434 Mr. R. Moon on the Equation of Laplace's Coefficients. 



and condenses, that the reaction of the gas is always equal to the 

 pressure upon the piston, whatever it may be. That this opi- 

 nion, however, is wrong is evident from this, that if the pressure 

 of the gas were always equal to the external pressure put upon 

 it by the piston, the gas would never yield and be condensed by 

 the descent of the piston, but would continue to sustain it. It 

 appears to me to be certain that the internal pressure of a gas 

 is never any thing else than that given by Mariotte's law as de- 

 pending upon the density and the temperature only, and that 

 any additional external force applied to its surface acts dynami- 

 cally, and not statically, upon it. 



LIII. On the Equation of Laplace's Coefficients. By & Moon, 

 M.A., Honorary Fellow of Queen's College, Cambridge*. 



THE equation of Laplace's coefficients has attracted the 

 attention of three distinguished English mathematicians, 

 all of whom within a comparatively brief space have passed from 

 the scene — the late Judge Hargreave, Mr. Boole, and Professor 

 Donkin, 



Having adopted Laplace's method of transformation and re- 

 duction, Mr. Hargreave gave, in the Philosophical Transactions 

 for 1841, the first solution of the problem in finite terms. 



One peculiarity of this method is that the equation actually 

 integrated by it is not that which was originally proposed, but a 

 derivative from the latter, the problem solved being in fact vastly 

 more general than that proposed for solution — a circumstance of 

 which the solution obtained by it affords ample prooff. 



Proceeding by the method of separation of symbols of opera- 

 tion from those of quantity, Mr. Boole, by each of three indepen- 

 dent methods, arrived at an expression for the integral ; and one 

 of great elegance, obtained on the same principle, was given by 

 Professor Donkin in the Philosophical Transactions for 1857. 



In a paper published in the Philosophical Magazine for July 

 last I showed that the equation 



where R, S, T, P, Q, U are functions of x and y only, in the 

 cases which are not amenable to Monge's method will always 

 have an integral in which z is represented by a series (finite or in- 

 finite according to circumstances) of the form 



z=z A X (u) +A l §dux{u)+A 2 §§du* x {u) + &c, 



* Communicated by the Author. 



t For instance, the expression thus obtained for the fiftieth coefficient 

 involves upwards of fifty arbitrary functions. 



