10 
Lagrange determines the ratios of A h A 2 , . . . B b &c. a priori. 
With respect to quartics, Hargreave seems to have thought 
that he had obtained new results, indeed a new solution. 
He sees an objection (pp. 100-101), and seeks to obviate by 
saying that there has been a change in the framework of the 
root. Far be it from me to contradict him dogmatically, 
but one cannot refrain from the observation that if we have 
x i -\- &c.=0, 
y—p-\-qx-^r . . +sx% 
and, consequently, by division and reduction 
<»=P+QH- • • 4- Sy 3 , 
we must not be surprised to find in the last expression 
for x any radicals that we may have introduced into p, q, 
&c., and which will reappear I suppose in P, Q, & c. and the 
accompanying y°, y, &c., but (as I take it) in such a manner 
as that when development takes place all those radicals will 
disappear from the final expression for x leaving only the 
usual radicals. Hargreave however relies (p. 100) on the 
“ special or singular forms of \p (x) which,” &c., (and see p. 
101). He does not miss or ignore the objection, but seeks 
to obviate it. I am only expressing a mere opinion of what 
the final expression would contain, and am unable to give 
you (what would really assist you if not too late) the actual 
calculation of the final expression. If n be the discrimi- 
nant of the quartic in x so that n = (x 0 — x^\x 0 — x 2 ) 2 . . . 
then the discriminant of the quartic in 0 or (p (x) is (conform- 
ing more closely to Hargreave’s notation, p. 63) 
| k(x l ^-x 2 ) + 1 | (x x — x 2 ) | h(x x -\ -x 3 )+l | (#!— # 3 ) 
rz:7r | k(x x -\-X^)-\-l j ^(Xy—'X^)— -7T "I 7c(X] -p Xo) j Q 
7 T denoting the product of the various values of the expres- 
sion which follows it (compare pp. 63-64). And we see 
that so far as the discriminant of the equation in 0 is concerned 
the new radicals multiply the old radical, and probably or 
certainly the old radicals will appear (considering for a 
