63 
resolvents is a direct process, and it is only fair to state that 
it has been gathered entirely from the correspondence with 
Sir James Cockle, F.R.S., &c., and the Rev. Robert Harley, 
F.R.S., &c. ; with whom originated the important invention 
of differential resolvents of algebraical equations. 
I am not sure whether the results in Arts. (4) and (5) 
have been published by either Sir James Cockle or Rev. 
Robert Harley. 
It is more than curious that Cardan’s formula should be 
reached as it has been, without the slightest assumption, by 
means of the second differential resolvent. And, the method 
which has been adopted is, in my opinion, very suggestive 
in the theory of higher algebraical equations. Especially so 
when several independent variables are made use of. 
6. If in (14) we put 
2 am 1 
*/ 4a 3 + 27 m 2 
where (r) is another function of x. 
2 ar 
J27 
■(24) 
Then, 
(m 1 ) 2 
27 
4a s + 27m 2 
Hence, there results by substitution 

dx 1 rdx dx 9 ^ ' ' 
which is the second differential resolvent of the cubic 
e~ frdx 2a 3 
r + ay + 
27 
J'rdx 
.(26) 
The value of (m) is found by integrating (24), and, solving 
algebraically with respect to (m), to be 
2a 3 
27 e 
frdx 
(27) 
7. If y and z are such as to satisfy the equation 
^x =ll2+ ^k ( 28 > 
Substitute this value of y in terms of 0 in (25), and it 
becomes 
r\ o 1 ( /'dry 4 r 2- ) . 
dx + & ~ i/3 { “ ^dxi^dx) + [rdx) + 9 } ( 29 ) 
