8 
changed its sign, and the difference is not obliterated by in- 
volution. Hence is not cyclical, and I think the method 
fails. Again, I do not assent to the principle contained in 
the words at p. 146, “that the involutions cannot destroy 
the symmetry in the roots.” For instance, the expression 
(x 0 -j- x x ) + (x 2 + x 3 ) = Sx is symmetric, but (x 0 + aq) 2 + (x 2 + x 2 ) 2 
= 2x 2 -j- 2(x 0 x 1 4- x 2 x 3 ), and is not symmetric but has three 
values. 
Next as to Hargreave’s Posthumous Essay. The British 
Association have paid a tribute to departed worth, and 
also to you in naming you to report on it. You will, 
of course, examine it carefully. I read it some time ago 
and made a brief communication to you about it. You as 
well as I seemed disposed to think that Hargreave recog- 
nised -J- and — as radical signs, where he excluded others. 
But it is necessary to be careful in criticising him, as he 
cannot now explain his meaning, and very likely where he 
uses + and — it is on the express understanding that the 
radical is capable of extraction. I looked at the Essay be- 
fore the last mail left on the chance of being able to render 
you some assistance, but in vain. Since then I have looked 
at it more narrowly. Many observations will have occurred 
to you in perusing it, and I can now only give you my pre- 
sent opinion respecting it, which after all is but a mere 
opinion, and subject to correction or modification. I can- 
not give assent to all his views. On arts. 9 and 10, pp. 12- 
14, every one must form his own opinion. I propose to go 
rather into special points. I cannot feel so strongly as that 
able man seems to have done, the necessity for uniqueness 
of expression (see pp. 45, 48, 51-2, &c., 77 and, probably, 
other places). There seems to me to be a capacity for re- 
storing the uniqueness, even when the quantic is conditioned. 
Thus, by way of illustration (the illustration being capable 
probably of any requisite extension). Suppose that we have 
given the fact that, a certain quadratic wanting its middle 
