114 Mn. STOKES, ON SOME CASES OF FLUID MOTION. 



Similarly, from the condition (5), we get 



Ao = bC'„, 



J„6-<" + '>-^'„6"-' = --C'„, 

 w 



R j-(-+i)_ B'„b-'= --D'„. 

 n 



It will be observed that aCo = bC'„, by the condition that the volume of fluid remain unchanged, 

 whicli gives 



ai'y(e')de' = bi"F(e')de'. 



From the above equations we easily get 



nd, changing the sign of n, 



nib-" -a") ' ' 



X= ,r! .^A b''^'C\-a''*^C„\, 

 n (b" - a-') 



with similar expressions for S„ and B',„ involving D in place of C. 

 We have then 

 ^ = aC„ logr + X-ib'" - «-'■)"' J[(6-"+' C'„ - a""*' C„) cosnf) 



+ (6-" + 'X>'„-a-" + 'A,) sinw0]a°-"i-"r-" 



+ [(6"+' C'„-a"+'C„) cos ?i0 + (6"+' i)'„- «"*'/)„) sin«0])-'} (6), 



which completely determines the motion. 



It will be necessary however, (Art. 2), to shew that this value of (p does not alter abruptly 

 for points within the fluid, as may be easily done. For the quantities C„, D„ cannot be greater 



than — / ±f(6)d6, where each element of the integral is taken positively; and since by 



liypothesis f{d) is finite for all values of from to Stt, it follows that neither C„ nor D„ can 

 be numerically greater than a constant quantity which is independent of w. The same will be 

 true of C'„ and Z>'„. Remembering then that r > a and < b, it can be easily shewn that the 

 series which occur in (6) have their terms numerically less than those of eight geometric series 

 respectively whose ratios are less than unity ; and since moreover the terms of tlie former set 

 of series do not alter abruptly, it follows that tp cannot alter abruptly. The same may be 

 proved in a similar manner of the differential coeflicients of <p. The other infinite series ex- 

 pressing the value of (p which occur in this paper may be treated in the same way : and in 

 Art. 10, where (b is expressed by a definite integral, the value of (p and its differential coefficients 

 will alter continuously, since that is the case with each element of the integral. It will be 

 unnecessary therefore to refer again to the condition (rf). 



If the fluid be infinitely extended, we must suppose C'„ and D'„ to vanish in (6), since the 



velocity vanishes at an infinite distance; we must then make b infinite, which reduces the above 



equation to 



a"*' 

 <p = a Co log r - 2" — - \C„cosn6 + D„ %mnd\ (■)• 



This value of <p may be put under the form of a definite integral: for, replacing Co, C„ and 

 Z)„ bv their values, it becomes 



