80 
ME. W. H. L. EIJSSELL ON THE CALCELES OF SY^ilBOLS. 
To integrate the differential equation, 
flu'll //?/ 
3^-(i-+iy £-i2x- 3)«=X. 
The symbolical form of this equation is 
§V^^^-§^(3■r®+7r — 1 )u=r.'K.x. 
The divisor of tt® is t only, the divisors of — 1) are tt— 1, 5r, “rff-l; hence putting 
'4 /uT=t— 1, we find the criterion of the symbolical quantity to 
become 
-(^-3)^ = 0, 
an identical equation. 
Hence qr-\-{‘7r—\) is an internal factor of 
§V^d-§^(37r^T-7r — l) + 3g('r^+l)4-^r(‘T^— 1) ; 
and the equation may be written, effecting the internal division, 
{f{‘^ — l)^4-^('rfi-l)(2‘T — 3)-l-?r(‘r+l)}(g'r-f-(‘r— l))%=Xa.’ ; 
or if §T+('T — l)^^=^q, 
{ + l)(27r — 3) +'r(T T 1) =X.r. 
The only divisor of (tt — 1)^ is tt — 1, the divisors of 7r(7r+l) are tt and tt ff"! j and by 
trial it is found that the divisor ^(tt — l)-|-(7rfi-l) satisfies the criterion, and is therefore 
an internal factor. Hence, effecting the internal division, 
(§(7r — 2)+7rX§(7r — l) + (7r+l))Wi=XA’, 
and the differential equation becomes 
(§(7r — 2) + 7r)(X7r— l) + (7r+l))(§7r+(7r— 1)}m=X^, 
or 
Hence, performing the inverse calculations, we find for the complete integral ; 
rdx{x+l)^l 
r dx j 
r xdx 
J J 
U+iJ 
1 (<*■+ 1 )^’ 
the three arbitrary constants being included under the signs of integration. 
In case this method does not succeed, we may sometimes resolve the symbolical 
function into factors by assuming ^^=(7^+|)v and proceeding as before, determining (a) 
from the criterion, as will be shown in the following examples : — 
To mtegrate the differential equation 
+ 1 X ^ fi- a( 4^® + 1 1 3) ^ + 2^® + 1 0^^ + 5a; — 3 = X. 
The symbolical form of the equation is 
§®(7r*4~3‘T-l-2)+gX3’r^ + 8’r+10)+X3'^^+77r+5)-|-'7^+2'r — 3=X. 
