127 
Ordinary Meeting, March 9th, 1880. 
J. P. Joule, D.C.L., LL.D., F.RSi, &c., President, in the 
Chair. 
Notes on the Biquadratic equation 
+ c = 0 (1)/’ 
by Robert Rawson, Hon. Member of the Manchester 
Literary and Phil. Society, Associ. of the Institution of Naval 
Architects, Member of the London Mathematical Society. 
1. This equation has received the earnest attention of the 
greatest Mathematicians of Europe during the last three 
centuries. There is one principle, however, common to all 
their researches on this subject, viz., the formation of a 
biquadratic, with indefinite coefficients, the roots of which 
can be determined by means of cubic and quadratic equa- 
tions. The form (1) can be readily reduced to the following 
form : 
ax^:^hx ■¥ c = 0 ( 2 ) 
where a, 6, c are real numbers. Under this form the 
biquadratic has been considered by nearly all writers on the 
subject. 
The various forms which have been successfully proposed 
for the formation of the biquadratic, with indefinite coeffi- 
cients, may be arranged in the following manner : 
By Harriot. 
{x — a){x — I3)(x — y){x — d) = 0 (A) 
By Des Cartes, Bombelli, Oughtred, and Harriot. 
(x^ + + l3)(x^ — ax + y) = 0 (B) 
By Ferrari j Newton, Waring, Maclaurin, and Simpson. 
(x^ + ^X + aY = fi.X + JyY (C) 
By Euler. 
x^ — (a + yY (D) 
The quantities a, /3, y, S are indefinite constant, which may 
be determined so as to satisfy four given conditions. 
Procebdinus— Lit. & Phil. Soc.—Vol. XIX.— No. 11.— Session 1879-80 
