64 PEOFESSOR A. R, FORSYTE ON THE INTEGRATION OF 



p* + j* = - &\ 

 p'p" + q'q" = - 60', 

 p"* + q"* = - ff* - 6\ 

 pp" + qq" = P, 



'" + qq" 1 = see', 



quantities which occur in the substitution. Now 



^ __ v > <?? _i_ p " * , 



dx " Srj " 3f 



d*v ., 3 2 F , 3 2 F 



/>Y* , 1 T */.>i ^i 



7 .> ~ I J o o r *'/' A' ^ ^i* 



//o>a -* *-i -* * *-Wi r\r 



similarly 



and therefore 



<P '2? 



<fy ~ 9 ft 



dy* r dy*' W fy 3? 



Again, 



so that, as 



^ _1_ / ?J- _L / '" _J_ "'\ ^ 



we have 



n^ I ,y /32 __ |_ /D2 _ [^ /32j 



./ ^, ,;^, "r* 1 ,j>.,7,,, " n,. :u* "i " ^.. ' ^ 



Taking account of the values of r) and , it is not difficult to prove that 

 & (zp'" + yq'") = -n (00" - 30'* - 0*) + 



consequently 



a-F 



= 



a a F ari 

 a?" 1 " aT? j 



and the equation satisfied by F becomes 



+ 300' + {T, (00" - 30' 2 - 0*) + 300'a 



