518 PROFESSOR KELLAND ON THE VIBRATIONS OF 



Now, it is evident that, since the medium is one of symmetry, the force put 

 in play is in the direction of the motion ; and is represented by 

 xfa. 



2 m 



xfa 



{r 2_ 2 (*-/) 



The co-efficient of a 2 " in this expression is as follows : 



( + l)w( 1) 

 3.5. . .(2 + l) n 3 . 5 . . . (2 + 3) 1.2.3 



2 - - 



r 3.5. . .( + 

 ~ 2 m [2.4.... 2n 



&c. 



every term of which involves (xf] as a factor. Hence this quantity is zero. 

 Again, the co-efficient of a 2n+1 is 



f 3.5... (2n + l) 1 3.6...(2n + 8) (n + 1) 2 (+/) 



12.4 ____ " 



2 r 2 "+ 3 2.4...(2 + 2) 1.2 



3.5... (2^ + 5) ( + 2) (+l)n(n-l) 2* (-/)* 

 2 .4 . . . (2w + 4) 1.2.3.4 r 2 + 7 



3.5.. . (2n + 3) f .. 2 (a;-/) 2 8 . 5 . . . (2'+5)_ (n + 2) ( + 1) 2(g-/) 4 

 2.4. . .(2w + 2)^ r 2K + 5 2.5 ... (2n + 4) 1.2.3 r 2 ^ 7 



3.5... (2^ + 7) ( + 3) ( + 2) ( + l) M (n-1) 2 5 (a;-/) 

 " 1.2.3.4.5 r 2l> + 9 



It remains that we find the sum of this series. To effect this, we remark that 

 the co-efficients can all be reduced by the result of Theorem 2. Applying this 

 theorem, the co-efficient is reduced to 



m ( 3. 5. . . (2w+l) _ 3. 5 ... (2rc + 3) (n+l)n 2* 

 i r 2 + 3 l2.4 ..... 2n 2 . 4 . . . (2n + 2) 1.2 3~ 



3.5 .. . (2 + 6) (n + 2) ( + !) (n-1) 2* 

 f 2.4.. . (2 + 4) 1.2.3.4 5 



3.5. ..(2 + 3) , 2 3 .5. . . (2 + 6) (n + 2) (n-^1) 2 3 



' 2.4...(2n + 2) ( ^ 3 + 2 . 4 . . . (2n + 4) ' 1.2.3 " T &C ' 



Now, we observe that the co-efficient of a 2n+1 in 



p-a . [8.5...(2n + l) 3 . 5 . . . (2it + 3) (n + l) 



(l-2ja + a a )i J -12.4 ..... 2n 2 . 4 . . . (2-t-2) 1.2 



