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)

[447 
447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
203 
26 
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point constituted by an intersection of the curves r, n — r. {Observe that the two curves 
have only this single intersection; viz., the remaining r (n — r) — 1 intersections are 
at points a/ + a 3 '... + a'of the principal system of the second plane.} We have thus, 
in the second plane, a series of curves, each of them having a new double point; 
viz., these are the several curves which correspond to the lines through a r in the first 
figure. Each of the curves is a fixed curve r together with a variable curve n — r. 
The new double point is an intersection of the two curves; that is, it is a variable 
point on the curve r. The locus of the new double point is thus the curve r; therefore 
the curve r is part of the Jacobian of the reseau of the second plane. Since each 
point a r gives a curve r, the curves in question form an aggregate curve of the order 
ofj + 2a 2 ...+ (n — 1) a M _ 1} = 3n — 3 ; viz., this is the order of the Jacobian; or, as stated, 
the curves r (that is, the principal counter-system of the second plane) constitute the 
Jacobian of the reseau of this plane. 
41. The numerical systems (a 1} c 2 ...a n ^ 1 ) and (a/, a/... are each of them a 
solution of the same two indeterminate equations 
Xr 2 a r — n 2 — 1, Xra r = 3 n — 3, 
but not every solution .of these equations is admissible; for instance, if r > \n, then 
a. r is = 0 or 1, for a r =2 would imply a curve of the order n with two ?’-tuple points, 
and the line joining these would meet the curve in more than r points; similarly, 
r>\n, Or is =4 at most, for a r — o would imply a curve of the order n with five 
r-tuple points, and the conic through these would meet the curve in more than 2n 
points; and there are of course other like restrictions. The different admissible systems 
up to n = 10 are tabulated in Cremona’s Memoir; and he has also given systems 
belonging to certain specified forms of n : these results are as follows: 
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