38 G Lord Rayleigh : Problems in 



character of the integrand, the limits may be taken a? — it 

 and +7T. Accordingly 



f +7r , C v 



l cos nyfr e p' cc >s xp d\jr = 2 \ cos nty e p' c °* ty d4r 3 



sin n-^r ep' G0S ^ d-fr=. ; and 



I cos n<j) . ep' cos (<l>- 9 ) d(f> = 2 cosnO I cos nyfr e p' cos ^ d-^r 

 J Jo 



.... (20) 



The integral on the right of (26) is equivalent to 7rl„ (/>'), 

 where 



z»I n ( P ') = J» ftO, (27) 



J w being, as usual, the symbol of BessePs function of order n. 

 For, if n be even, 



nrr ' fir 



I cosmjr e p' cos $ dyjr = ± 1 cosnyjr (ep' cos ^ + e~p coatydyjr 

 Jo Jo 



= COS ?lifr COS (tp' COS ^r) d^r = 7Tl~ n Jn (ip l ) — IT I n (p) ; 



and, if n be odd, 



cos n-^r ep' costy dty— — J 1 cos nyJr (e - p' cos ^ — e p' cos ^} dty 



Jo 



= — z I cos n-yjr sin (/// cos yjr)dyfr = 7r I n (p). 



Jo 

 In either case 



f 77 



I cos n-tyep' cos ^ dyjr = 7T I K {p' 



Jo 



). . . (28) 



Thus 



/i27T 



1 cos n<f> ep cos (0-0) d$ = 2ir cos ?*<9 I n (p f ), (29) 



Jo 



and (24) becomes 



qa cos n0 T fap\ _«!+o 2 ±^ 



This gives the temperature at time £ and place (p, s) due to 

 an initial instantaneous source distributed over the circle a. 



The solution (30) may now be used to find the effect of 

 the initial source expressed by (22). For this purpose we 



