On Non-Linear Coresohents. 29 



which is a quadratic in b then (16) after division by N may be 

 put under the form 



(a + L) 3 = Q + L3 (18) 



and the determination of a depends on the extraction of a cube 

 root only. 



5. It follows that by solving a quadratic and extracting a cube 

 root, the solution of a cubic may be connected with that of a non- 

 linear differential resolvent of the form (15), and that by solving 

 a cubic, the solution of a quartic may be made to depend upon a 

 similar differential resolvent. 



6. Eeplacing P in (15) by its value as given in (3) equation 

 (15) becomes 



a + Ip + J^i= o (19) 



y 



assume y = ef vdx then 



p =yu,q -y J? + y *i> 2 (20,21) 



whence, substituting these values in (19) and dividing the result 

 by y, we have 



— +Iu + (J + l)u 2 = o (22) 



dx 



or, dividing by « 2 



1 ■ 0& + ll+ J + 1 = o (23) 



u 2 dx U 



Hence, if 



1 = - v and .-. i *! = * (24, 25) 



u v, dx dx 



equation (23) becomes 



^ —Iv + J + l = o (26) 



dx 

 a linear differential equation of the first order in v. 



7. Thus in the case of cubics and biquadratics we have, through 

 a non-linear, been conducted to a linear differential resolvent, the 

 latter being, moreover, of the first order only. At a meeting of 

 the Queensland Philosophical Society, held on Monday, July 30th, 

 1866, I read a Paper " On the Inverse Problem of Coresolvents," 

 which was printed in the Queensland Daily Guardian of Tuesday, 

 August 7th, 1866, and in which, after calling attention to the 

 theoretical possibility of determining the foregoing resolvents, I 

 suggested that further results might spring from the study of 

 non-linear differential resolvents. Some three months ago I sent 

 to England a Paper in which I pointed out certain conditions 

 which, when satisfied, enable us by means of a non-linear differ- 

 ential resolvent, to obtain a linear differential resolvent of a 

 quintic which is of the third order only. By the last (April 1867) 



