Harmonics of the Second Type. 9 



The simplest proof is probably by use of the formula 



1 P n (cos cos 0' + sin sin 6 cos (/>) dcf> 



= irF n (cos0)F n (cos0). 



Writing 0=0'= j, 



i 



| 7r p, l (cosc/))^ = 7r[P n (0)] 2 , 

 which is zero when n is odd. Writing cosc£ = yu,, we have 

 11 P.(m) 



j: 



v/1-/* 



.^ = 7r[P n (0)] 



leading immediately to the expansion. 



If a direct proof be adopted as follows, we obtain 

 important properties of the Q functions. For, whether 

 P stands for P or Q, we have 



■<*+ 1) Jp. w * = - vr^f + j ^ f y* 



and therefore, applying this formula twice, if 



we have J Vl-/* 2 



w(w + l)M n — (n— 2)(n— l)w„_ 2 



=Jvfe(f-^)*-^'(t-^) 



= _(2«-l) vr-7 2 P„_ 1 + (2n-l)r HSL^ 



! VI — A 1 



= -(2n-l) i /T^?.P n _ 1 + nw ft +(n~l)« w _ 2 



or n 2 Un — (n — l) 2 u n _ 2 = — (2n— 1) a/1— /z 2 P ?z _i. 



Between limits + 1, the right side vanishes whether the P 

 or Q function is concerned, and 



u n / u n . 2 = (n — 1 / n) 2 . 



