210 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
[447 
45. It is to be remarked upon the tables—first, as regards the lines: if we add 
the numbers in each line, reckoning m p as mp, (that is, each multiple point, according 
to "the number of branches through it,) the sums for the successive lines are 
2, 5, 8, 11, 14, &c.; that is, each line passes through 2 points, each conic through 
5 points, each cubic through 8 points, each quartic through 11 points, &c. But if we 
add the numbers reckoning mP as m.^p(p + 1), (that is, each multiple point according 
to its effect in the determination of the curve,) then the sums are 2, 5, 9, 14, 20, &c., 
that is, all the curves are completely determined, viz., the line by 2 conditions, the 
conic by 5 conditions, the cubic by 9 conditions, &c. Secondly, as regards the columns, 
if for any column, reckoning m p as mp, we multiply each number by the corresponding 
outside left-hand number, add, and divide the sum by the outside number at the head 
of the column, the successive results are 2, 5, 8, 11, 14, &c.; this merely expresses the 
known circumstance that the Jacobian passes 3r — 1 times through each point c^. 
46. The analogous tables showing the passage of the Jacobian through the 
principal system, in the solutions belonging to certain special forms of n, are 
Table n=p. 
“i 
a p-1 
II 
II 
2p-2 
1 
af = 2p — 2 
1 
1 
a p-1— 1 
2p — 2 
p-2 
Tables n = 2p. 
«1 
II 
3 
«2 
II 
2p-2 
1 
II 
0 
a P 
II 
0 
a 2p- 
II 
1 
2 
«1 
II 
2p — 2 
a 2 
II 
0 
a p-l 
II 
1 
a p 
II 
3 
a 2p-2 
II 
0 
af = 2p - 2 
1 
1 
af = 3 
2 
a.; = 0 
a/ =.2p-2 
1 
1 
3 
a 'p-1 = 1 
2p-2 
\P~ 2 
O-'p-1 = 0 
< 
II 
05 
2 
2p — 2 
Ji)- 1 
a p = 0 
a 2p~2 — 9 
a 2p~2 = 1 
2p — 2 
p-2 
3^-1