XL On some points of the Integral Calculus. By Augustus De Morgan, of 

 Trinity College, Secretary of the Royal Astronomical Society, and Professor of 

 Mathematics in University College, London. 



[Read Feb. 24, 1851.] 



This Paper contains some investigations on singular solutions, the integration of differential 

 equations by elimination, the question of the general form of solution, and the general 

 test of integrability. 



Section I. 



On the singular solution of a differential equation of the first order. 



The equation <p {x, y, c) = belongs, in a plane, to a family of curves, the individuals 

 of which are distinguished by their values of c. And x and y must have real values ; but 

 these of c may be imaginary. 



Let c - a + b y/ - 1, and let <p(x + y ^/ - 1) = A + B <y/- 1, where A and B are two 

 functions of x, y, a, and b. When <p = 0, we have A = 0, B = ; which signify a number 

 of isolated points determined by the intersections of the curves, A = 0, B = 0. In certain 

 cases only does it happen that, for one value of a, and one of b, both A = and B = can 

 be satisfied by any point on a certain curve. One of these cases, of course, and the most 

 important, occurs when b = 0. 



It is impossible that A and B should have a common factor* containing a only, or 6 only : 

 for then d> (x, y, a + b y/ — 1) = would exist under a relation between x and y and one of 

 the two, a or b, independently of the other. Neither can A and B have a common factor 

 containing both a and b in a real form, as two distinct constants. For if M be the most 

 complete common factor, and if <p = M (P + Q -y/ - l), so that P = and Q = are not 

 satisfied by any but determinate values of x and y, the only differential relation between 

 x, y, y, y", under which <p always vanishes, is the second differential equation deduced from 

 M = 0. But this is contrary to what we know of <p (x, y, c) = ; which is satisfied by an 

 infinite number of relations between x, y, y , and y". 



* By A and B having M for a common factor, in the widest 

 sense, it should be meant that neither A-^-M nor B-i-M 

 becomes infinite when M vanishes. When it is not so, I 

 generally call M, in M x (A~M), a fictitious factor; and 1 

 find it useful to have language to express the difference of these 



cases. This definition of the word factor is more extensive 

 than that of common algebra : and so is that of the word homo- 

 geneous when we define an homogeneous function of the nth 



degree as that which has the form x*tp 1 — 1 . 



38—2 



