( •'544 ) 

 Thf potential eiiorgy is givoii by 



r /i-(/( + ly- <f~ 1 1 r 



L 2 /i -|- 1 cv' «- J ^/ 



If WO put 



2 /) --j- 1 cc a- 



and 



Bu = h (/< + 1) -^V , 



tlicse formulae may be replaced by 



and 



§ 5. AVe shall now take for h a determinate number and consider 

 only vibrations corresponding to harmonies of this order. These 

 motions of the common frequency «/i iH'^y differ from one another by 

 the position of the poles of the harmonic Yj,. Moreover, vibrations 

 depending on ditferent functions Y/, may be superposed with any 

 amplitudes and phases we like. 



However, not all of these modes of motion are mutually independent. 

 Since any surface harmonic of order h may be decomposed into 

 2 /' + 1 particular harmonics of the same order, there are only 

 2 /t + 1 equivalent degrees of freedom, for which the frequency is 

 w/r As for those 2 A + 1 fundamenial haimonics, as we shall call 

 them, they need only satisfy the condition that none of them can be 

 expressed in terms of the other ones. After having chosen these 

 functions, which I shall denote by 



i'/i] . J'as , J'a.'i etc., 



