Motion of Oscillation. 219 



/ x , 3 x- \ . 



V 47 J^J ' 



by substituting this value in the expression for d J, namely, 



ds 



and reducing, we obtain 



g 4a 32 d 2 



As we know already the integral of the first term, we shall 

 confine ourselves to finding that of the two last. Representing it 

 by % f", we shall have 



" = -i [F (f^ 

 \ g \ 4 a 



32 a 



To obtain the integral of this equation, we have recourse to 

 the method laid down in the Calculus, articles 128, &c., and put 

 i 1 ' equal to the following expression, namely, 



The co-efficients are then determined as follows, namely, CaU2$ 



A- 3 D- J_ 9^ C- 



~' ~4a""l28a 2 ' 8 a 



Now the integral of a? * dx (h a?) 2 ? or o 



r 1 



or of X 

 | h 



is -- X arc J3JIF ; 



