48 Rev. S. Earnshaw on the Finite Integration of Linear 



variables. With your permission I will now fulfil that pro- 

 mise. 



In the communication referred to it was shown that these 

 two classes of equations can by a change of variables be reduced 

 to the following fundamental forms : — 



/i\ c ^ u _ / >\ d* u —du (<xx d 2 u _d 2 u 



(i) dldi/-™'* W U -ch, 1 {o) d^"d? ; 



d?u 



du 



dx 2 



~dy> 



du 



di'' 



(5) 



, . d?u __ du ^ x-v d?u _ d 2 u du # 

 ^ ' dxdy dz ' ^ ' d# dy ~~ ^ 2 c?^ ' 



so that if we can succeed in integrating these forms, then we 

 may consider the general linear partial differential equations of 

 the second order, with two or three variables and with constant 

 coefficients, to have been integrated. When certain relations 

 among the coefficients exist the finite integration is easy ; but 

 in other cases the difficulty of finite integration was never over- 

 come. I propose to point out the cause of the difficulty which 

 occurs. 



1. If U be an integral in finite terms, then are also -y- i -v— ; 



and if U be differentiated any number of times with regard to 

 the independent variables, the results will be integrals also. 

 Consequently from any one integral we can deduce an unli- 

 mited number of other integrals ; and all these must be included 

 with U in the general integral ; and each must stand therein 

 multiplied by a separate and independent arbitrary constant. 

 And if so, how is their sum to be gathered into a finite form ? 

 Surely the arbitrary constants will be an insuperable barrier 

 to finiteness of expression. 



2. Again, if U contain in the body of it an arbitrary con- 

 stant c. then will —7-? -nr> -r^? . . . be cointegrals with U, and 



dc dc 2 dc 3 ' & 



their sum, each multiplied with an arbitrary constant, must 

 enter into the general integral ; so that, as before, we have what 

 appears to be an insuperable barrier to finiteness of expression. 



3. If U be expanded in a series in powers of c, each coeffi- 

 cient of its powers will be an integral, and we have again the 

 same difficulty repeated. Thus there are three impediments, 

 any one of which seems to be an insuperable barrier to finite- 

 ness of expression. 



An example will make this clear. Let us take the equation 



-r-g = -j-. A known integral of this equation is 



F(a?,y)=Ae* + «* (1) 



If this be differentiated any number of times with regard to 



