130 mh. stokes, on some cases of fluid motion. 



■ sin n 6. 



d<p ^„ n J„ 



rdO ' 7" + ' 



Equating the above expressions to the velocities along and perpendicular to the radius vector 

 given by equations (29) and (30), when ss is put = kc, and then equating coefficients of cor- 

 responding sines and cosines we have 



(1 -k)Q„ + kn'^= -^ (32), 



(1 _A;)-^ + A;Q„=-^, {S3), 



c y 



when ft > 0, and equating constant terms we have 



(l-/t)Q„ = ^, 

 7 



from which equation with (32) and (33) we have, putting 7 = (l + k)c, 

 — = ^,, Q»=--^, when n>0, and «„ = - . 

 Substituting these values in the expression (31), it becomes 



^:{l(n^^)in-2)^:-'^^+l(n^.)(n^2)^]cosne-,^-'^cos,0. 



In the case of a circular cylinder the quantities A„, J^, A^, &c. are each zero. In tlie present 

 case therefore they are small quantities depending on e. Hence, neglecting quantities of the order 

 e^ in the above expression, it becomes 



—^ H r— cos 3 1^ - ii —rrr cos W 0, 



c c' c 



which must be equal to C \{\ - e) cos + e cos 39|. Equating coefficients of corresponding 

 cosines, we have 



^. = -C(l -e)c\ 



A,= - Cec', 

 and the other quantities A^, A,,, &c. are of an order higher than e. Hence, for the part of 

 the fluid which lies without the radius 7, we have 



(p= - C {(1 - e) - COS0 + ^ cos 36} (34), 



and for the part which lies between that radius and the ellipse we have from (28) 



<p = - C'f {(1 - e) cos + e cos S6\ + C {(1 - e) cos 9 + 3e cos 39] z 



C 



cosflar^ (35). 



c 



The value of (p given by equation (35) may be deduced from that given by equation (34) 

 by putting r = c + z, and expanding as far as to «^. In the case of the elliptic cylinder then it 

 appears that the same value of (h serves for the part of the fluid without, and the part within 

 the radius 7. If the cylinder be moving with a velocity C' in the direction of the minor axis of a 

 section, the value of (p will be found from that given by equation (34) by changing the sign of e, 

 putting C' for C, and supposing 9 to be measured from the minor axis. 



