DIFFERENTIAL EQUATIONS OF THE n"' ORDER. 35 



and then the equation (58) becomes 



i dx + n dy + g C„_.., dz,^_,] = , 



a, 



because by (57) ßdx = a,dy . This case of (58) is evidently the result 



of the total differentiation of (54). The root ^, tlierefore, must be rejected, 



a, 



because it only reproduces the given differential equation with an arbitrary 



constant instead of its right member, 0; and accordingly the solution of 



(54) is to be obtained by means of tlie auxiliary system (57) , (58) and 



S(-l)' c ^^" ' = (60), 



from which we finally choose two systems, produced by putting alternate- 

 ly /S = and Ä = . These two systems then will belong to the two 

 equations between (54) and (55). These auxiliary equations may, if desi- 

 rable, be simplified by the equations 



dz,._ij = Zr-i+i.i dx -^ ',._ij+i dy (61) 



(i = 0, 1, 2, ..., r and r ^ 0, 1,2, ..., n— 1) . 



This is the general method to integrate partial differential equations 

 of the n"' order with one dependent and two independent variables, and 

 may, of course, be applied as well to linear as to non-linear equations. 

 We proceed to shew some special cases of it. 



1) The method of Charpit to integrate the equation 



F(.v.y,z, z,,, ^o,i) = (62). 



The equation (GO) becomes here Ci o '" ~" C i = ^ ' ^^'^^"ce (57) 

 becomes 



dx dy 



We have further W^^ = <* Ci ' ^^i = '^ Ci o ' '^^ consequence of 

 Avhich (58) assumes the form 



*^^2,o + ^^^0,1 _ dx 



Now putting here alternately ß =■. and a, = , we get the system 



