mm 
ON THE HOMOGRAPHIC TRANSFORMATION OF 
The entire theory depends upon what may be termed the transformation of a 
surface of the second order into itself, or analytically, upon the transformation of a 
quadratic form of four indeterminates into itself. I use for shortness the term trans 
formation simply ; but this is to be understood as meaning a homographie transformation, 
or in analytic language, a transformation by means of linear substitutions. It will 
be convenient to remark at the outset, that if two points of a surface of the second 
order have the relation contemplated in the data of Sir W. Hamilton’s theorem (viz. 
if the line joining the two points pass through a fixed point), the transformation is, 
using the language of the Recherches Arithmétiques, an improper one, but that the 
relation contemplated in the conclusion of the theorem (viz. that of two points of a 
surface of the second order, connected by a line touching two surfaces of the second 
order each of them intersecting the given surface of the second order in the same 
four lines) depends upon a proper transformation; and that the circumstance that an 
even number of improper transformations is required in order to make a proper trans 
formation (that this circumstance, I say), is the reason why the theorem applies to 
polygons in which an even number of sides pass through fixed points, that is, to 
polygons of an odd number of sides. 
Consider, in the first place, two points of a surface of the second order such that 
the line joining them passes through a given point. Let x, y, z, w be current 
coordinates 1 , and let the equation of the surface be 
(a,...)(x, y, z, wf = 0, 
and take for the coordinates of the two points on the surface x 1} y 1} z lt w 1 and 
x 2 , Vi, w 2 , and for the coordinates of the fixed point a, /3, 7, 8. Write for shortness 
(a, ...) (a, /3, 7, S) 2 =p, 
(a,...) (a, /3, 7, 8) (x u y lf z 1} w 1 ) = q 1 , 
then the coordinates x 2 , y 2 , z 2 , w 2 are determined by the very simple formulae 
2a 
x 2 = x l — q u 
p 
V* 
2/3 
27 
w 2 =w 1 - 
28 
P 
<P- 
1 Strictly speaking, it is the ratios of these quantities, e.g. x : w, y : w, z : w, which are the coordinates, and 
consequently, even when the point is given, the values x, y, z, w are essentially indeterminate to a factor pres. 
So that in assuming that a point is given, we should write x : y : z : w = a : ¡3 : 7: S; and that when a point is 
obtained as the result of an analytical process, the conclusion is necessarily of the form just mentioned: but 
when this is once understood, the language of the text may be properly employed. It may be proper to explain 
here a notation made use of in the text: taking for greater simplicity the case of forms of two variables, 
(l, m) (x, y) means lx+my, (a, b, c) (x, y) 2 means ax 2 + 2bxy + cy^; (a,b,c) (£, y) (x, y) means a%x + b (£?/ + yx) +cyy . 
The system of coefficients may frequently be indicated by a single coefficient only: thus in the text (a, ...) (x, y, z, w) 2 
stands for the most general quadratic function of four variables.