566 Cambridge Philosophical Society. 



Mr. De Morgan proposes the following geometrical illustration of 

 the connexion between the primaries and the singular. Let c be the 

 third ordinate of a surface (usually denoted by 2) having the equation 

 ^(.r, y, c)=0. The projections upon the plane of ay of sections 

 parallel to that plane are the primaries : the singular solution is the 

 base, upon the plane of sy, of a cylinder perpendicular to that 

 plane, and which always touches the surface. By means of this 

 illustration, it maj^ be made manifest that certain cases of singular 

 solution which have always been discarded as unmeaning, are 

 limiting cases of the kind which are admitted in analysis so soon as 

 the way up to the limit is clearly seen. 



Taking a general equation with two arbitrary constants, so that 

 a relation between those constants selects and designates a family 

 of curves, it is shown generally (without examination of exceptional 

 cases) how to find the families which have with their singular curves 

 contact of the second order. The equation of these singular curves 

 is a differential equation of the first order : but its singular solution 

 is the singular curve of a family of curves which have with it a con- 

 tact of the third order. 



2. Solution of differential equations by elimination, — This is an idea 

 derived from the method which Mr. De Morgan communicated 

 (vol. viii. part 5) relative to partial differential equations, and which 

 he found, after his paper was finished, had been given by M. Chasles, 

 as he supposed, from knowledge of the restilts of Monge. But it 

 afterwards appeared that the authority for Monge having obtained 

 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 

 ]\Ionge. 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 c quatioh 



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 ^=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 : — 



^=Y' ' X=y' 



y=XY'-Y Y=V-y 



y=X Y'=^ 



// 1 V" 1 



.III 



y-- 



Y"3 



Y"'=— -i— 



