212 Proceedings of Royal Society of Edinburgh. [sess. 
axle of this crown wheel can he moved round the common axis 
of the vertical side wheels and clamped in any required position 
by means of the set-screw and overhead arch, as shown. Each 
of the side wheels carries an eccentric pulley, and to these eccentric 
pulleys the “ summation wire ” descends. As the side wheels are 
of equal diameter (3 inches), they must revolve at the same speed ; 
and also the throw of their pulleys is made to he the same (J inch). 
Hence the resultant displacement of the summation wire is that due 
to two S.H.Ms. of equal period and amplitude. How the whole 
point of the device is this, that the relative phase of these S.H.Ms. 
can be altered at will by moving the axle of the crown wheel. 
That this is so may easily he seen by supposing one of the side 
wheels to be fixed, when it is apparent that if the axle of the 
crown wheel be unclamped and rotated through an angle a, the 
other side wheel will have to rotate through an angle 2a.* 
Mathematically we have then, ij being the displacement imparted 
to the summation wire, 
y = J cos t + J cos ( t + 2a), 
y = cos a . cos (t + a) ; . . . (1) 
that is, we obtain by means of this arrangement a S.H. dis- 
placement of the wire, the amplitude of which may be varied from 
0 to 1 by turning the crown-wheel axle through an angle of 90°. 
In fig. 4 are shown two traces, each of a S.H.M., the amplitude 
of which was thus altered, the axis of the crown wheel having been 
slowly moved round by hand as the machine was running. 
If the above mechanism is to be used merely for setting the 
amplitudes before the machine is put in motion, it is quite satis- 
factory ; but if the crown-wheel axle be moved ivhen the machine 
is in motion, then in equation (1) a is no longer a constant, but 
becomes a function of the time; that is (1) no longer strictly 
represents a S.H.M. of periodic time r/t, and indeed in general 
it would not represent a S.H.M. at all. Of course in practice 
most likely it would only be wished to change the amplitude 
slowly, in which case the error would be very small. All error, 
however, can be removed by means of the following modification of 
* The matter is fully discussed in S. Dunkerley’s Mechanism , p. 97 (edition 
1905). 
