AND THE ACTION OF TWO VORTICES IN A PERFECT FLUID. 
497 
„ , 0 . . clx dy dz 
3rd. Small terms im —, —» TV 
ds ds ds 
u= ?r\ —sin n6(a sin 0—p sin i//)[A 0 +A 1 cos (#—1//) + . . . ~]cidQ 
Ztt J q ci 
met 
2lT 
2ir 
J 0 
^u/3 n (cos (n-\-l)0— cos (n— l)0)+-^/> sin i/> sin wd[A 0 +^i cos (0—^)~\d0 
n/3 u (A tt+1 cos A n _ 1 cos (n — l)r//) + —' l p sin \jj sin mpA 
=^ma^|cos (n-{-l)\p(^A n+1 — -A^ — cos (n— lJt/^A^ — j 
v— — n/3 u sin n0(ci cos 0—p cos rjj)[A 0 -{-A 1 cos (0—tp)—]ad0 
ZlT J Q 
mmfk | sin (n +1 )\jj[A n+l — - P -A^j + sin (n— 1 )^(a ;/ _ 1 — ~A,)j j 
— i 
— 4 
m [ Zn j na n . . , 
W ~~2tt) 1 —~ 0 smn 0\ a cos 0—p cosxp) 
'0 
a 
+—- cos 0 sin n0(a sin 0—p sin xfc) mAq+A]^ cos (0 — xjj) — )ad0 
Ct I 
772 f 271 * 
— 9 ^ a «l Up cos ^ (cos (n—l)0— cos (n-\-\)0) 
—^p sin i/f (sin [n-\-l)0-\- sin (n— 1)^)(A 0 +A 1 cos ( 0—\p))ad0 } 
—^rrijia^p cos \jj( A w-1 cos (w— l)t//—A„ +1 cos 
— lp sin t//(A« +] sin (^+l)t/;+A„_ 1 sin (w— 
=iwma )i /)(A )J _ 1 - A w+1 ) cos nift. 
4th. Small terms in x—x, y'—y . 
u— — — a cos 0 cos n0/3 n ( A 0 +A 1 cos (0—x //)+ • • . )d0 
Zttj q 
= — -J$»{cos (^—1)^+ COS (ft+l)^}(A 0 +A 1 COS (0 — t/r)+ • • • )<A7 
27TJ 0 
= — J»ia/3j,(A fI _ 1 cos (n— 1)^+A* +1 cos (n+l)\jt) 
772 f 2?r 
v= — —a sin # cos n^ w (A 0 +A x cos (0—$) + . . . )d0 
Z-TT-Jq 
=Jma#»(A„_ 1 sin (n— A» +1 sin (n + 1)^) 
l 2w 
(cos 0(a w cos n# cos 0) -j- sin 0(a„ cos sin 0)} (A 0 +A x cos (0—\jj))ad0 
o 
2tt 
a» cos n0(A o -f-A x cos (#—i//) + • • • )cid0 
0 
— \macL ll A n cos mfj , 
3 s 2 
m [ 
~27TJ 
m 
2tt, 
