+ /^;&o (2) 



Partial Differential Equations ivith constant Coefficients. 49 



the independent variables no new integral results from it, and 

 consequently the integral before us evades by its form the diffi- 

 culty to finiteness stated in (1). But if it be differentiated 

 with regard to c, we obtain an infinite series of dissimilar inte- 

 grals ; and when each has been multiplied by an arbitrary 

 constant the sum of the series will not be expressible in a finite 

 form ; so that the difficulty stated in (2) holds good here. 



Again, if the above integral be expanded in a series accord- 

 ing to the powers of c, the coefficients of these powers will be 

 unlimited in number, and each of them will be an integral and 

 finite in form, as follows : — 



1 x x 2 y x z xy x* x 2 y 



' V l72 + l ; l7273 + r71 ; 1.2.3.4 + 1.2.1 



r\ 



1.2 



Hence the general integral of the equation must include the 

 following infinite series of independent integrals, 



and it is at once evident that this series cannot be summed, by 

 reason of the arbitrary multipliers of its terms. 



Thus it is clearly seen to be a hopeless task to seek for a 

 finite integral of any of the five equations mentioned above ; 

 and it is evident that the difficulty arises directly out of their 

 property of linearity. 



Some few years ago I discovered that the equation, above 

 quoted as an example, admits of an integral of the form 



FOvy)=A«r*e *a\ (3) 



and thus we have two integrals, (1) and (3), of one equation 

 which are as distinct in form as can well be conceived ; besides 

 which we have an unlimited number of other distinct integrals 

 of the same equation in (2). It has, in consequence of this 

 curious abundance of distinct integrals of the same equation, 

 appeared to me a worthy object to ascertain the nature of the 

 connexion existing among them, and to what their abundance 

 and distinctness are due. I believe I have completely suc- 

 ceeded in this object, and that I have been enabled thereby to 

 find the various integrals which each of the five fundamental 

 differential equations can have in finite forms. The investiga- 

 tions are too long for your Magazine ; and I propose therefore 

 to commit them to the press for private circulation among such 

 mathematicians as may desire to possess them. 



Sheffield, May 31, 1876. 



Phil. Mag. S. 5. Vol 2. No. 8, July 1876. E 



