1888.] in a Liquid when the impulse reduces to a couple. 277 
Taking the formula for decomposition mto simple elements 
ag ees 2 (= 4s) oxy (La, + ta,) w 
ih = jeer an a) exp (ta, + ta,) u| ...(20), 
and in it writing @, =@,, @, = wg, we find that there is a differential 
equation of the form 
a w = = constant, 
viz. this equation is 
1 dv 6 vO(v — w,— wp) © (w, — 8) 
in Ai == eXP (Np. ary Nap) 6 (v mye) )6 5 (v a We) 60, Go, ) 
decomposing the right-hand side into simple elements, we find 
1 dv 
wedge 0.) = £(0 = a9) + boa be 
Ii ow g'v 
=>5 Sa eles fathead syiainetie@ tate 21). 
oa) = | i dethcads (21) 
It thus appears that the differential equations (15) for A, p, v 
as functions of u are satisfied by 
© (u—a@,) © (v—a@,) 
Pee ANS Sea exp (0+ 0) 
ae Bly — 
pt 5 (u ae = @p) exp 1g (v + U — ap) ie mol 
Cee eas) 
= ae exp — 7y(u +u+o,) | 
provided 
< R  e Lns ek 2 
21 QU— 6," 20 Qu — €’ ee ie se 
9. From these and equations (13) we can obtain the constants 
on which the elliptic functions depend, viz. we have three such 
equations as 
(«B® — w”) (e«C” — a’) 4 ee) 
ies eee ee Se Nh 
@ 4 (QU — ég) (Qu — ey) 
= = ME (OU = Cad w cain sare solnnve nee (24), 
