10 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



contain a derivative with regard to u of the arbitrary function in question, of 

 higher order than any other term in any of the equations. Now the equations must 

 be satisfied identically when the value of v is substituted in them ; hence the term in 

 /8j must disappear in and by itself in each case, that is, we have 



the same conclusion as before. The remaining parts of the equations must also 

 vanish ; their forms are already given. 



5. The quantities, which have to be determined for the present purpose, are 

 a, b, c, f, g, h, I, m, n, v, z, viz., eleven in all. They are functions of x, y, u. 

 Omitting those equations in which derivatives with regard to u occur, the eleven 

 quantities are to be functions of x and y. Constants that arise in the integration 

 are constant because the variation of u does not appear explicitly ; that is to say, 

 the constants are functions of u. 



The equations for the determination of the eleven unknowns are partial differential 

 equations of the first order ; their aggregate is constituted as follows : 



First, for the equations defining quantities, we have 



dv 1 dv 



= I -j- np, := m -\- no, 



ftX CLlI 



dl dl 



~r~ a + gp, ~r = h + w. 



dx dy yy.' 



dm , dm 



= ll + fp, - : 



dx dy 



dn dn 



- = q + Cf>. - : 



dx dy 



eight in all. 



But there are certain relations among derivatives that must be satisfied. We 

 have 



d /dv \ d /dv \ 

 dy \dx j dx \dy / 

 that is, 



or, since 



dp d fdz\ d fdz\ da 



j I ) = ~ I 



dy dy \dxj dx \dy J dx 



we have 



in the solution, is finite. If the solution is not expressible in finite terms, the inference is not neces- 

 sarily jnstified in the present connection ; we should then fall back upon the first of the two arguments. 

 An example will be found in 41-43. 



