64 On the Graduation of the Sonometer. 



But we have the relation for zonal harmonics, 



£ 



.„*•**" n(n+l) dp 



Substituting, we obtain 



1.3.5 b 6 d?< 



2.4.65c 5 dfi 



&c. j 



The following are the values for the coefficients (Ferrer's 

 Spherical Harmonics/ p. 23), both in terms of //, and when 



/Q2 



we substitute fi 2 = 1 — ^ '• — 

 dfi~ > 



f=i< 5 ,.-.,_!( 4 -4), 



^P 7 _ 3003^-3465^ + 945^-35 _ 4 48-3024g + &c. 

 d/x "16 16 ' 



^P 9 _109395^ 8 -180180^ 6 + 90090^ 4 -13860^ 3 + 315 

 dfjL ~~ 128 



= 5760+&c. 



" 128 

 Substituting these values and putting c 2 = a 2 +/3 2 , 



-, . /3 2 b 2 (l 3 6 2 + /3 2 , 15 ^ + 3W + ^ 4 

 a 6 1 2 4 a 2 16 a 4 



ss^+e^+e^+zs 2 



The more useful form is obtained by retaining c. If we take 

 the two circles of equal radius (i. e. b = /3), we obtain 



A 2 bUl 3 5 2 ,15 5 4 35 b 6 2835 5 s s \ 

 M =" 47r ?i2-4? + y?--8? + ^56-^- &C -r 



