2G4 PEOFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
mine f for arbitrary values of u, i\ u\ v' (Art. 3). These four scalars may be selected 
in any way we please. 
58. A single condition connects two quartics of the second class'*' when it is possible 
to transform one into the other. 
The equation of transformation is 
fiahedeft, 1)^ = {a'Vdd'c'Jiit -f- if vt + vf .(238), 
and five equations of condition may be'written down analogous to (232). 
Let the relations connecting the sets of five quaternions be 
x^a + Axf + Gxp + 4xpl + aqe = 0, ypi' + igf' + 6y.c' + 4ij./l' + yp' = 0 . (239); 
then, as in Art. 56 (234), we obtain the equation 
XoA + 4Xi?/ + GX,c + 4 X 31 /' + X/ = 0 .(240), 
where 
Xy = {xQxp‘.;,x.,x.,Xvvy .(241), 
and where Xj, Xo, X 3 , and X^ are its successive polars to u'v\ 
On comparison of (240) and (239) the equality of ratios 
5) ^ X , ^ X^ ^ X 3 ^ X, 
2/0 Vi y-2 2/3 2 /r. 
is seen to be necessary. This is equivalent to four quartic equations in the homo¬ 
geneous variables n, v, u, v, and the resultant of these four equations equated to 
zero is the single condition in question. 
SECTION X. 
COYAPJAXCE OF FUNCTIONS. 
59. The eight types of covariance. .. 054 
60. Special cases in which the types coalesce.. 
61. Second general method of obtaining covariant functions.266 
62. The Hamiltonian invariants and the method of arrays.266 
59. The subject of covariance naturally arises in connection with the various 
transformations lately considered, but as the principles laid down in the note on 
Invariants of Linear "V ector functions printed in the Ajipendix to the new edition of 
Hamilton s Elements apply with but slight modification to the more general case of 
quaternion functions, it does not seem desirable to go into any great detail. 
A quartic of the second class is the partial intersection of a cubic and quadric surface, and only one 
quadric can be drawn through it. 
