1904-5.] Mr T. Oliver on Diameters of Twisted Threads. 489 
ing the twist on the throstle frame, the threads are under 
considerable tension, which strains the cylindrical singles. When 
the stress is relieved, after the thread passes away from the 
throstle, the friction between the rough surfaces of the singles 
prevent to some extent the natural elasticity of the material from 
bringing the thread hack to its original form. The single threads 
no longer present a circular cross section, hut elliptical, with the 
minor axes of the ellipses everywhere at right angles to the line 
or surface of contact. The mean projection width is now more 
difficult to find, since the integral is of a higher order. Section 
C is now as shown in fig. 6. 
Let BE = «, BE = b , BA = r. 
The polar equation to the ellipse when 0 is the angle CBO or 
angle of twist is 
1 _ sin 2 0 cos 2 0 
r 2 a 2 b 2 
ah 
J b 2 sin 2 0 + a 2 cos 2 6 
b 
\/ cos 2 0 ~f- ^ sin 2 0 
a 2 
b 
\/l- 
b 
J 1 - e 2 sin 2 0 
Therefore AB or CD = ^ 
J 1 - e 2 sin 2 6 
and BC = BO cos 0 = 2 b cos 0 
But the projection width = AD = AB + CD + BC 
= 2r+2b cos 0 
= 2b( = ===== + cos 6) 
\ - e 1 sm 2 0 / 
Then the sum of all the sections between the limits 0 = 0 and 
0 = 90° or J radians =2 b [ Y ( — — + cos 0)d0 
- 1 J o \ Jl - e 2 sin 2 0 J 
l W/* OOBflcW) 
\J o D 1 - e 2 sm 2 0 J o / 
1 - — ■) sin 2 0 
whp.rp _ 1 
b 2 
