64 



continue to be cinematical, in the largest sense of which this word 

 (so far as relates to local motion) can admit. 



June 5, 1848. 



Methods of Integrating Partial Differential Equations. By Prof. 

 De Morgan. 



This paper contains a sketch of two distinct methods. In the first 

 (x, y, z, p, q, r, s, t, having their usual significations) the given equa- 

 tion is supposed to be of the form <p(x,y,p, q) = 0, and this is made 

 the result of elimination between two equations involving a new 

 variable v. From these two, and their four differentials of the first 

 order, p, q, r, s, t are eliminated, and an equation of the first order 

 results between x, y, v. This last equation is often more manageable 

 than the original one. 



The process is rendered very simple when the given equation can 

 be reduced to depend on two of the form 



p=<p(x,y,v) q=^/(x,y,v). 



The second method was completed, Mr. De Morgan states, and out 

 of his hands for transmission to the Society, when he discovered that 

 Mouge had communicated it to the Institute, by which body it was 

 never published. But M. Chasles found it among the manuscripts 

 of the Institute, and stated it a few years ago in one of the notes to 

 his Aperni Historiqne .... des Methodes en Geometrie. Its occurrence 

 in the voluminous additions made to a work which itself treats only 

 of geometry, seems to have prevented it from becoming known to 

 any writer on the differential calculus. Certain particular cases 

 appear in the writings of Legendre and Lacroix. 



Let the equation be <p(x,y, z,p, q, r, s, t)=0. Change x into p, 



y into q, z into px + qy — z, p into x, q into y, r into , s into 



rt — s 2 



t into . If the equation thus resulting can be inte- 



rs — s 2 rt — s 2 



grated, let its solution be Z = \J/(X, Y). Then the solution of the 



original equation can be obtained by eliminating X, Y, Z from 



Z = *(X,Y) *=§ y=§ ,=,X + yY- Z . 



In both methods the most effective mode of proceeding is to find 

 what Lagrange calls a primary solution, containing two arbitrary con- 

 stants, and then to use that primary solution. 



On some new Fossil Fish of the Carboniferous Period. By Fre- 

 deric M'Coy, M.G.S., N.H.S.D. 



The author having premised that the species of fish of the carbo- 

 niferous limestone enumerated in the third volume of the Poissons 

 Fossiles of M. Agassiz are for the most part still unpublished, being 

 without definitions or figures, states that through the kindness of 



