[136] PROFESSOR DE MORGAN, ON SOME 



ADDITIONS. 



Section 1, page 8 (of the memoir). It should have been noticed that when ^ x = co 

 or jk, = co produces a result which does not satisfy y = ^, the equation which it does satisfy 

 is y = ^ — X, where X is the value of P-^Xi/' 



Though Clairaut was the first who entered upon the theory of the singular solution, yet 

 it must be observed that (as noted by the late D. F. Gregory) the existence of such solutions 

 was first seen, and the name given to them, by Brook Taylor (Method. Increment, p. 27). 

 His instance is y 2 - 2xyy +(l + a? 2 ) y* = 1, of which his general solution is y = a+x^/(l -a"), 

 and his singularis qumdam solutio is y s = 1 + /J? 2 , obtained by differentiation. 



Section 2. The method given in my former paper for partial differential equations, and 

 first imitated in section 2 with reference to equations between two variables, and then extended, 

 may now have the corresponding extension made with reference to partial differential equa- 

 tions. Let <p (x, y, z, X, Y, Z) = be an equation connecting * with the variables x, y, and 

 the constants X, Y, Z. Let p, 9, r, s, t, have their usual significations. From <p = 0, 

 <p, + <p* ■ P = °, y + 0*-9 = O, we may obtain X = £ (#, y, z, p, q), Y = r, (x, y, z, p, q), 

 Z = %(x, y, z, p, q) : from which, if Z be a function of X and Y, we obtain the partial differ- 

 ential equation of which d> = 0, X and Y being constant, is Lagrange's primary solution. 

 Instead of this, let all six letters be variables, and let z be a function of m and y understood, 

 whether determined or undetermined : then X = f, Y = rj, Z = £ imply that Z is a conjugate 

 function of X and Y. These last relations are substitutions for transformation, by aid of 

 which any partial differential equation may be transformed into another of the same order. 

 For, P, Q, R, &c. being differential coefficients of Z, <p m gives 



($. + d)zP) dx + (d> y + <p x q) dy+ (<p x + d> z P) dX + (<p Y + (p 2 Q) d Y = 0, 

 whence since d> x + (p x p = and <p y +(f> z q = even when X and Y are variable (for under 

 these relations they were made to vary), we have (p x + (p z P = 0, <p Y + <p z Q= 0, that is, P 

 and Q are expressible in terms of x, y, z, p, q. Carrying this on, we may shew that R, S, T 

 are expressible by aid of no higher differential coefficients of z than r, s, t. And the analogies 

 alluded to in section 2 between the differential equations in which the small letters are con- 

 stants, and those in which the large letters are constants, exist also in this case. If then, ob- 

 taining from the above 



P = w(x, y, z, p, q), Q=k (x, y, z, p, q), 

 we have an equation to be integrated, 



/{?(*> y>*» P. ?). v(--), £(••). 7sr(. •). *(••) }=°» 



and if we can integrate f(X, Y, Z, P, Q, ...) =0, it follows that we integrate the original 

 equation, by eliminating X, Y, p, q, between the five equations X = £, F= r/, Z= £, P= w, 

 Q m k, in which Z is now known in terms of X and Y. If we choose that X, Y, Z shall be 

 reciprocally related to x, y, z, we have but to take a form of d> in which x, y, z are simul- 

 taneously interchangeable with X, Y, Z, as in « + Z - Xx + Yy, the case treated by M. 

 Chasles, and by myself in my former paper. 



