A MECHANISM FOR THE SOLUTION OF AN EQUATION
OF THE rim DEGREE
BY W.
PEDDIE.
In the well-known system of pulleys illustrated in the diagram below, the free
end P of the last cord moves down through a distance 2n — 1 if the bar ad be moved
up through unit distance. Here n is the total number of pulleys including the fixed
one. Suppose now that ad be fixed, while a, b, c, d, are drums on which the
Fig. l .
400
W. PEDDIE
respective cords are wound. If a length a be let off the drum a, the free end of the
last cord descends by the amount 2na, the number of moving pulleys being n. If, in
addition, a length b be unwound from the second pulley, P descends farther by the
amount 2n~1b. If finally, after the various lengths have been let off, or wound on,
the first n drums, the (n •+ l)th drum be adjusted so that P retakes its initial position,
the equation
a.2w + 6 . 2 w - 1 + . . . = 0
is satisfied.
Thus the arrangement satisfies the conditions imposed by the relation
axn-]-bxn-1 + ... = 0
in the particular case in which x = 2.
drum a.
The fixed pulley may act the part of the
If bed be vertical, the drums b, c, d, must be capable of sliding ; and, if they are
so slid that the parts of the cords adjacent to them are kept horizontal, the equation
is satisfied in the particular case in which x = 1.
If the line bed be fixed at an angle 6 to the vertical, the drums being slid into
positions in which the parts of the cords adjacent to them are perpendicular to bed,
the corresponding root of the equation is 1 + sin 6. The plus sign occurs if bed
slopes downwards to the right ; the minus sign occurs if it slopes downwards to the
left. Either arrangement may be made the basis of construction of an instrument
for finding the roots of an equation. It is convenient to make the axes of the pulleys
slide in parallel slots on a metal arm to which the arm bed is hinged, and the action
of gravity on the pulleys is conveniently replaced by the control of cords wound on
spring drums. To secure inextensibility thin steel wires may be used instead of
cords.
4
If bed be a fixed rigid arm, mechanical necessities prevent the axes
pulleys coinciding with the axis of rotation of the arm carrying the pulleys.
if the original displacements, measured parallel to 6, of the centres of the
from the axis of rotation be a, ß, etc., the lengths of wire which have to
off the corresponding drums are a —a, b — ß, etc., instead of a, b, etc.
of the
Hence,
pulleys
be let
If desired this may be avoided by the following construction. Let pqrstuv
represent the rotating arm ; qr, st, and uv, representing the lines along which the
axes of the first, second, and third movable pulleys slide. The distances pq, rs, and
tu, are of course equal to the radii of the sliding pulleys ; more strictly, they are
equal to the radius of a pulley plus the semi-diameter of the cord (or fine wire)
which is wound upon it. The fixed drum a has its axis coincident with the axis
of the hinge p by which the moving arm is attached to the fixed arm pw along which
the drum b slides. Arms rxy and tz are hinged to the moving arm at the points
r and t ; and are compelled to remain parallel to pw by links wx and yz respectively.
The cord which passes over the first movable pulley and is wound on the drum b
is guided by a pin carried by the free end of an arm m which is rigidly attached to
the axle of the pulley. The pin m is at a distance qp from the line of qr. Thus the
free part of the cord between m and b is, when the drum b is slid into position, equal
to pm sin 0 ; and a similar arrangement is made with each of the movable pulleys.
A MECHANISM FOR THE SOLUTION OF AN EQUATION OF THE ?2TH DEGREE
401
The mechanism must be so arranged that, when the whole link-work is closed up,
9 being zero, the pin m is on the prolongation of the axis p ; the pin attached to the
Fig. 2.
second sliding pulley coincides with the prolongation of the axis r ; and so on.
A length a is then unwound off the drum centred at p, and the first pulley slides out
by the distance pm = a. So also does the second pulley ; but, when the rotating arm
is turned through the angle 9, the second pulley is pulled back by the amount a sin 9.
And it moves out through the additional distance b, if that length be unwound off
the drum b. Thus the necessary condition is satisfied for the second pulley; and
similarly for the others.
The free end of the last cord being wound on a spring drum fixed at the outer
end of the movable arm, the angular position of that drum is noted when, 9 being
zero, all the adjustments of the pulleys have been made, i.e. when m coincides with
p, etc. The lengths a, b, etc., having been let off the respective drums, which have
been slid if necessary to their appropriate positions, if the drum at the end of the
movable arm has retaken its initial angular position, one root of the equation at least
is unity. If the end drum is not in its initial position, the angle 9 is increased, the
drums being slid correspondingly, until the initial position is retaken. The value of
1 — sin 9, which can be indicated directly on a scale by a pointer attached to the
movable arm, is then a root of the equation. By appropriate graduation of the end
drum, the sum of the terms of the function, for any value of the variable from 0 to 1,
can be directly indicated.
I t is presumed that the radius, p, of any pin m, is so small that rO is negligible
so fir as scale readings are concerned.
If no root of the equation lies within the limits 0 and 1, or at either limit, the
equation must be altered so as to realise that condition.
M. c.
26
402
W. PEDDIE
Lastly, the instrument can be also used to solve an equation one degree higher
than the number of movable pulleys. Thus one constructed to solve a cubic can
Fig. 3.
solve a quartic. Each term being divided by x, the last term will be, say, 1/x. If
squared paper be fastened on the baseboard of the instrument, and the scale
indicating x=l-sin9
be marked on it, as also the curve xy=l, any value of x,
at which the reading of the end drum exceeds its initial reading by the value of y
given by the hyperbola, is a root of the quartic.
It may be noted that the instrument, if set to solve a given equation with the
left hand side equated to zero, will also give the solution of the equation with the
right hand side equal to any constant different from zero, by altering 9 until the end
drum has a reading differing from its initial reading by the value of that constant.
Thus it can be used to trace the value of the function.