474 
THE REV. B. BRONWIN ON THE SOLUTION OF 
If we put f' these will take the more convenient form 
"^1 = AX, ““"2= - AX ; 
which are only of the first order, since is of that order. The equation (19.) may 
therefore be considered as integrated. 
By the conversion of symbols (18.) and (19.) will be changed into 
^ 20 .) 
These may be solved by means of the formulae {d^ and (/*.) in exactly the same way 
as (18.) and (19.), and every step of the solution in the one case may be derived from 
the corresponding step of the solution in the other merely by the conversion of sym- 
bols. But every solution of these equations will not be a practicable one, or be 
susceptible of interpretation in finite terms. The operations however can be 
performed if 
%(D) being a rational function of D, and the constant q being positive, negative, or 
nothing. By differentiating the logarithms of each member relative to D, this will give 
or 
4>~(D) + (m + 7-)x(D) _ gp(;(D) +x'(D) 
f(D) “ x(D) 
by putting for 
1 
^5(D) 
its value, and dividing the equation by >^'(D). 
Therefore 
T-(D) 
_ g%(D)+ x'(D) 
“ X(D)A'(D) 
(&— ^^(D)') — (»i+r)X(D). 
Such is the value which T(D) must have in order that the solution may be prac- 
ticable, X(D) being at the same time a rational function of D. 
The value of T(D) is too complex for practical utility. But if we make 
X(D)=flDd-c, 
we shall have 
j 
V(D) = — 2^(«D+2c). 
If therefore %(D)=aD-|-2c, then we find 
T(D) = — \ciq ^^ — ^am-j-ar-j-c^+^a^D — (cm+cr). 
