of Differential Equations, 535 



been proposed to investigate the singular solution of the equation 



^{x, y, z, &c.) -^ + ^{x, y, z, &c.) ^ +X(a7, y, z, &c.) ^ + &c. 



it might be supposed that, generalizing the above theory, and 



eliminating ^-, -j-, ^-, &c. between this equation and its derivees 

 ax ay az 



with respect to these same quantities, we should obtain as its 



singular solution, 





^)™-' + &c.=l. 



But, upon deriving from this equation the values of the partial 



differential coefficients -^, -r, -r, &c., and substituting them in 

 ax ay az 



the equation proposed, we find that it is not satisfied ; conse- 

 quently the expression which we have obtained, though the result 

 of the required elimination, and perhaps in a certain unrecog- 

 nized sort an integral, is not the singular solution properly so 

 called. In fact we may regard the failure of the method as in 

 itself significant, indicating that the qviestion admits of no sin- 

 gular solution ; or, in other words, that the geometrical property 

 symbolized by the differential equation and the curve or surface, 

 as the case may be, symbolized by its general integral, are incom- 

 patible with the condition of the admissibility of an envelope. 



3. The theory above stated as attributed to Laplace, is rather 

 his by implication than expressly. The form in which the dif- 

 ferential equation of the first order was regarded and discussed 

 by him was that of 



and Laplace argued, that if we derive from this equation the 

 value of 



dp _M. 



Ty-W 



then all the factors of N=0, which satisfy the differential equa- 

 tion, are singular solutions ; but the existence and significance of 

 those factors which do not satisfy the differential equation are 

 not explained. The theory previously quoted is an obvious con- 

 sequence of that which has been now stated, but it has omitted 

 to account for or interpret the condition, unproductive, indeed, 



