[447 
447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 217 
55. We may, in the second figure, in the place of a line consider a curve of 
the order k'. If the equation hereof is (*$V, y', z') k ' =0, then the corresponding curve 
in the first figure is 7, Z) k ' = 0 ; viz., this is a curve of the order k = nk'. If, 
however, the curve in the second figure passes once or more times through all or any 
of the points a/, a/, ... u'n-i, then there will be a depression in the order of the 
corresponding curve in the first figure; and, moreover, this curve will pass a certain 
number of times through all or some of the points a 1} a 2 , a 3 ,The diagram of 
the correspondence will be : 
trans- 
ons on 
ations, 
y' • z> 
Second figure. 
> 0l 3 , ... CC n—i 
First figure. 
ts are 
CLi Clo (l 3 Qj n —i 
\ ^ h 
^ ^ n-i [ curve order k' 
curve order k 
ctively 
where a 1} b 1} c x ... denote the number of times that the curve of the order k passes 
through the several points respectively, (viz., the number of the letters a 1 , b 1} Cx... 
is =a 2 , any or all of them being zeros,) a 2 , b 2 , c 2 ... the number of times that the 
curve passes through the several points a 2 respectively, (viz., the number of the letters 
a 2 , b 2 , C-2... is = a 2 , any or all of them being zeros,) and so on; and the like for the 
curve in the second figure. 
)bable, 
56. By what precedes, it is easy to see that, if the curve k' passes through a 
point a/, then the curve k throws off a line, and the depression of order is = 1; so, 
if the curve passes 2 times, 3 times, ... or a/ times through the point in question, 
then the curve throws off the line repeated 2 times, 3 times,... a/ times, or the 
depression of order is =2, 3,... or a/; and the like for each of the points a/; so 
that, writing for shortness a/ + W + c/ + ... = 2a/, the depression of order on account 
of the passages through the several points a x is = 2a/. Similarly, for each time of 
passage through a point a/, there is thrown off a conic; or if a/ + b 2 +... = 2a/, then 
the depression of order is = 22a/, and so on; and the like for the figure in the 
other plane; and we thus arrive at the equations 
k = k'n — 2 (a/ + 2a 2 + 3a 3 ... + n 1 a i) 
k' = kn — 2 (cL\ + 2a 2 + 3a 3 ... n 1 cin—i )■ 
57. The simplest case is when the curve k' does not pass through any of the 
points a/, a/, ...a' n _i. We have then 
a/ = 6/ = c/ ... — 0, a 2 — b-2 ... — 0, a n —y — b ... — 0 , 
irough 
passes 
consequently k = k'n. And, moreover, it is easy to see that 
ai = ¿i... — k, a 2 = b. 2 ... = 2k, 0L n —i = b n ^i ... = (n 1) k ; 
C. VII. 
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