Note on linear transformations of 𝑛-ics in 𝑚 variables
Alan Campbell · Bulletin of the American Mathematical Society · 1929
Let us consider the w-ic in m variables (1) F(xi,x2, • • , xm) = 0. If we subject (1) to the linear transformation pxi = auXi + #12^2' + aizXa + • • • + aimXm , (2) px2 = a2iXi + a22x2 + • • • + a2mxj , • • • , pXm = am\X + dm2X2 + dmZXi + * * • + dmmXm , we obtain F(anxi + ai2x2' + anxi + • • • + a\mXm ,a2XXi + a22x2 ( 3 ) + 023*3 + * ' ' + d2mXm , ' ' * , dm\x{ + am2X2 + amzxi + • • • + ammxj) = 0. Note that in the expansion of (3) the coefficient of the term in x!, 0* = 1, 2, 3, • • • , w), is F(aUl a2i} a3i, • • • , ami). A necessary and sufficient condition for this coefficient to vanish is that the point Pi(au, a2i, • • • , ami) shall lie on the geometric locus of (1). To obtain the coefficient of such a term as xl xJ~ in the expansion of (3) we can put