122 PROFESSOE G. H. DAE WIN ON THE INTEGRALS OF THE 



In " Harmonics " the definitions were 



P, 1 M = 



o 1 ^) = (l - /3 cos 2^)* cos <f> = 



cos 



o 



In order to make the two definitions agree we must take 



= 3 



so that / 3 </W' 3 = 3 3 ^ + ^ = 3~ (1 + 3/3 + 4,8-). On multiplying (24) by this 



1 p 



factor, we have 



(26) 



8 i (cos) = *" M . 3 (1 + 3/3 + 4/3-). 



O 



agreeing with the result on p. 549 of " Harmonics " with i = 2, s = 1, type EOC. 



(3) The Tesseral Sine Harmonic. 

 This is defined thus, 



P 2 i (p) = sin e cos e=f. K cos (K- - /c 2 cos- 6)*, 



So 1 ((^>) = sin (f>([ K'~ cos" <) 4 = r/ . v / - 1 K ' sin ^ (/r 



(27) 



:- sn- 



, 1 1 



where / = , , <j 

 - 



Squaring ^ 2 ! we find 



1 



A (> = 0, ^j = /c e , ^1. = -- 



whence, on putting f : g-K 4 x'' 2 = 1, 



In " Harmonics " the definitions were 

 $.,i (|LI) = P.^ (p.) = 3 sin cos 0, 



S, 1 (</>) = sin <> 1 - /3 cos 2 < ) i = 



5 sin 3 y8 



(28). 



sn 





 1 +P 



Therefore, to make the two definitions agree, we must take 



cos 



(29). 



