653] 
ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS. 
77 
We have thus z = k\J(a<x)u, no constant of integration being required, viz. u is a 
mere constant multiple of z: and the first and second equations then give s and 6 
as functions of u, that is, of z\ but it is obviously convenient to retain u instead 
of expressing it in terms of 5. As regards the form of these integrals observe that, 
writing sn u = X, we have 
dX 
and thence 
du = 
dd = 
ds = 
Vii-xM-ZrX 2 }’ 
k 2 \d\ 
jl - X 2 .1 - № • 1 - (l +¿wj 5 
k \J(clol) dX 
v 7 ! 1 - 
X 2 .l- 1 + - k 2 X 2 
each of which is in fact reducible to elliptic integrals, but I do not further pursue 
this general case. 
In the particular case a = a, we have 
1 — ^1 + ^ k 2 sn 2 u = cn 2 u, 
and the equations become 
T/1 k 2 sn u du 7 ka. dn u du 
do — , ds — . 
cn ll cn It 
which admit of immediate integration; viz. we have 
n . k°-. dn u + k' 
e= h' l0 «s^k- 
or determining the constant so that 6 may vanish for u — 0, say 
and 
a . k 2 , /dn u + k' 1 — k'\ 
e = h' lo x[d^WT+v)’ 
•-f^CriSS 
viz. to verify these results we have 
dd 
du 
1& 2 , 2 il 1 Ì 
h y> .rsnM cn U -, . — T tA , 
i k' (dn u + k dn u - k'\ ’ 
and 
k* sn u cn u _ , „ sn u 
dn 2 u — k' 2 ’ cn u ’ 
ds 
du 
= kka . cn u dn u -L— h -—-— i, 
(1 + snw 1-sn U) 
ka cn u dn u _ ka dn u 
cn u 
1 — sn 2 u