PARTIAL DIFFERENTIAL EQUATIONS. 613 



The equation ar + bs + ct. = {rt - s') .f(p, q) 



transforms into at - bs + cr = f{x,y), which, a, b, c, being constants, is integrable. 



Since rt - s' transforms into (rt - s^)-\ the equations rt - s' = f{p, q), and rt - s^ = 

 {fi^tV)}' depend each upon the other. 



The failure of this method in the case of developable surfaces may be illustrated geometrically, 

 as follows. Let the equation ^ = be that of a surface, and for each point (.v, y, x) of that 

 surface, taive another point having for its co-ordinates p, q, p x + qy - x. The surface which has 

 the second point for its locus is conjugate with the first ; that is, what properties soever connect the 

 first with the second, the same connect the second with the first. This conjugation cannot exhibit 

 any absolute geometrical properties, for the conjugate surface depends, as to what it shall be, not 

 only on the primitive surface, but on the position of the axes of co-ordinates, and also on the linear 

 unit chosen. Thus it will be found that the conjugate surface of a given sphere is a double 

 hyperboloid of revolution, having for its real axis the diameter of the sphere which is parallel to 

 the axis of ss, and for its imaginary semiaxis the linear unit. Now when the first surface is 

 developable, its conjugate surface becomes a cylinder described by a straight line parallel to x, 

 guided by the curve f(p, q) = on the plane of xy. There is then no relation which involves all 

 the three co-ordinates. 



It may be worth while to notice, that we can at pleasure obtain forms for elimination which 

 reproduce the function originally given, by assuming an equation which is its own tranformation. 



A. DE MORGAN. 



UNrvEEsrrY College, London, 

 April 27, 1848. 



June 1, 1847. I had finished the foregoing Paper, as here written and dated, and it was in the hands of 

 a friend for transmission to the Society, when I happened to have occasion to turn over all the Notes of M. 

 Chasles's Aperfu Hlstoriqiie .... des methodes en Gcomctrie, tliat I might collect all that has reference to the history 

 of Aritlimetic. To my surprise, at Note xxx. p. 376, under the head iS'tir les Courhes el Surfaces reciproques 

 de Moiif/c, being an account of an tiripiMished memoir of Monge in possession of the Institute, I found tlie 

 second of these methods fully described. But to judge from all elementary writings, as well as from tlie apparent 

 resources of those who liavc had to use modes of integration, this method is not known; and therefore 1 do not 

 abandon my intention of commumcating it to the Society. 



A. DE MORGAN. 



Vol. VIII. Paut V. 4E 



