532 



MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



16. We have now to consider the complementary functions. 

 Ih equations (65), (60), write 



(2 + * 



1), 



. . 



then these will be the equations of tesseral harmonics of orders a, ft respectively. 



Calling T^ (/*) the solution which does not become infinite for p. = 1 , we have 



U = 



\A(1- 2) A f T o( \> 



To find V we have 



so that 

 Hence, 



V = (i _ ^j {T 0*)} 



0*). 



j 



cos , 



(69) 



17. Properties of 1 (/*). 

 The differential equation is 



(70). 



and for any value of a, real or imaginary, this is satisfied by the integral 



f {/* cos < v/(/x. 2 1)}" cos stj> d$.* 



Jn 



Also, if we put 



* HEINE, ' Handbuch der Kugelf anctionen,' pp. 225 st seq. 



