Mb. stokes, ok SOME CASES OF FLUID MOTION. 113 



Since for any particular radius vector between a and h the value of 9 is a periodic function 

 of Q which does not become infinite, (for the motion at each point of each bounding surface is 

 supposed to be finite), and which does not alter abruptly, it may be expanded in a converging 

 series of sines and cosines of Q and its multiples. Let then 



= P„ + Sr(P„cosw0+ Q„sinra0) (1). 



Substituting the above value in the equation 



— r-^ + — i = (2). 



dr \ drj de- *■ ''' 



d I dP„\ 

 ,_(^,__)_,-p„ = 0, 



which (j) is to satisfy, and equating to zero the coefficients of corresponding sines and cosines, 

 which is allowable, since a given function can be expanded in only one series of the form (l), 

 we find that P„ must satisfy the equation 



d ( dPA 



of which the general integral is 



Po = ^Jogr + B, 



the base being e, and P„ and Q„ must both satisfy the same equation, viz. 



of which the general integral is 



P„ = Ci--" + C'r". 



We have then, omitting the arbitrary constant in cp, as will be done for the future, since we 

 have occasion to use only the differential coefficients of <b, 



(p = JJogr + 2r{(^„»-"" + A'y) cosnO + (fi„)-" + B'„r") sinnfi] (.•3) 



with the conditions 



— =/(&) whenr = a (4), 



-j^ = F(e) when r = 6 (5). 



Let / (d) = Co + 2"(C„ cos 7ie + D„ sin nO), 

 F(fi) = C. + 2r(C'„cosn0 + Z)'„sinw0) ; 



so that C. = ^ ffiO') dd\ C„ = - r'fiff) cos nO'dff, D„ = - ['"/(O) sin n&dO', 



with similar expressions for C,,, &c. Then the condition (4) gives 

 — + Sr«{(-A«-*"'"' +-<'„<«"-') COSW0 + (-£„«-'"+'> + 



= C„ + Sr(C„ COS nd + D„ sin nB) ; 

 whence. 



An = nC'o, 



A«-'""''-A«"-' = --c„, 



B„a''-<'>-B\a"-' = --D„ 

 n 



Vol.. VIII. 1'akt I. 



