POINTS OF THE INTEGRAL CALCULUS. [119] 



may and does happen that a branch of the singular solution which the first form only yields 

 out of <p, m oo , may in the second form be given by \^ c = 0. And I believe that in every 

 case a form may be obtained which will allow \|/- c = to represent the complete singular 

 solution. If this be so, the inclusiveness of the preceding reasoning is much augmented. 



Section II. 



On certain solutions of differential equations which ultimately depend on elimination. 



Some time ago I communicated to the Society a method (printed in Vol. vm. Part n.) 

 of which I afterwards found, as I acknowledged in a note* at the end, that it had been given 

 by M. Chasles, from an unpublished paper of Monge, as I supposed. On looking again at 

 the account given by M. Chasles, (Apercu Historique, $e„ p. 376.) I find that he had not seen 

 the memoir of Monge (which was never printed, though intended to appear in the Memoirs 

 of the Institute for 1808): but that he had only the hint of the reciprocal surfaces men- 

 tioned at the end of my paper. From this M. Chasles infers that Monge must have used 

 these surfaces in the integration of partial differential equations. So that it by no means 

 follows that Monge did invent the method : this remains with M. Chasles until a prior right is 

 established. It seems unlikely that Monge's memoir is now in existence. 



The most striking peculiarity of the method is the perfectly reciprocal character of the 

 substitutions ; in transforming the equations, x and p, y and q, &c. are interchanged. The 

 following extensions will shew that this reciprocity is not an essential part of all similar 

 methods. Their peculiar feature really is, that they are founded upon the circumstance 

 noted by Clairaut in his celebrated solution of y = y'x +fy, namely, that there are cases of 

 differentiation in which the only new differential coefficient introduced appears as a factor 

 of the whole. 



Let y {n) express, as usual, the n th differential coefficient of y with respect to x, and let 



V m>n = f(j>x . dy M = <px. y»» - <f>'x . 2/*"- 1 ' + <p"x . y< 9 - 2 ' - ... 



<px being a rational function of the degree m. When m exceeds n, integrals of y will enter : 

 we need not exclude these cases from the reasoning, though they will be of no use. Accord- 

 ingly, y' n+I > is a factor of V' m „, whence it follows that V m n is constant if y be any rational 

 function of x, not exceeding the n th degree. 



Let/(F mn , F m>n , r m . „,...)«= 0, a differential equation of the n th order. 



Assume y = C + C x x + . . . + C n x", having n + 1 constants. Substitution reduces V m „, &c. 

 and therefore /, to constants, and this, even though the rational functions entering in them be 



* The date of this note is misprinted; it should be 1848, not 1847. 



