6 



PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



The following relations, free from derivatives of the third order and not involving 

 derivatives with regard to u, subsist among the derivatives of a, b, c, f, g, h, viz. : 



dh da 

 dx dy 



dq 



db dh _ df df I 



d-^ = *P-0rf = P;-2: > 



df dij _ 

 dx ~dy ~ P' 2 + 



Further, from those equations, we have 



dy 



dr, dc 



P i " 1 T" 

 (HI ax 



' '1 

 <i <1 



p Ida dh 



dh, 

 dx 



(hi 



pq 



A, = - 



'1 



p dx 



dh 

 dx' 



pq (/// ' 



7o = 



1 



7 



db dk 



Other expressions are obtainable, but they are equivalent to this set in virtue of 

 the three relations above given. We take the value" of ft to be 



dh dz 



du ' du 



Substituting these values in the three equations, we find 

 _ A 4- X -1- A -- A fda *V ~ dh 



: _ i n J _ "" n 



y + ' i Ty = Q > 



