40 On Linear Partial Differential Equations of the Second Order. 



if P= — , where /is constant ; 



_ \k+h{h-l)\{k+(h*-l)(k~ 2)-/} x^ 



Similarly we shall arrive at 

 _\k + h(h-l)\ { h+(h-l)(h-2)-l\ \k+(h-l){h-2)--2l} _rf 



3 K-« 2 ) 3 "'1.2. J 



&c. &c. 



^ \k + h{h-l\\k+(h-l)(h-2)-l\...\k+(k~r + 2)(h-r + l)-{r-l 



{a,-a 2 Y 



Ji-3 



' 1 . 2 . . . r 



If A is fractional or negative, we shall, on the particular as- 

 sumptions above introduced, always have an integral of the 

 assumed form, the number of terms being finite or infinite ac- 

 cording to circumstances, though as to the practical value of the 

 integral so obtained in the latter case I am not prepared to ex- 

 press an opinion. 



The condition to be satisfied in order that A r may vanish and 

 that the expression for z may have a finite number of terms 

 when h is not a positive integer, is, that we have 



= k+(h-r + 2){h-r+l)-{r-l)l, 



the only conditions limiting the quantities h } k, l> r being that 

 they are all constant, and that r is, and h is not, a positive 

 integer. 



When h is a positive integer and r = A, we shall have A r con- 

 stant ; whence it follows that upon this supposition the series 

 will always terminate when U = 0. 



The well-known equations 



n _ **,**_ n(n + l) 



U "~ da* dy* \x\ ' 



d 2 z %d 2 z 2n dz 

 dx* dy 2 x dx 



are readily solvable by the foregoing method. 



It remains to be remarked with respect to the first term of the 

 series for z, that since (3) is of precisely the same form as (1), 

 any value assigned to A must be different from any derivable 

 from the general expression for z — as, for instance, a solution 

 obtained upon a particular hypothesis not necessai \ly implied by 



