624 
PROFESSOR SYLVESTER ON THE REAL 
functions, 
ru h -\-sv 5 — t{u-\-v) 5 ’, rstuv(u-\-v ), 
and conversely, 
J=A, 
K= — 
*+he 
2 8 
1= — il — ^ aJi+ b a!i 
TJnliappily a farther step is . wanting to bring M. Heemite’s results to the final test of comparison ; for the 
value of AA' (p. 192) does not agree with that given for AA' (p. 186) by simply changing J 13 J 2 into J 2 , J 3 
respectively ; a further change of A into 2A becomes necessary to make the ratios of AA', BB', CC' (p. 192) 
accord with the ratios of the same quantities at p. 186. Finally, even after making this change the expression 
for 16I 2 (p. 203) does not accord (even to a constant coefficient prbs) with that with which it is meant to be 
identical, viz. 16I 3 (p. 187); so that after great labour I am still baffled in my attempt to ascertain the agree- 
ment or discrepancy of my conclusions with those of my precursor in the inquiry. As will appear hereafter, 
the two sets of conclusions are undoubtedly discrepant in form ; but whether they are so in substance or not, or 
rather whether they are or not in contradiction to each other, requires a close examination to discover, the more 
especially because, as will hereafter be shown, there is a certain necessary element of indeterminateness in the 
scheme of invariantive conditions which serve to fix the character of the roots. It is greatly to be lamented that 
so valuable a paper as M. Heemite’s should be to some extent marred, in respect of the important end it would 
serve as a term of comparison, by the existence of these numerical and notational inaccuracies. I have spent 
hours upon hours in endeavouring to reconcile these several texts of the same memoir, and, after all my labour, 
the work is left unperformed without which the truth as between the two methods cannot be elicited. I feel, 
however, as confident of the correctness of my own conclusions as of the truth of any proposition in Euclid. 
( b ) It is worthy of notice that there is a failing case in H. Heemite’s process for finding I 2 in terms of A, J 2 , 
J 3 , just as there is one in mine for finding Gr in terms of J, K, L, — the failure of the process, however, in neither 
case entailing any corresponding defect in the results obtained. The process employed in this memoir fails 
when L=0 : for then the general form ru 3 -\-sv 3 -\-tw 3 is superseded by the supplementary one, au 3 + 5euv 4 +fv 5 , 
M. Heemite’s fails when J (the J of this memoir) = 0 ; for then the quadratic invariant becomes a perfect square, 
and the substitution of its factors in place of the original variables becomes inadmissible, since the two former 
coincide. 
(°) It may be as well here to notice the form which M. Heemite’s two linear covariants assume when 
referred to the canonical form above written. The quadratic covariant being rsuv + stvw -f- trwu, if we operate 
with the correlative of this obtained by writing in it ^ > L—lL in lieu of u, v, w, viz. 
dv du du dv 
* s LI -stL(±-L]+trl(L-L) 
du dv du\du dv) dv\du dv) 
rst 
upon the primitive, we obtain to a factor pres the canonizant rstuvw, which has been already obtained ; repeating 
the process, it is easy to see that the first linear covariant of the fifth degree in the coefficient assumes the simple 
form rst(stu + trv + rsw), or rst(^u+ <xv -\-rw). Taking again the correlative of this, viz. 
d _ d / d _ <A\ 
%dv ° du \du dv))’ 
and operating with it upon rsuv+stvw+trwu, it will be found without difficulty that the second linear covariant 
of the seventh degree in the coefficients becomes 
rst{((r— r)((r+r— f)w+(r— f)(r+f— <r)v+(f— o-)(f + (r— r)«/}, 
which is distinguishable in species from the former one by its s ymm etry being only of the hemihedral kind. 
( a ) It may not be out of place to notice here that the Hessian of the canonical form will be found to be 
vhc 3 + tiv 3 u 3 + tu 3 v 3 . 
