Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 7)

216 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
[447 
where (P', Q', R', P/, Q x , P/) are linear functions of (x, y, z'), we have 
that is to X', Y', Z', where X' = 0, Y' = 0, Z' = 0 are conics each passing through the 
same three points in the second plane. 
53. The lineo-linear transformation is thus the same thing as the quadric trans 
formation. It is, moreover, clear that the equations must, by linear transformations on 
the two sets of variables respectively, and by linear combination of the two equations, 
be reducible into forms giving the before-mentioned values of x : y : z and x' : y' : z' 
respectively. Thus, in the general case, where in each plane the three points are 
distinct points, the lineo-linear equations will be reducible to 
xx' — yy' = 0, XX — zz' — 0 ; 
in the case where B, G in the first plane, and B', C' in the second plane respectively 
coincide, the forms will be 
XX - yy' = 0, yz -y'z = <0\ 
and in the case where A, B, C in the first plane, and A', B r , C' in the second plane 
respectively coincide, the forms will be 
xy — yx = 0, xz — yy + zx = 0. 
The determination of the actual formulse for these reductions would, it is probable, 
give rise to investigations of considerable interest. 
The General Rational Transformation between Two Planes {resumed). 
54. Consider, as above, the first plane or figure with a principal system (a 1} a... a n _ x ), 
and the second plane or figure with a principal system (a/, a 2 "... aV-i). To a line in 
the second plane there corresponds in the first plane a curve of the order n passing 
1 time through each of the points a u 2 times through each of the points a 2 , 3 times 
through each of the points a 3 , &c.; or, as we may write this: 
First figure. 
Second figure. 
Points oíj a 2 a 3 ... a, 
Points a/ a/ a 3 '... a 
1 23 n-1 
123 n-1 A 
h curve order n 
1 : 3 n-1 
0 0 0 n — 1 \ 
000 n—1 ui 
L curve order 1 
0 
viz., the l’s denote the number of times which the curve of the order n passes through 
the several points vl x respectively; the 2’s the number of times which the curve passes 
through the several points a 2 respectively; and so on.
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.