But, ifx^ is not present in A, (so that an - 0), we may suppose a,,
different from zero* and then method (ii) of Art. 7 is to be applied
to ^. We obtain then

yi = (Oai^i + «a^2 + ««^8) - i ^ (^"•^a + ais^s),



where



1 X 1^1



♦ For, provided that x, is actually preBent in A, one or other of «„, a„ muat
not vanish ; and we suppose the suffixes chosen so that a,, is not zero. Of course
Tx, is not present in A. the problem depends only on the reduction of forms in
two variables x^, x,.
13, i 1]



REDUCTION: THREE VARIABLES



33



Substitute now x^^^y^- (ais/^M) ^s in {cA - S) and we obtain the
forms

A = 2yiy2-^ax^\ cA-B=fi{y2y x^) (3).
The geometrical meaning of this transformation is that P is now
on the conies A4 - J? = 0, and y^ =^ is the polar of P (that is to say,
the common tangent to the conies at P) ; while ^i = is the tangent at
the other intersection of the line Xs = with the conic ^ = 0.
We must next examine (3), to see i/x^ is present in (cA -B) ; if so,
we may apply method (iii) of Art. 7, and obtain*

CA-B = b^y^^ + ftg (^3 + %2)2 = ^23^2^ + 63^3^.
Substitute x^ -yz-hy^ in A and we get

A = 2yiy2 -^aiy^- ky^f = 2^,>2 + ays',
where y/ = ^i + 1 ali^y2 - aky^.
