

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 113 



If da- be ,in element of surface of the ellipsoid, and p the central perpendicular ou 

 to the tangent plane, it appears from the formula at the foot of p. 257 of " Stability " 

 that 



P dv . - ^' 3 cos ft cos y * 2 cos2 

 ' 



AF 



where A 2 = 1 - /c 2 sin 2 0, F 3 = 1 - K'- cos 2 tf>. 



In the previous papers I have expressed the two factors of which a surface 

 harmonic consists by $,*(/*) or P,-'(/x), and <f ($), d s (<f>), S/ (<) or S,'(<), one of the 

 two P-functions being multiplied by one of the four cosine or sine functions. 



Taking a pair of typical cases, the integrals to be evaluated are 



> &f* do- and < &' da: 



As it will be convenient to use an abridged notation, I will write these integrals 

 //(cos) and //(sin), according to an easily intelligible notation. 



These functions involve integrals of even functions, and therefore we may integrate 

 through one octant of space, the limits of 6 and <j) being ^TT to 0, and multiply the- 

 result by 8. 



It is clear then that 



j,/ cos) __ 8 cos cosy 

 sin :! /3 



r 



Similar expressions are applicable to all the other forms of function, but we may 

 proceed with this form as a type of all the others. 



This formula shows that the variables are separable, and since we might substitute 

 5-77- x// for (f> without changing the result, the (f) integrals are of the same type as the 

 6 integrals. 



It^ias been stated above that two of the roots of the cubic equation are proportional 

 to K- sin 2 6 and (1 /c' 2 cos 2 </>). By the nature of the harmonic functions it follows 

 that if [^; s (/x)]~ is proportional to a certain function of /rshrfl, [(/*(</>)]' is proportional 

 to the same function of (1 /c' 2 cos' 3 ^)). 



It follows that if ($,*)- = F(K~ - - sin 2 0) = F (K cos 2 6), 



where a is a constant, which for the present we may regard as being unity. If then 



we must have 



[<,' ()]2 = A n - A lK '* sin 2 j> + /V sin 4 $ - A^K'* sin 6 

 VOL. CCIII. A. 9 



