SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 125 



On substituting these in the above expression, and noting that K z f-y~ will be unity 

 with the definition adopted, I find 



(38). 

 / 3 2 (cos) = the same with the sign of D changed. 



If these expressions be developed in powers of K', and if the factor ic f-y" be re- 

 introduced, I find 



= -?M ()*(! -40 + ^0*). </V 



ro/ \ 4irP COS /3 COS y /,/, . 31 



:! (C 8) = 7 sin" ' 5 ( K + ' " 



= ^ M . ^ ^ ( I - 2/3 + /8' : ) . 



In "Harmonics" I defined, 



P 3 (/,) = P ;s (/z) - i /8/V (/.) = sin [f sin 2 ^ (1 

 In order that our previous definition may agree with this we must have 



Now 



whence //c = f(l +I/3 + 5y8 2 ), and this value of / satisfies the second equation. 



With regard to C s (<j>) there is a mistake in the table (the only one I have detected 

 therein) on p. 556 of " Harmonics," for the coefficient of the second term should r.ot 

 be 3yS but f /8. The mistake obviously arose from my using the formula for j), 2 

 instead of that for p' 3 as given on p. 490. 



With the corrected coefficient the definition is 



C 3 (<) = (1 - /3 cos 2^)* (1 - f ft cos 2<) 



In order that the previous definition may agree with this we must have 



<* = 1 + /8)* 1+1=1 + 3^8 4- f /3 2 , 



