Ara THE ACTION - OF TWO YOBTICES IN' A PEBFECT FLUID. 
501 
except when n= 0 when 
1 f 
—< 
2 rr 
d x 
Now 
27tJ 0 (a 3 + p 3 + £ 3 — 2pa cosy) -5 
a- 
! +p 3 +£ 2 —2p<x cos y=(a+p) 3 -}-£ 3 —4pa cos 3 ^=((a+p) 3 +£ 3 )[l — k 3 cos 3 y 
where 
/c 3 = 
4: pa 
(« + /)) 3 +^ 3 
Now in the case we are considering p is very nearly equal to a and £ is very small, 
hence k is very nearly equal to unity 
where 
and is very small 
Therefore ^ 
Now since k 1 is very small 
1 — k 2 cos 3 /c^-j-K 3 sin 3 ^ 
O, 9 (p-ay + Z" 
1 (p+ciy+t~ 
»2 fl ¬ 
eets %yc£y 
t rj ((a + p ) 3 + £ 3 ) ! (^ 3 + ^sin 3 | 
I 
3,r cos n% d X 
i w 
K,y + /c 3 sin 3 — 
will be very large, and the large part will arise from very small values of y, or from 
values of y very nearly equal to 27r, the parts arising from small values of y and from 
values near to 27 t will evidently be equal; hence the integral will be approximately 
where e is large compared with K b this integral 
4(1 + 2% V) 
% 
8 n 2 
log (y+A/^+y 3 ) 
(icy+xfy 
or, since e is large compared with k 13 this is approximately 
4(1 + 2^ V) n 91 2e 
-8" 8 log- 
/v -1 /v-i 
or retaining only the more important terms 
4 
=- 1 +8re 21 °gK 1 
ACt 
