282 
Mr. Wilson’s Essay on the 
tigated, and what new forms they have taken, or what different 
functions of them are used in the operation. 
17. If we resume the general indeterminate equation 
(x v — px n ~ x -J- qx n ~ 2 — rx n ~ 3 &c. = o), and assign the pro- 
gressive values (2, 3, 4 &c.) to the index (re), in the first case 
it will become the quadratic (a: 2 — px -j- q = o). Now, as this 
equation has two roots, in order to obtain a general formula for 
its resolution, the first step that suggests itself is, to inquire 
what is necessary to construct a general representation of two 
quantities in a simple equation. Two quantities are known to be 
generally expressed by means of their sum and difference ; that 
half their sum added to half their difference gives the greater, and 
the same quantities subtracted, the lesser. The sum being al- 
ways the coefficient of the second term of the equation, is given 
in all cases , and here the difference is readily found; for, the 
square of the difference of any two quantities differs from the 
square of their sum by a constant quantity, viz. four times their 
product or the coefficient of the third term. If ( a ) and (6) 
be called the roots of the equation ( x 2 — px -f q = o) ; then 
(p = a - f h ) and (p z = a 2 -f- 2re& -f- b 2 ), 
(qr = ab) and ( — 49= —4 ab ), 
. r the square 
whence (a 2 — 2ab-\-b z z=a — b\=p 2 -*- ^ql ofthedif- 
[ ference. 
The difference itself is therefore ( s/p 2 — 4^). And now, being 
possessed of the parts required to construct a general repre- 
sentation of the two quantities, we can at once complete the 
formula of general resolution of equations of this degree, viz. 
= EH). 
This, as I observed before in Art. 6 , is however the same qua- 
