1905 - 6 .] Mr T. Oliver on Tahe-up in Twisted Threads. 193 
Differentiating as before, and equating to zero, 
dy 7r 2 (i 2 /19 n 
dx 2L \ 6 
which vanishes when y has a minimum value, 
, 19 n A 
i. e. when — x + - = 0 
6 2 
3 
„ x = 19 w • 
Substituting this value of x in the expression for y gives the 
minimum value of the take-up or the maximum elongation, 
^ r 2 d 2 f 3 2 3 « 
Vmin ‘ = ‘2LW " 
ttH 2 3 o 
” 2L ' 76^ ’ 
i.e. T¥ of the elongation which would result from opening out 
all the twist from one of the singles separately. 
In general, if D = pd 
V = 
7r^V2 nx + x 2 
2L 
+J? Z & 
y is zero when x — 0 and when x = - — _“h- _ n 
ip 2 + 1 
y is negative between these limits 
y is positive beyond „ ,, 
The maximum elongation = • -j — ~ — 
8L \.p 2 + 1 
or — of the 
4p 2 + 1 
elongation which would result from opening out all the twist from 
the singles separately, and it occurs when 
dy rr 2 d 2 fn + x 0 9 \ a 
A 01 2l{— + 2PX] = ° 
i.e. when x = — — - . 
4^2+1 
It may be shown by geometry from diagrams similar to fig. 7 
that p = J2 or 1*4 or a four-fold thread, 
4 
or 1*7 for a five-fold thread, 
J10-2 
= 2 for a six- or seven-fold thread. 
PROC. ROY. SOC. EDIN., VOL. XXVI. 13 
