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can be inserted pegs P,, P, and Q, Q, P,-P, are fixed at 
equal distances from the centres in B, By: Q, Q, in C, Q. 
Now as B,, B, revolve through equal angles in equal times, 
the line P,P, will always move parallel to itself, and similarly 
Q,9.. 
Hence, if O be the intersection of the lines joining the 
centres, and P of P,P, and Q,Q,; and straight lines in the 
plane of the disks through O be taken as axes of co-ordinates, 
the equation to the locus of P will be 
x =a COS (mé + a,), 
y= bsin (n@ + 8,), 
where 8, which can be varied at pleasure, is the distance of the 
peg P, from the centre of B,, and a of Q, from the centre of C.. 
min are the velocity ratios of rotation of the disks Q, 
and B,. This ratio may be altered at pleasure by shifting the 
band along the cones. Either m or n is negative if the bands 
are crossed. The above equation to the locus of P may be 
written (as in Donkin’s Acoustics) : 
2a 
ry 
== asin ( 
1 
. (2a 
tra), y=dsin (248), 
T, 
where ¢ is the time, 7, the time of revolution of the disk Cc. 
7 
T, OF B,: a= Sta, B=8.. 
If 7, and +, are commensurable then the locus of P is 
a re-entering curve; if not, not. 
If 7,, 7, are nearly in the ratio of two small numbers m 
and », the curve though not re-entering after the time mn may 
be expressed by the equation 
>) - 9 
w=asin(“ttatit), y=bsin (“"e+8), 
where % is a small quantity, so that a+kt represents a slow 
