492 Proceedings of Royal Society of Edinburgh. [sess. 
the subject of experiment. The results of the author’s experi- 
ments give reasonable ground for the belief that the law of 
compression is such that a + b = d to a first approximation if e 2 
is not >'6, where a and b are the semi-major and semi-minor axes 
respectively of the elliptical section, and d the original diameter 
of the unstrained single thread. In any case it is instructive to 
work out the results for this hypothetical case. This is practi- 
cally equivalent to reckoning the perimeter constant if e is not 
large. 
Proof . — The perimeter of an ellipse = 4 aj~ - g 2 sin 2 6)d6 
which is a complete elliptic function of the second order, values 
of which may be obtained from tables for values of e and 0. But 
as the compressibility of the material is not known exactly, it is 
unnecessary to work with exact values. 
*yi - e 2 sin 2 6 = 1 - |e 2 sin 2 6 — -g-e 4 sin 4 0 ... . (by Binomial 
Theorem). 
Integrating term by term between the limits 6 = 0 + 6 = ^ radians. 
The perimeter =2 va (1 - Je 2 - ¥ 3 T e 4 . . . .) 
Neglecting all powers of e of the fourth and higher degree 
is small, and if a + b = d 
then 7 r [a + b) = ird a constant, viz., the original circumference of 
the single thread. 
Substituting for a in a + b = d 
= 77 -(a + b) v b = aj 1 - e 2 
approx, when e 
b 
.'. mean projection width of strained thread = - (Fj + 1)6 
7 T 
