L’i)8 rKOFES.S<Ji; C. J. .T<JLY OX QUATEKXKJXS AND PKOJECTIVE OEOMETKV 
Moreover, wlieii we s})ecially select the points a, h, c as the united jioints of the. 
function fijJi/j), and when we form successive polars of with respect to JT, Ah and 
Ag, we find (Art. 97) in terms of the latent roots t.^, tg corresponding to a, h and c, 
S/>ol).A=0, S27 oD . Ah = (h + tg) A, {H/jqDY . (464), 
1 because 
= h («»/(l>«), h, /( pc)) + h {a, /{ pa), /( ph), c) .(465), 
and similarly in the other cases. 
«/ 
Thus the equation of the sextic may be wnhtten in the form 
A Y Z 
Ag Fg F, 
= 0 
(466), 
Ah Y, Zg 
(467). 
= 0 
(468); 
= 0 . 
(469) 
and the third polar of the point Pq is 
ih h) ih h) (h ^o) A I Z = 0 . 
Thus the tangent cone at the triple point breaks up into three planes. 
In the same notation the double curve is represented by 
A T Z 
I Ih Zo 
and forming the polars, the point 2^0 is seen to l^e triple and 
A Y Z 
(h “h ^3)A, (h + h)^ ’ (h “k 
represents the system of tangents at the triple points—the lines of intersection 
of the planes A, Y and Z. 
We may add that the equation of the cone, vertex yq, standing on the curve is 
{t,-t,)XYZ,+ {t,-t,)XYZ, + {p-tYXYZ, = 0 . . . (470). 
119. This surface resembles a Steiner’s quartic in many particulars, but it is a 
degraded case of the general surface 
P = {^!P)^ .(471), 
where {xyz)‘^ is the general quaternion function of three homogeneous scalar 
parameters oc, y, z. The general surface is of the 16th order. The Steiner quartic 
may be written 2> — a general cpiaternion quadratic function of x, y, :. 
Surfaces of this type arise from the general transformation 
P =f{<P Y ■ ■ ■ <l) . 
of tlm 7/fli order, being the transformations of planes. 
( 472 ) 
