642 
ME. A. CATLET ON THE ANALYTICAL THEOEY OP THE CONIC. 
8. Suppose now that 
(a, I, c,/, g, hjx, y, zf 
breaks up into factors, or say that Ave have 
{a, h, c,f, g, hXx, y, zf=2{ax-\-^y-\-yz){a!x-\-^'y-\-yz), 
the values of the coefficients (a, ..) then are 
(a, h, C, f, g, /i)=(2aa', 2/3/3', 2yy', 'ya'-{-'y'ci, a(B' + u'j3), 
and forming from these the inverse coefficients (A, . .) and the discriminant K, we find 
(A, B, C, F, G, H) = -(i37'-/3'y, a(3'-a'(5f. 
K=0. 
9. The last-mentioned equation, K=0, is the condition in order that {a, ..Jx.y^zf 
may break up into factors ; and when it does so, we have 
(A, ..XI, n, — [(/3y'— ya'— y'a, ocfB'—a'PX^^ 
that is, (a, ..Xx,y, zf breaking up into factors, (A, ..XI, ??, is a perfect square; and 
equating it to zero, we have 
[(/3y' — /3'y, ya' — y'a , a/3' — a'/3XI, ??, QJ = 0 ; 
AA'hich, (I, <X) being line-coordinates, gives (as a twofold point) the point of intersection 
of the lines (a, /3, y), (a', /3', y'), that is, the lines a^+^3^+y2=0, a'iT-f /3'3/-f y'z=0. 
10. If («, . .X^’, y, zf is a perfect square, then a' : ,8' : y'=a : /3 : y ; whence not only, as 
before, K = 0, but the coefficients (A, B, C, F, G, H) all vanish (this implies the first- 
mentioned condition, K=:0); and the line-equation (A, . .XI, becomes the mere 
identity 0 = 0. 
11. Conversely if K = 0, then (a,. fXx.y, 2 )"breaks up into factors; andif (A,B,C,F,G,H) 
all vanish, then (a, . .X^, ^f is a perfect square. The conclusions stated ante. No. o, 
are thus sustained. 
12. I assume, first, that [a, . .X^’, y-> ^f is a perfect square (No. 13) ; and secondly, that 
it breaks up into factors (No. 14) ; and I proceed to inquire how in the one case the 
root, and in the other case the factors, can be determined in a symmetrical form. 
13. Considering the before-mentioned identical equation 
{a,..Xx, y, zf.(a, ..Jx', y', z'f-[_{a, ..X^, y, ^X^', y’, z')J={A, . .Jt/z' -y'z, zx'-z'x, xy'- 
if («, ..Xx,y,zf is a perfect square, then by what precedes, the right-hand side ol the 
equation vanishes, and we have 
r/fl, ..Ya?, y, zTsc', y', z')f 
and the root of {a,.^x,y, zf is thus seen to be 
, { a, ■ .X^> ^X^'> y'> ^') 
“ ^{a, ..X^,y',z’f 
an expression which invoh'es the quantities {x', y', z'), the values whereof may be assumed 
