503] THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 125 
Adding thereto a fourth term - S. afty, the value of the sum would be = afiy8, or 
the sum of the three terms is = a/3y8 + 8. a/3y, where the symbols represent deter 
minants. But in each case the determinant a/3y is = 0, as containing the column 
h a , hp, h y , the terms of which are each =0: thus %g. agh is = gagh — g$. agh, where 
in gagh the suffixes are a., ¡3, y, 8, and in agh they are a, /3, y: that is, we have 
'Zg. agh = gagh. And the whole expression thus is 
= x 3 (gagh — Xaagh) 
+ x l y (gahf—Xaahf+ gbgh — Xabgh — fagh - Xbagh) 
+ xy 1 (gbhf — Xabhf — fahf — Xbahf—fbgh — Xbbgh) 
+ V 3 ( ~fW ~ 'Xbbhf), 
where gahf denotes the determinant 
g, a, h, f i, with the suffixes a, ß, y, S, in the 
four lines respectively, and so in other cases: the terms, such as gagh, which contain 
a twice-repeated letter, vanish of themselves; and in the coefficients of x 3 y and xy 3 , the 
terms which do not separately vanish destroy each other in pairs, gahf—fagh — 0, &c.; 
whence the factor vanishes, being = 0 1 ; there are two such factors (viz. the zero term 
fpaS.pbS.p^a/Sy may be taken with the sign + or — at pleasure), and the norm is thus 
= 0 2 . 
40. But the line is tacnodal, each sheet of the surface touching along the line in 
question the hyperboloid p^afiy. To prove this, write 
A = X8 x +Y8 y + Z8 z + W8 w ] 
we have for the hyperboloid, writing z = 0, w = 0, 
Ap-aßy = (afg .x + bfg.y)Z+ (abg.x- abf .y)W\ 
and it is to be shown that 
A (fpaa.. pboL .p^ßyS — Vpaß . pbß. p 2 y8a + fpay. pby .p 2 8aß + fpa8 . pb8 .p aßy) 
each contain the factor Ap 2 cißy; or, what is the same thing, that 
A2 Vpaa. pba .p 3 ßy8 
contains the factor in question, % denoting the sum of the first three terms of the 
original expression. The value is 
_ 2 ('P acL ' ^ a p>ßy8 + fpaa .pba. Ap 2 ßy8j 5 
V 2 Vpaa. pba 
where Paa, =Apaa, denotes what paa becomes on writing therein X, Y, Z, W for 
#, y, z, w\ and the like as to Pba. Substituting for paa and pba their values z a I a and 
Zi I a , and multiplying by z a z b , the expression is 
= % {(z a Pba + ZbPaa)p 2 ßy8 + 2z a zj,la Ap 2 ßy8), 
(Surface dbcdaß.}