149 
Each of the three roots of (2) is a particular solution of 
(20), and a solution of (20) is, therefore, a root of (2). 
The above values of P, Q, R, agree with the results of 
Wm. Spottiswoode, M.A., F.R.S. (See Manchester P. So- 
ciety’s Memoirs, Yol. II., third series, page 230.) 
The reader of Mr. Spottiswoode’s paper above referred to 
must make the following corrections, viz., page 231, for 
( a}) 2 a 2 b read ( a}) 2 a 2 d in line 10 from top; in Line 11 from top, 
for Sa^ad 2 read cdb\Sad 2 -9bcd). Mr. Spottiswoode has 
pointed out the line to be pursued to obtain the resolvent 
of the second order. (See also Rev. Robert Harley’s Report 
on the Theory of Differential Resolvents to the British As- 
sociation for the Advancement of Science, 1873. 
The problem, viz : to find the second differential resol- 
vent of a general cubic is exceedingly complex and tedious ; 
it was completely solved by me some three or four years ago 
and the results sent to the Rev. Robert Harley, F.R.S., &c., 
in whose possession they still remain. 
8. The equation (20) is soluble when c, m are constants, 
and am 2 = Sbcm~2c 3 . Hence, the root of the following 
cubic is obtained by integration, viz : 
(3 hem - 2c 3 )?/ 3 + 3 bm 2 y 2 + 3 cm 2 y + m 3 = 0. 
9. When 6 = 0 then, a — - 3a(4c 3 + am 2 ) 
and, aP = - acma 1 + 2>a 2 mc l - 2 a 2 cm} (26) 
and, aQ = - (6c 3 + am 2 )a} + 6ac 2 d + a 2 mm l (27) 
and, aR = - 2 c 2 ma}< + Qacmc 1 - ±ac 2 m l (28) 
The further condition, c 3 = Ci<xm 2 , where Ci is an arbitrary 
constant, will cause P, R, to vanish, thereby rendering 
equation (20) soluble, and determine by integration a root 
of the cubic 
c 3 y 3 + 2>c x cm 2 y + c x m 3 = 0 (29) 
10. With a view of obtaining a root of a general cubic 
by integration it will be necessary to examine the condi- 
tions of solubility of equation (20) which are given by Abel. 
(See Abel’s works, vol. 2). 
