13] 
ON THE THEORY OF LINEAR TRANSFORMATIONS. 
87 
In a following number of the Journal I shall prove, and apply to the theories of 
Maxima and Minima and of Spherical Coordinates, (I may just mention having obtained, 
in an elegant form, the formulas for transforming from one oblique set of coordinates 
to another oblique one) the more general theorem, 
“ If k be the order of the determinant formed as above, the determinant itself 
is a quadratic function, its coefficients being determinants formed with the coefficients 
rs, its variables being determinants formed respectively with the variables x and the 
variables y; and the number of variables in each set being the number of combi 
nations of k things out of m, ( = 1 if k — m\ if k>m the determinant vanishes).” 
I shall give in the same paper the demonstration of a very beautiful theorem, 
rather relating, however, to determinants than to quadratic functions, proved by Hesse 
in a Memoir in Crelles Journal, vol. xx., “ De curvis et superficiebus secundi ordinis;” 
and from which he has deduced the most interesting geometrical results. Another 
Memoir, by the same author, Crelle, vol. xxviii., “ Ueber die Elimination der Variabeln 
aus drei algebraischen Gleichungen vom zweiter Grade mit zwei Variabeln,” though 
relating in point of fact rather to functions of the third order, contains some most 
important results. A few theorems on quadratic functions, belonging, however, to a 
different part of the subject, will be found in my paper already quoted in the Cambridge 
Philosophical Transactions [12]; and likewise in a paper in the Journal, Chapters in 
the Algebraical Geometry of n dimensions [11]. 
I shall, just before concluding this case, write down the particular formula corre 
sponding to three variables, and for the symmetrical case. It is, as is well known, the 
theorem, 
“If 17= Axr + By 1 + Cz- + 2 Fyz + 2 Gxz + 2 Hxy 
be transformed into 
+ Wv 2 + + 2 §yd + 2©£d + 
by means of x = a i; + /3 g + y 6, 
y=a'Z + /3'ri + ry' 0, 
z = a"i; + j3"g + 
then Gaft® 23<£ 2 - <&W + 2 $&№) = 
(a/3'y" - a/3'V + a'£"Y - a'/fy" + a 'W “ «"£'V) 2 ( ABC - AF 2 - BG 2 - CIP + 2FGH)." 
(B) Let n = 3, and for greater simplicity m = 2; write 
a— 111, e = 112, 
b = 211, /=212, 
c = 121, # = 122, 
d — 221, /¿ = 222, 
U = a x^y^i + b x.,y l z 1 + c xgjzZx + d x. 1 y. 1 z l + e x-gj^Zo, +fx. 2 y l z 2 + g x{y 2 z 2 + h x. 2 y. 2 z. 2 . 
so that