A RIGID BODY IN ELLIPTIC SPACE. 
317 
ct~oj ] -f- (1 — b~ — c 2 ) (wgWfj — (UgWg) = 0 
lroi. 2 -\- (1 —c 3 —a 3 )(<u 3 <y 4 —&j 1 &j 6 ) = 0 
c z o) 3 -\-(l—a* — w a w j ! ) = 0. 
From these we immediately deduce the equation 
crwy 
1 —A 2 —e 2 
5 2 &) 2 3 c 2 o > 3 2 
1-Y 2+1 - T~ T 2 = a constant. 
1 —c 2 —cr 1—a- —6“ 
Solution of the equations in terms of the double theta-functions. 
54. The equations of motion of a solid body under the action of no forces can be 
solved in terms of the theta-functions of two variables, one of the variables being a 
linear function of the time. The solution, however, is not complete, owing to a 
deficiency in the number of arbitrary constants. 
The notation employed will be that given by Mr. Forsyth in his “ Memoir on the 
Double Theta-functions,” published in the Philosophical Transactions for 1882. 
The definition of a double theta-function is expressed by the equation 
f /"X n\ I m=CO 51=00 (2 rii+n) 2 (2n+v) 2 (2 m+iJ.)(2n+v) 
' H )x, y\— t . 2 ( — l) mK+np p i q 4 r 2 ^+fO W jf@»+»> 
l\M'> J m= — oo n= —oo 
in which y, p, v are given integers (afterwards taken to be each either zero or unity) 
\P> v ) 
and 
is called the characteristic; x, y are the variables; p, q, r, v, w are known 
constants, called parameters (in our case arbitrary constants) ; and the double summa¬ 
tion extends to all positive and negative integral values (including zero) of m and n. 
There are sixteen different double theta-functions distinguished by suffixes 0, 1, 2, ... 15. 
These are written 3- 0 (x, y), Sfx, y), . . . , or when the variables are easily understood, 
simply S 0 , f, . . . 
The characteristics of the sixteen functions are given in the following table :— 
n 
\0, Oj 
CD 
CD 
/°,1\ 
\o, o) 
(So) 
(”) 
CD 
CD 
^2 
dlO 
S 8 
f 
dll 
$ 9 
CD 
CD 
CD 
n. ox 
Vo, 0 / 
CD 
r—1 r-H 
H3 
CD 
d 5 
d 7 
dl 5 
dl 3 
d 6 
dl 4 
dl 3 
The functions & 10 , & n , S 5 , S 7 , S 13 , S 14j are odd functions, the rest even. 
