380 On Cubic Equations. [May; 
On comparing this with equations (4), (6), (13), in order to ob- 
tain a formula involving only « and known quantities, we have the 
option of two such formule, viz. 
a Boast (Bab) Osi b igen at oe See eee (II.) 
2 ee 
and 2? + Ra + = =O oh an echemuenae <tee oi o be we Get lees) 
The solution of these equations gives us 
x=—-1tR+ivV4b-—3R 
=—1R#1 2 4c 
= rR+34/R_—*% 
Perhaps the most familiar way of considering the general relation 
of the roots is this: R the greatest, r decreasing, and @ increasing, 
in their progress from the nascent case, in which r = R, and g = O. 
The correct interpretation of the solution just obtained will, there- 
fore, be 
pie= (Li se +f 4b 3 RY sad “(v) 
@ =. 2 (REPOS SR) 
T\ | 
Or pT Tie 
R J wi ae RS) eee (W.) 
e= 2 R¥\/R-X)I 
the upper signs obtaining in the irreducible, and the under signs in 
the reducible case. 
In equations 5, 8, and 7, 9, we have the solutions of two curious 
diophantine problems; the common condition of limitation in the 
results being given in equation 2. 
The analogies traced in this paper being distinct from the general 
theory of cubic equations—on which, if agreeable, I propose to 
send you a memoir ona highly condensed but comprehensive plan— 
are offered in the form of a detached essay, as the most suitable to 
their character. Their utility, of which I have specified two in- 
stances only, will abundantly appear on applying them to other in- 
cidental cases, or to any particular form of a reduced cubic, such as 
»—3pPac—2p?q =O, 
used by Cotes in his Logometria; or 
a —dx—d=O, 
which Mr. Lockhart has so ingeniously employed in his method of 
approximation. 
W. G. Horner. 
P. S. Having still a vacant space, I am tempted to put in a word 
on the curvature of the circle, which has been so much agitated 
lately in the Annals. No mathematician certainly has ever regarded 
the circle as a polygon of any finite number of sides: all the inge-, 
