[124] PROFESSOR DE MORGAN ON SOME 



If we want to have equations of the same invertible character as F= xy — y, X = y', 

 y = XY' -Y t x = Y", &c, we must choose a form of (p (x, y, X, Y) = 0, in which X and Y 

 may be interchanged with x and y, without alteration of the relation. The instance just cited 

 comes from y + Y = xX; if we choose y Y, <=■ x + X , we get 



X = %-x, Y=-,, Y'=~, r' = ^.4,&c, 

 y y y if y 



wkh power to interchange large and small letters. If we choose X'y* + Y'x 2 = I, we have 



y{y-yx) x(y -yx) tfy" 



with the same power. 



The following problem thus becomes of interest. Given <p {x, y, y , y", ... ), required 

 means of determining all or any of the modes by which it may be transformed into the given 

 function y\,(X,Y,Y\ ... ), where X = £ (x, y, y), Y=n(x,y,y'), these equations being two 

 primitives of the same equation of the second order.] 



Section III. 

 On the number of constants which the complete primitive of an equation can contain. 



It has never, till very lately, been thought necessary to prove that the number of con- 

 stants in the primitive of an equation of the «th order amounts to n : and I am not aware 

 of any attempt to prove, generally, that the number cannot exceed n. It was, at one time, 

 usually taken for granted that the restoration of the primitive form requires n integrations 

 and no more. Now, first, it was never shewn that <p (x, y, y',...) always admits of a 

 factor which renders it integrable, independently of relation between x and y. Secondly, 

 in the absence of direct methods, and of proof of their possibility, it might be suspected that 

 in certain cases, more than n integrations would be requisite, followed by changes of form, 

 and then by differentiations which, in consequence of the changes, would not destroy the 

 previously introduced constants. Thus the notion that y' y = (p (x, y, y, y", y") must yield 

 y by four integrations would not stand, on its own a priori probability, against any one who 

 should assert that it required twenty integrations, the introduction of a factor, and sixteen 

 differentiations. 



That the solution does introduce as many as n constants is proved without difficulty. 

 Taking a given value of x, and assuming values of y y'...y^ n ~ y) , independent of each other, we 

 may construct a solution geometrically, as the limit of a polygon of curved sides, or alge- 

 braically, by forming subsequent differential coefficients from the equation. M. Cauchy 

 prefers to shew the approximate construction of the solution of n simultaneous differential 

 equations of the first order, and its introduction of n constants; after which he reduces the 

 single equation of the rath order to n such simultaneous equations. 



