[110] PROFESSOR DE MORGAN ON SOME 



The differential equation is y = —$> x : 4> y , say y' = %(x, y). Not confining ourselves to the 

 supposition that c is constant, we have dc = <b x dx + <i> y dy, whence 



(y - y)dx = — = - *_ — . 



Now if either of the two, <£ x and (£> y , be made infinite by a relation between x and y, <1> 

 remaining finite, it can only be in one of three ways: (l), <l> a = oo , 4> y = oo ; (2), 3^ = co only, 

 and a? = const. ; (3), <J> y = as only, and y = const. For if <t> x = x , 4> r dj? + <J>j,dy cannot be 

 an ordinary differential, except either <J> y = co , or dx = 0. On any other supposition, (J> would 

 have a permanently infinite differential coefficient, which a finite function cannot have. 



Accordingly, except by dc = 0, giving the ordinary primitive, y — ^ = is solved only as 

 follows. Either <& y = co , <J) r = co ; or <J) y = co , y = const., which gives ^ = 0, y = ; or 

 $ x = co , # = const., which gives y' = co , ^ = °°' 



That is, every mode of satisfying either of the following systems, 

 $, = co , 4> y = co ;<!>*= co , and not i> y , x = const. ; <l> y = co and not <J> t , y = const, 

 is a singular solution : and by this mode all the singular solutions are obtained. 



And since $ a = — (p„ : (p c , $ y = — <p v '-<t> c > these last fractions may be substituted in the tests. 



Any two solutions which have a point* in common have a contact of at least the first order 

 at that point, since y is ^(*, y) for both. But it often happens in analysis, that a result which 

 is d priori evident from the conditions of the question, cannot be universally verified, by reason 

 of the difficulty of the extreme cases. If we make c a function of te and y, no doubt 



<p(x,y,c) = gives 

 d) , d). (dc dc ,\ , d>c (dc dc ,\ 



If then we would have curves contained under y' — ^ = 0, but not under c = const., we must 

 look for them by determining c from (p c :d> y = 0. But if, which may happen for aught we can 

 see to the contrary, the same value of c which is thus obtained, substituted in <p(x,y, c) =0, 

 leads to a relation between w and y under which dc : d.v and dc : dy are infinite, we can no 

 longer be sure that y — ^ = is satisfied. 



The following geometrical interpretation will add to our power over the meaning of the 

 singular solution. Let w, y, c, be the three co-ordinates of a point in a surface, of which the 

 equation is (p (x, y,c) = 0, or c = <$>(x, y). Let the plane of xy be horizontal. Then, c being 

 constant, c = <i> is, in the horizontal plane, the equation of the projection of a horizontal section 

 of the surface. The projecting cylinders of these sections may be called vertical. Of other 

 vertical cylinders we have especially to consider the one we may call the tangent cylinder, 

 which has the same tangent plane with the surface at every point of concourse. Its equa- 

 tion gives $ x = co, <t>j, = co, except when it is a plane parallel to xss or yz, in which cases 

 its equation gives <t> y = co , y m const, and <J>, = co , x = const. 



• It is here understood that if x(f,y) be of multiple value, we speak of one of the values. 



