g 6 Mr. Woodhouse on the Independence of the 
XL In the irreducible case of cubic equations, the root, it is 
said, may be exhibited by means of certain lines drawn in a 
circle. There is, however, independently of all geometrical con- 
siderations, a method of analytically expressing the root ; and, 
from the analytical expression, although it is not the formula 
which from the time of Cardan mathematicians have been 
seeking, the value of the root may in all cases be arithmetically 
computed ; but, previously, it is necessary to shew what are the 
different symbols that may be substituted for z in the equations, 
x = (zy/ — i)*“ x jfsv'zi — and s/ (1— x 1 ) = 2"" 1 
1, and 7 T be the value of z that 
| and 
-f -j- < c — iV — 1 1. 6 Let x 
answers the equations 1 = (a\/ — i) —I » 
o = a— T -f which value of tt may be numerically 
computed from the expression ,, 7 rr:s: = 1 r-|- -jy + -jj- + 
(x = 1). 
Hence, c ; 
• • c 
\S%d - — 1 __ 
—.47 rV ■ 
= £ 
— x 67 rV — 
- 7rv '- 1 = \/~ 
Sird . 
=ZTtV - 
•} £ 
1 &c. (for since £ 
.Sir'd — ] 
1 
and 
V —»l mird— I 
— £ 
2 mv 
d—\ 
Again, since £ 
£m 7 cd — 1 
4^-1 __ j anc j 8»V_i 
). 
£tnir 
d: 
L £ 
12 nrd c 
1; and 
tions such as x‘ | a+bx-{ cx*+dx s ^j. 2 -fy* + j* 2 — .0, 
because no analytical expression or equation of a finite form has hitherto been in- 
vented, from which, according to the processes of the differential Calculus, such diffe- 
rential equations may be deduced. To find the algebraical expressions which answer to 
these equations, recourse must be had to what are properly to be denominated artifices. 
For such, see Mem. de Turin. Vol. IV. Comm. Petr. Tom. VI. VII. Lagrange, 
Fonct, Analyt. p. 80. Lacroix, Calc. diff. p. 427, &c. 
