268 MR. hargre'ave on the resolution of linear equations in 
and since in the present case and are constants, we have 
Mp=aF-\-{p— — — — 
whence the particular value of is a series of terms of the form 
G,, 4" 1 "h ( G^_ 2 + + 2a G ^_3 + ( ^^ + 3a^6^ + G ^_4 + ( a® + 4 a®Z»^ + 3^ G^._ s + . 
The difference in character between the solution proposed in this paper, and that 
which would result from a perfect analysis of the general equation, may be exempli- 
fied by the present instance. 
The perfect solution of this equation is known to be 
«-/3 
Gsr + 2 
where a and j3 are the roots of — at—h-=0, and each 2 introduces a constant. 
Now if the constants be made equal, and the expression be written at length, we 
shall obtain the form derived above. 
The expansion of the two terms within the parenthesis gives 
G^+ 1 -f aG^ -f a"G^_ 1 + . . + ' Gp -f . . + ca^ 
and 
G.^.+f3G.+f3^G,_2+..+f3^^'G,4-...+c^^; 
and if the difference of these be divided by a— 13, we get 
Gr 
a -/3 
'G 4 ,_i+... 
aP+i—/3P+i — (S-^ 
which is the same in effect as the result which we have obtained. 
JE.v. 2. Let the equation be 
Then 
,=cr(x+ 1 )[i 2'"i{STT)) + • •} 
cr(j ^-|- 1 )|^l a ^ -{-a 2.3 (ar — 2 )(iE’— l)ar "J 
=c|r(j7-i-i)-«ra?+^r(^— i)— ^r(a?— 2)-i-..j ; 
which may be put under the form of the definite integral 
If we add a second member G„ the equation for determining is 
Mp= {x—p -\- 1 )Mp_i+ aM^_ 2 ; 
