4 
(See Boole, Diff. Eq., 2nd ed., pp. 206-7). In like manner, 
by eliminating a dependent variable between the third and 
fourth equations, we shall obtain a result which, combined 
with the second equation, will give us a linear differential 
equation of the first order for determining the particular 
integral of the second equation. Use the particular inte- 
grals thus obtained in the formation of an integral of the 
first equation, and substitute the result therein, giving 
where necessary proper values to the arbitrary constants. 
Thus an integral of each of the three equations will be found ; 
in the case of the Boolian integrals, the process admits of a 
simpler application, which is not however in all cases so 
simple as in that of the cubic. I shall illustrate this 
application. 
Consider the Boolian cubic in y. Transform it into an- 
other cubic in 0 , wherein z=y m — 1, m being within Boole’s 
limits, and having no relation to the m hereinbefore men- 
tioned. Then the differential resolvent of the cubic in 0 is 
a linear biordinal which may be written thus : — 
d 2 z dz 
d? + L 'Tx 
+ z. 
*« = Z • . 
(14), 
and Boole has shown that one of the three values of 0 satis- 
fies an equation of the form 
2 -/(“ - P) dv > 
the integration being within the limits zero and infinity 
and a and /3 being, each of them, proper fractions of the 
third class. But the integrals of such fractions satisfy 
linear differential equations of the second order. Hence, 
putting 
J'adv = a, Jfidv — b, 
we may write 
d 2 a da 
* +Al & + A * a = x(“> = A ■ • • < 15 > 
<p& 
dx" 
B,^r + B Jj = Mb) = B 
(16) 
