520 PROFESSOR KELLAND ON THE VIBRATIONS OF 



The same equations give likewise, 



&c. &c. 



By substituting these results in the value of c 3 it becomes 



d)'r fr 2 



~ " 



3.51.2 5.71.2.3.4 



~ &c ' 



Of this expression the first line is not dependent on expansion, the second and 

 third are. It is desirable to keep them distinct. 



(p r T 



To sum the series which multiply </> r and ~ respectively under the symbol 2 m. 



sin kr 

 ~kr~ 



which is the multiplier of r. 



r*_ k*r* k*r* 



Let 3 " 3.5.1.2 + 5.71.2.3.4 



f u r 2 k 2 r* * 4 r 



then J o ~ dr-.-. -3- T - 3 + T T - 



3 r* tfr 5 



'" ' :C 



sn 



r d sin k "r 

 k 3 dr r 



1 /sin kr \ 



= 73- { - A cos k r I 



** V r ) 



which is the co-efficient of ~- 



By substituting these values in the expression for c 2 it is reduced to 



sin k r cb' r /sin k r cos kr 



- -- 2 m J- - - 



kr r 



/sin k r cos kr\ . . 



( - --- - I ( a \ 



\ k 3 r k 2 J *> ' 



This is a remarkably simple expression for c 2 . 



