11 
moment only the term Q y of x) both in Q and in y. But, 
this is what I suppose might in any circumstances be ex- 
pected (compare Serret, 2nd ed., p. 562, equation (1) and 
lines which follow). If it should turn out that Hargreave’s 
second solution of the quartic does not essentially differ 
from the old one, will not the validity of his solution of the 
quintic be disprobabilized ? In the instance given at p. 4 
can we not extract the roots of the algebraic formula so as 
to obtain 2, and 3+ J — 1 (see Gamier, Analyse, 2nd ed., 
ch. 15, p. 321)? As to linearity of results, Hargreave uses 
the principle of articles 9 and 10 (p. 12 to p. 14) to account 
for the linearity. But the fact of the principle accounting 
for certain results in the case of cubics and quartics or per- 
haps of quadratics, scarcely proves the universality of the 
principle. Hymers (Equations, 1837, p. 54) speaking of the 
equation x n — qx+r— 0, says, it ‘which has necessarily n — 2 
imaginary roots, will have two real roots or none, according 
as 
(l) 7 or ^Gr~l) ( e ^ v ^ e ibid. p. 52 and pp. 99-100, 
and compare Hargreave, p. 115). But the existence of a 
cubical function of this kind does not appear to have sug- 
gested an algebraic solution of the equation — though per- 
haps we might (probably enough) conjecture that such 
functions would enter into the expression for the root even 
were such expression transcendental. Moreover, I am at 
present unable to attach the same importance as Hargreave 
does to the system of 3 cubics, (p. 28), of 2 quartics (p. 41), 
(p. 53), (p. 99 ?), and of the 5 quintics (p. 72). Hargreave 
does not overlook objections (see p. 32, foot note, to which 
references I would add one to Waring’s Misc. An., pp. 38-9, 
to my paper on Approximation, &c., in Diary, arts. 56 and 
57, and elsewhere). Supposing that we adopted a system 
of 5 congeneric quintics, should we account for all the va- 
lues of the root ? Suppose, for example, that the root of an 
imperfect quintic is a sum of 4 quintic surds, Waring ob- 
