46 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



the second order. These conditions are the simultaneous partial differential equations 

 of the first order determining u. 



Thus, dealing with the case of 11, when the compatible equations of the second 



order are 



, . 21 m n 



a 4- f a h - , 



II + z 



a - 2g + c = (y + z}<j> (x + y, z), 



we know that the first has no intermediary integral. As regards an intermediary for 

 the second, substituting for a, b, <?, from the derivatives of the ?<-equation, the result 



- 2h - H '-k 



'/ "/ '"/ / \ ". 



must, qti.d equation iny, g, h, be evanescent ; hence we find 



(u,,, -\- u/) 2 = 0, v,, = 0, -' - - = (y + z) 



'I' I V'M 



that is, the equations for u are 



en 

 da 



(ili 



^ = 0, 



(ill, cu cu , . en n 



5 -- + (I m] 2 - (y + z) - 6> = 0. 

 c.i' c// ' (.:/ ' cm 



Tlie system is a complete system ; hence it possesses four functionally independent 

 solutions. "Writing 



^ = , r /ni 3g n . 2 + 3f/, #,3.,, 



these four solutions can be expressed in the form 



z,x+y, 

 l-m-^ + z) (g n - 2g u + ry 33 ) - ^ + ^ 3 , 



and another involving v, which would require either the arbitrary constant or the 

 arbitrary functional form to which the third would be equated. (The fourth is, in fact, 

 a primitive of the compatible equation under discussion, though it is not necessarily 

 the common primitive of the three simultaneous equations.) We thus infer that 



} (g n - 2(/ J3 + g K ) - (Jl + ff , = $ ( x _|_ y> 

 is an intermediary of the second of the equations. 



