Differential Equations of the Second Order, 39 



so that, in effect, we may in the above values of A, A 1? &c. put 

 c=l, and c! = c 2 =&c. =0; i. e. we shall have 



A = €-/**», 



A 1= = -e-/^* §dx . 7 l€ /^ 



A 2 = - € -/^ j&r . y 2 e/^ 

 &c. &c; 



as to which it is to be observed that no constants are to be in- 

 troduced in the integrations, all such having been accounted for. 

 Take the case where 11=1, and S and T are constant, and 

 we get 



u = y + a l %, or u = y~\-a 2 x ; 



the first of which values gives 

 If we take 



we shall have 



a x — a 2 





a l — a <2 



h 



00 



— I $dx = I - d%= logy??'' ; 



.-. k=se h . 



And the substitution of this value in (8) gives us 



h(h— l)x h - 2 + 'PhaB h - 1 + \J3t! h 

 7i=— 



= k + h(h-l) ^ 

 a 1 — a 2 

 if we assume 



where k is constant ; 



.*. A 1 =—x h J'dx.€~ h y l 



_ k + h{h-l) x h ~\ 

 Also flj— « 2 1^ 



_ *+_^-l) . ^ (A -i) ( A-2)+P(ft-l) J + U^ .^ 

 «i — H 



,. *+**l" . ((MKS-8) +i-P'} ,£p 



