82 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



We have 



X = K *l, Y = *X Z = /c 2 n. 



Thus the characteristic invariant equation is 



from which we have, as before, 



z = u + xp (u) + yq (u), 



so that p and q are parametric for differentiations with regard to x and y. The 



three subsidiary equations are 



K~l + - p -', - <l T~ 



fte ' (/,'.: ay ay 



dli df db df 



K~m + p ', + -, ? T~ == 



// ' // /' //'J/ //?/ 



j 

 ay 



and we .have the equations of definition and derivation as before. 



5 L. One simple mode of proceeding is to use the six equations involving derivatives 



of /, in, n, with regard to x and y they are equivalent to five independent equations 



in order to express a, b,f, g, h, in terms of c and those derivatives. These being 



obtained in the form 



dl dn tin 



a = ^-P^ + i rc ' v = ^--v c ' 



dm dm , , tin 



b - -- (/ - --- \- (re, / = -- qc, 



d)/ * dy J dy 



dm dn dl dn 



h = - -- p + pqc = -- -- a- -\- pqc, 

 dx ^ dy dij a dx 



we substitute them in the subsidiary equations. The third of the latter then 



becomes 



d* ,\ / d d 



TT + K "= 2 PJ - + 9'T - 



dy~ / \- dx dy 



the second of them, on using the first of the two values for h and also the relation 

 between n and c just deduced, gives 



id- d* A / d? d- d- ,\ 



^ K " m = ~ 2 . < ~ K n 5 



o . , 



df j \ dx 3 f dx dy 



and the first of them, on using the second of the two values for h and also the 

 relation between n and c, gives 



= (P 



2 







df J \ 2 dx* ^ dx dy ^ dy 



