62 
PROFESSOR W. M. HTCKS OX VORTEX MOTION. 
or, in more general terms, 
Integral = C | 
where C and a are arbitrary constants. 
J and Y ma}" be expressed in infinite convergent series. Thus 
T / \ _ Sin ?/ 
J {y) = y - cos y 
{2n + 1)! 
y- 
2m + 3 {2m + 1) 
+ 
(14) 
Y(y)s'!^ + sin 2 / = i + ij/ + . . . + (-)“■ + .. . 
also, 
and 
1 ^ +12/=+ ... +(-r> y /■■ + ... 
y 
<u Y) 
<^\iy) 
sm y 
cos y 
,U 
y 
Y 
y 
o 1 r t 
}/ 
sin y 
cos y I 
V' ! 
Y ' - J — 
,Jy //// 
(^5), 
(16). 
Clearly the functions J refer only to space excluding- infinity ; Y to space excluding 
the origin. 
16. For the problem in question the stream-functions ai'e, therefore, 
inside, 
outside, 
i/'i = — y r- slid 6 -f 
^■2 = ^ 
Applying the surface conditions that when x = 1, xjji = xfjo, and d\\jjdx = c/i/zo/c/ai, it 
follows that when 
n >2, A„ = = 0, 
when 
n = 2, 
Y 2 , A T' - 
— ^ + AnJ — , 
A." Ci 
2y 
a~ 4- Ao 
dx 
a 
7 
where J' and d^'jdx mean the values of J and d^jdx when x = 1, that is 
