresponding sections of decade attenuators RA1, RA2 attenuating by a factor of 10 per section, and the outputs from the respective sections are connected over corresponding contacts CO1-C41 and CO2-C42 of relays LRO-LR4 together with the output of a calibrated potentiometer RP controlled by the output dial OD, through mixing resistors RL1; RL2, R3 to an amplifier AM and phase detector PD feeding a galvanometer G. Relays LRO-LR4 are selectively energized from supply PR through sliding contacts D1, D2 respectively mechanically linked with the movements of resistances RL1, RL2 over contacts A10-A14 and A20-A24 and through normally closed contacts C03-C43 of the respective relays so that the energization of any relay disables all subsequent relays in the circuit, while signal lamps X1-X10<SP>5</SP> energized over relay contacts C04-C44, indicate the power of 10 to be applied to the output indication of the galvanometer; multiple switch SR operating to remove a portion of the mixing resistor RM3 to multiply the current from potentiometer RP by 10 and at the same time adjust the lamp bank circuit to indicate the appropriate power of 10. Traversing the respective logarithmic, resistances RL1, RL2 over a single contact of the series B10-B14 and B20-B24 in accordance with the input parameter r produces a range of voltage change corresponding to a decade variation of r, while traversing the resistances from one contact to the next in the series operates a selected relay system to vary by a factor of 10 the attenuation applied to the derived voltages, so that the voltages applied to the mixing resistances from the respective logarithmic resistances are progressively variable with the variation over several decades of the value of r appropriate to the polynomial term represented by each logarithmic resistance while the appropriate power of 10 applicable to the solution is indicated by the lamp bank. The range of r is scaled from unity, and resistors R1<SP>1</SP> and R1<SP>2</SP> are adjusted to give a potential on terminals xx, yy corresponding to unity. Where any terms of the polynomial involve complex numbers, phase-shift networks may be introduced at P1 or P2 in the circuits of the appropriate terms, and also in the circuit of potentiometer RP, whereby both magnitude and phase of the balancing voltage are adjustable for a null reading of galvanometer G. Successive terms of a polynomial may be represented by further assemblies of logarithmic resistances movable over decade contacts in response to variations in r, whose outputs may be mixed with those derived from logarithmic resistances RL1, RL2. Each logarithmic resistance is enclosed in a carrier 16<SP>1</SP> (Fig. 7) movable up and down a pair of parallel vertical guides 171 by a roller 18<SP>1</SP> engaging the upper edge of a corresponding arm A<SP>1</SP> pivoted about an axis 10 corresponding to that shown in Fig. 4 and carrying a cursor 22 co-operating with a vertical scale 21a graduated in terms of x. Each pair of guides 171 is carried between upper and lower saddles 19<SP>1</SP>a and 19<SP>1</SP>b sliding on horizontal guides 20<SP>1</SP>, the upper of which has a linear scale calibrated in terms of r co-operating with a cursor 22<SP>1</SP> on saddle 19<SP>1</SP>a to measure the resistance e