276 Mr A. LE. H. Love, On the Motion of wa Solid  [Oct. 29, 
Taking elliptic functions of an argument wu, the quantity 
pu—e, is the square of a uniform function of wu, e, being either 
of the roots e,, ¢,, @, which occur in the fundamental equation 
eu = 4 (au — e,) (Qu — @,) (Qu — e,) 5 
it follows that the quantity (@w—e,)/(@v' — e,) is a perfect square, 
say, and then, taking v=v'+ @,, we have 
_ © (U= 4) 9 (Y = a) 
us 6u Ov 
EXP Na (UV + U — Wa)eeeeseeee (16). 
Now (@v' —@u)/(@v' — ea) =1—X=(1—A)(1+A): but this 
quantity admits of decomposition into two factors, each a uniform 
function of wu in the form (A'+7n") (A’— 7X”), and thus A, dr’, A” 
are the direction-cosines of some line; these factors are 
Vv +1." =0 AU DISC Na (V+ U — @,) 
A Ou Ov 17) 
_ 1 O(u—v+ a4) Go ro 
XN —tr ae EXP Ta (V— U— a) | 
1 
where C, is an undetermined multiplier. 
If in the expressions (16), (17) we change @, into any other 
half-period of the functions, we shall obtain in the same way 
sets of direction-cosines, (u, uw’, “’), (v, v', v’), and Halphen proves 
that the lines thus determined are coorthogonal. 
The set of lines thus determined is congruent or incongruent 
with the axes according as the quantity 
= (Ne =X fe) vo eee (18), 
and Halphen shows that if the periods be so chosen that 
@, + @g + w,= 0, 
then J =1 exp (NaWg — NpWa) = (— 1)FtI.... ee (lg): 
since Nag — NeWa = $(2k + 1) tr, 
where & is an integer. 
8. The quantity X so far as it depends upon uw is a special 
case of the simple type of Hermite’s doubly periodic function 
of the second kind, and products of the quantities A, wu, v are in 
like manner doubly periodic functions of the second kind. 
