of Differential Equations. 537 



should at the same time reader 



-i_ > or < 0, 



this condition being similarly interpretable as above in the case 

 of a single independent variable. We should look, too, for ana- 

 logous conditions in the case of partial differential equations of 

 the higher orders, but none such are given. 



4. As regards partial differential equations of the second order, 

 it may be said that, either in the way of theory or illustration, 

 little or nothing has as yet been done ; the field may be I'cgarded, 

 not merely as unexplored, but as almost wholly unknown. The 

 contributions to the subject which have been made by Poisson 

 are scanty, and too much of a tentative character, whether they 

 be regarded as illustrative of this interesting department of science, 

 or suggestive of the means for its development and improvement. 



Proceeding indirectly from the integral expression 



y=^ + boo+f{a, b), 



where a and b are arbitraiy constants, Lagrange has discussed 

 the differential equation thence derived, and which is typical of 

 a class, namely 



y=^ivp- 1- ji' +/( p', p - ocp') , 



dhi 

 where p' =■ y-j, and has pointed out that this equation admits 



of a singular solution derived from the elimination of p between 

 this equation and 



3 dp' 



By adopting the same indirect method, we should obtain a corre- 

 sponding theorem for partial differential equations. 



It is suggested, as deserving of examination, whether the 

 Laplacian theory for the determination of singular solutions of 

 total differential equations of the first order should not be recon- 

 structed. 13oth upon analytical and geometrical grounds, the 

 condition with which it is embarrassed, and which is indispen- 

 sable to its existence as a theory, appears to be unnecessary and 

 troublesome. But if this examination result in its confirmation 

 and retention, then must an analogous condition be laid down 

 for partial differential equations of the first order, and corre- 

 sponding conditions for all differential equations, both total and 

 partial, of the higher orders. 



It may save some useless labour to know that there arc no 

 theorems in partial differentials of the second order corresjiond- 



Vhil. Mag. S. 4. No. 103. ^uppl Vol. 15. 2 N 



