SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 135 



advantage of using only one type of function would not compensate for the loss of convenience in the 

 result. Accordingly I do not think it worth while to undertake the very laborious task of revising all 

 the analysis of "Harmonics " from this point of view. 



I may mention, however, that I have gone far enough in this direction to feel pretty confident that, if 

 this new form of developing the cosine and sine-functions were adopted, the remarkable coincidence, 

 mentioned in the footnote on p. 547 of " Harmonics," as to the form of the integrals of the squares of 

 surface harmonics would become explicable. 



POSTSCRIPT. 



[December 2<llh, 1903.] 



Mr. HOBSON has shown me how these integrals may be evaluated by a simpler 

 method of analysis, without the intervention of elliptic integrals. As an example oi 

 the method he suggests I take the integral /., (cos) evaluated above. 



The solid ellipsoidal harmonics are given, except as regards a factor, in 3 of 

 " The Pear-Shaped Figure." 



In (19) of that paper we find 



S, = 19 3 (") $ 2 (M) ffj (ft = A ^fx- + (1 - -2q) if - q*3* + </Y~J , 

 where A is the factor to be evaluated so as to agree with the definitions 



The ellipsoid over which we desire to integrate is defined by v --.- , and the 



K sin y 



extremity of the c axis is defined by ju, = sin = 1, ^> = ^TT. 

 Hence at this point 



S. 2 = (cosec 3 y (f] (/c 3 q"} q' 2 . 



But at the extremity of the c axis 



x 0, y = 0, - =.f = -. . 



K sin 7 



Therefore 



(cosec 2 7 - cr) (K- - </) </- = 7^-1 f - - - $~ - + <r l "} = - A k ~'l - (cosec- y - q*~). 



\ K~ sur 7 K V / /c" 



Therefore A = T , arid 



A;" 



f -, .T 1 n W 3 /. o 3 J 



