108 



such results is only a candid supposition of M. Chasles himself, and 

 that no memoir on the subject, written by Monge, has been traced. 

 All that M. Chasles had to proceed on was the title of a memoir 

 mentioning a certain mode of generating conjugate surfaces, from 

 which he thought it very likely that the solution of partial differential 

 equations which he himself thence found, had really been found by 

 Monge. Under these circumstances, Mr. De Morgan is of opinion 

 that the method must be attributed to M. Chasles as its first dis- 

 coverer, at least until something further appears. 



Mr. De Morgan proceeds to make use of the equation 



fi.AJ 



W_p y W_p/ y (»-D + i 



~Y^"")~° 



to form various cases of equations which can be reduced to lower 

 orders, and which can finally be solved by elimination. Of these, the 

 most simple specimen, being the one suggested by thinking on the 

 method above alluded to, is as follows : — 



If x=Y' and y = XY' — Y, Y being a function of X, whence y is 

 a function of x, we have the following sets of correlative equations : — 



x-Y' X=y' 



y=XY'~Y Y=xy'-y 



y'=X Y'=x 



y" = — Y"=— 



* Y" y" 



Y'" «'" 



w '" = „__ Y'"=—£— 



y Y" 3 y" 3 



and so on. If, then, (p(x, y, y', y", y'",.. )-=0 be a given differential 

 equation, and if it be found that 



*(y\ XY'-Y, X, ± f , 



can be solved ; it is seen that the original equation can be solved by 

 eliminating X between x=Y' and y = XY'—Y. 



The general method of which this is a particular case, is as follows. 

 Let f{x, y, X, Y)=0 have its differential equations of the first order 

 formed on two suppositions : first, that X and Y are constant ; se- 

 condly, that x and y are constant. Let these differential equations 

 be 



X=®(x, y, y') x=(p(X, Y, Y') 



Y=V(x,y,y<) y=^(X, Y,Y'). 



These equations may be used instead of the first two pairs of cor- 

 relatives in the preceding example : and each differential coefficient 

 of Y is expressible by means of the same and lower differential 

 coefficients of y ; and vice versd. To get convertible forms, as in 

 the instance above, fix, y, X, Y) must be chosen so that x and y are 

 simultaneously interchangeable with X and Y. 



Mr. De Morgan gives a similar extension of the method as applied 

 to partial differential equations. 



3. On the constants of a primitive equation. — It is usually left to 



