jVIE. w. h. l. eussell on the calculus op symbols. 
81 
Let u=[7r-\-z)v, and the equation becomes 
^ ( T + 1 ) ( TT + 2 ) ( T 1) y ^ ( 3 -j- 8 T + 1 0 ) ( T -f- 1 ) v 
+ f + 7 t+ 5)(‘r + (t — 1 3)('r -J- |)y = X. 
Let \J/,(5r)=^r+2, \po(T)=‘r+t, then the criterion of §'(‘r+2)+(’r+|) become 
(t* + 2 t — 3)(Tri-|) — ,^ j:^( 3 t^-}"'^+ 1)(’*'+I~1) 
_^ (7r+g(^ + g 
(7r+ l)7r 
(7r + ^)(7r + l)(7r + ^ — 2) 
(^^-39r+2X’r+|-3) — U. 
Put T—O to determine and we have |=0 as one value of which on trial is found 
to satisfy the proposed. 
Hence is an internal factor of the symbolical function 
g V(t + 1 )( 7 r + 2 ) + §^( 3?^ + 1 Ot) 
+ g(39ri+7'^+5^)+9r(7r — l)(7r-j-3). 
Wherefore, effecting the internal division, the equation becomes 
fyV+X2T+3)ri-7r-f-3)(T— l)fy(7r+2)+T)y=X, 
whence performing the inverse calculations, we have 
1 r? / 1 ^Cdx{x+l)^ Cdx.x'^ .'X. 
(07+1)0 ^5 (a^+l)5 J 
df 1 1*7 / , ^Sdx{x + \Y Cdx.x^X'\ 
where the arbitrary constants must be reduced to two. 
Next consider the differential equation 
(^‘+2x»+a:’) A“ _6(a:=+ii-) g +6(x+2)«=X : 
the symbolical form of this equation is 
— l)ic-^2§{7r — l)('r — 3 )w-|-(t — 3)(5r — 4)w=X. 
Let w=(T-l-|)y, and the equation becomes 
oV(7r — l)(7rd-^)y + 2X7r — l)(7r — 3)(7rri-|)y + (7r — 3)(7r — 4)(7r-h|)y=X, 
Let •4',(7r)=7r— 1, ■v^o(7r)=7r — 3, and the criterion becomes 
(ir— 3X>r — 4X!r + 5)— y;|{2(!r — 2X!r — 4))(x+|— 1) 
+|^r||4J{(— 2X— 3)}(-+l-2)=o. 
Putting 7r=0 in this equation, we have i=0, and this value renders the above equation 
identical, 
... — l) + (7r-3} 
