158 Sir William Thomson on the 



r = e cos m: sin (nt — id) when approximately r = a/ 

 r = c cos m: sin (tit — 16) „ „ r = a. 



c, c, m, n s a, a 1 being any given quantities and 

 any given integer. 



The condition T = ft»' simplifies (9) to 



, . < f ( . \ dw 2io) ) 



( "- !B) {l B - !a, )<F-T w } 



(12) 



p= 



»j{4&> 2 — (n— ia>) 2 } 



(» 



■){ 



n dw i(n — i(o) 



zco ~j~ ' -co 



dr r 



} 



(13) 



m{4&) 2 — (n— ico) 2 } 

 and the elimination of p and t by these from (8) gives 

 d 2 w 1 dw 



dr 2 + r dr 



— + m 2 — , • A2 w = 0; (14) 

 r 2 (n— ico) 2 } v y 



or 



where 



or 



cl 2 w . 1 dw i 2 w 



dr 2 r dr 



¥ +v*w=0, 



/4:CD 2 — (n—ico) 2 

 V (n—zcoY 



d 2 w 1 cfoy 



) ! 



— <7 2 w = 0, 



(15) 



where 



/(n—ico) 



(16) 



r (ft — Zft>) 2 — 4ft> 2 



Hence if J;, $,- denote BessePs functions of order i, and of the 

 first and second kinds* (that is to say, J { finite or zero for in- 

 finitely small values of r, and $} finite or zero for infinitely 

 great values of r), and if I t - and ff £ denote the corresponding real 

 functions with v imaginary, we have 



w = CJ i (v7*) + €$ i (vr), 



(17) 



or 



w=GI i (&r) + €$ i ((rr), .... (18) 



where C and C denote arbitrary constants, to be determined in 

 the present case by the equations of condition (12). These are 

 equivalent to p = c when r = a, andp = c when r = a, and, when 

 (16) is used for iv in (13), give two simple equations to deter- 

 mine C and C. 



* Compare ' Proceedings/ March 17, 1879, " Gravitational Oscillations 

 of Rotating Water." Solution II. (Case of Circular Basins). Phil. Mag. 

 August 1880, p. 114. 



