Harmonics of the Second Type. 15 



fundamental integrals are therefore 



I 7r Q 2 « (cos 0) cos (2n + l)0d0 1 



c* l ; * ' (37) 



I 7r Q 2 , + i(cos6')cos(2n + 2)(9^) 



and to evaluate these, we need a recurrence formula by 

 which the index of Q can be depressed to zero or unity. 

 This formula, ultimately simple, is not easily found. 

 Starting with 



T C* /dQ„, dQ n - 2 \ . 



(WrfQ, dQ n _ 2 \ 



= — i I —j j- — I sm 6 sm md dd 



= - (2n - 1) I Q„_i sin 6 sin md dd, 



Jo 



by the properties of Q functions, we also have by parts, 



I=r(Qn-Q»- 2 ) sin m6>~T- i " m (Q n - Q n _ 2 ) cos mddd, 

 L J o Jo 



where the part in square brackets is zero. 

 Thus 



\ Q )l . l sin d sin md dd= - -\ (Q n — Q rt _ 2 ) cos mddd. 



Jo 2>l ~ 1 \ 



But also, by a property of Q, 



I Q n - i cos 6 cos mO dO 



Jo 



1 fir 



= „ . \ (n Qn + n — 1 . Q ;i _ 2 ) cos mO d0, 



zn — ±j 



or if u n —\ Q n (cos 0) cos m0 d0, . . . (38) 



Jo 



we have by subtraction and addition 

 \Q n - l cos(m + l)0d0 



r 



=*=«- ; -! (»— w»)M»+(n + w— l)ttft_ 3 t , 



--?A -+■ 1 (^ j 



rQ w _icos(>n--l)0d0 



= e, n + i { 1 + m ) u n + (^ — »l — 1) W» -2 f • 



