&aT 
sSSS¡l®íi? 
ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS. 
[653 
z = 
0X1' 
and the equation becomes 
or say 
il' 2 
il 2 — a 2 a 2 a 2 
V(il 2 -a 2 )’ 
=^i-i 
aadkl 
= d0, 
V(X1 2 — a 2 . il 4 — a 2 a 2 ) 
viz. this equation determines il as a function of 6, and we then have 
ds = kldO, 
dz = V(il 2 — a 2 ) dd, 
x = a cos 0, 
y = a sin 6, 
equations which determine x, y, z, s as functions of the parameter 6, and give thus 
the edge of regression of the torse in question. 
It is clear that the formulae are very much simplified in the case a = a, where 
the radius of absolute curvature a is equal to the radius a of the cylinder; but it 
is worth while to develope the general case somewhat further. 
Considering the elliptic functions sn u, cn u, dn u, to the modulus k (= k') = , 
assume 
then 
il = 
*J(aa) dn u 
k sn«’ 
_ V( aa ) cn u du 
CtlL — ; . 
k sn 2 u 
il 2 -« 2 = 
aa 
and hence 
il 4 — a?a* = 
dd = 
k 2 sn 2 u 
aa 
№ sn 2 u 
a?tf 
là sn 4 u 
a-a 2 
^dn 2 u—^№ sn 2 u^j , 
L 1 “i 1 + 
(dn 4 u — kà sn 4 u), 
a 2 a 2 
ds = 
j, — (1 — 2& 2 sn 2 w), = j 
là sn 4 u v ' là sn 4 u 
№ sn u du 
v^l 1 _ Í 1 
k V(aa) dn udu 
cn 2 u, 
\Z| l -( l+ “)* isn,M } 
dz = k \J(aa) du.