280 Mr KELLAND, ON THE MOTION OF 



and we obtain 



lit- I-' r^ 2 \ r- 



_3 / (^1+7!) /, _ l^^sin^^ +^4/3^^sin'^)!. 

 2 V r' V r- I 2 3 '^ r' 2 /J 



^^ 1 / 35M^ . .,Mx 



6/32-; {(/3^ + r) (l - -/-) sin^'— + ^/^^Trsm'— (. 



Now a term of the form /3^ is expressed by 



li> 

 6'cos'(m<- *a;) = - {3cos.(m^- kx) 4- cos3(«/- kx)} 



= P^/3 + 



but /3 is supposed, in fact, to consist of an infinite number of terms, 

 the type of which is the above, hence, omitting all terms which do 

 not come under this type, for /3' we shall write Jft'/3. 



Similarly for ^y, we shall write -—-; and we obtain by substitution, 

 putting also b = c, 



so that on this hypothesis 



2 I (fl _ ^) . sin^^- -l^il- "^ + m sh." !^l 

 r* IV 1 I X 4 V r r' ) \ j 



r- = 



2^' 

 X- 



whicli shews that the introduction of this term will affect the third 

 term of the expansion for v", but not the second. 



