524 The Rev. R. Canuichael on the Singular Solutions 



one (as there should be), since the equation may be divided off 

 by a. In the present case we see that there are in reality but 

 two independent arbitrary constants in the general solution; 

 and as one of these may be represented by an arbitrary function 

 of the other, the complete solution of the differential equation 

 proposed may, with Monge, be represented by the system of 

 equations, exhibiting one arbitrary function, 



0=x + blJ + ^{b)2-Y^{\, b, <^{b)],- 

 0= y + cp'(b)z~(^-^), 



where b is any arbitrary constant. 



In general, we see that for all partial differential equations, in 

 any number of independent variables, of the type 



du du du du . _, /du du du du ^ \ „ 



the form of the general (or primary, in the nomenclature pro- 

 posed by Professor De Morgan) solution is 



ctx+^y + r^2+8w+hc.—Y^{oL, B, y, B, &c.)=0, 

 where a, /3, 7, 8 are n arbitrary constants, as many in number 

 as the n variables x, y, z, &c. ; that the singular solution is ob- 

 tamed by eliminating «, fi, 7, &c. between this general solution 

 and the system of equations 



du 



^ dy "^"' 

 &c.; J 



and that the complete solution is given by a system of equations 

 analogous to the last, and apparently exhibiting (w— 2) arbitrary 

 functions. This latter point will deserve attention, as it appears 

 that the number of arbitrary functions which the complete solu- 

 tion of any partial differential equation of the first order can ex- 

 hibit is not necessarily limited to one, and is in certain cases 

 dependent on the number of independent variables. 



A similar method of solution would appear to be applicable to 

 partial differential equations of higher orders. 



The utility of the observations now made will be best exhibited 

 by their application to some examples ; and it will be observed 

 that it is only as stated in their primary form that the theory. 



