108 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and the contraction on B will be 
y 2 = kd^x(2n 2 + x) + 4M) 2 # 2 
Assuming that corresponding points on the axes of the singles move 
in a circle, then D = - 1 *— 2 . 
Then y, = kdfie(2 + x) + 4 1 & + a? 
y 2 = M 2 2 a;(2n 2 + a) + + ^) g2 
Writing the ratio ^ — d 2 = /, 
4/ 1 = M 2 2 ^{(2/ 2 + 2/+1)x+2/ 2 tz 1 } . . . (1) 
y 2 = hd^x{ (/ 2 + 2/+ 2)a? + 2w 2 } . . . (2) 
When the single threads are of different sizes, corresponding points on 
their axes may not move in the same circle. Assuming that the centre 
of area of cross-section is also the centre of revolution, corresponding 
points on the axes will revolve in circles of different sizes. 
Let 2 be the distance of the centre of area from the axis of the thick 
thread A ; then, by equating moments about G, the centroid of area, 
?s 2 ' 
d, + d 
2 _ 
_ a 2 y 1 + A ,) 
_ ./ + 1 ^2 
^ 2 +l ’ 2 
where f=d 1 /d 2 as before. 
