200 ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. [447 
30. We have thus a transformation in which the first set of equations is 
x : y' : / — X : Y : Z, where X — 0, Y = 0, Z = 0 are curves in the first plane, of the 
same order n, having in common a x , OL*...a n - x points which are simple points, double 
points,... (n — l)tuple points respectively on each of the curves; these numbers satisfy the 
conditions 
«I + 4a 2 + 9a 3 ... + (n— l) 2 a n _ 1 = n 2 - l, 
a 2 + 3a 2 + 6a 3 ... + \n{n — 1) a n _! = | (n 2 + 3n) — 2 ; 
conditions which give, as above, 
a 2 + 3a 3 ... + \ (ft - 1) (n - 2) a n _ x = £ (n -1) (n - 2), 
and also 
oq + 2a 2 + 3a 3 ... + (n — 1) a n _i = ~ 3 ; 
so that the relations between a x , a 2 ... a n _ x are given by any two of these four equations. 
31. The second set of equations then is x : y : z =X' : Y' : Z', where X' = 0, F^O, Z' = 0 
are curves in the second plane, of the same order n; and it is clear that these must 
be curves such as those in the first plane ; viz., they must have in common a/, a/, .. a' n _ x 
points, which are simple points, double points, ... (n — l)tuple points respectively on each 
of the curves, the relations between these numbers being expressed by any two of the 
four equations 
a/ + 4a/ + 9a/ ... 4- (n - l) 2 a' n _ x = n 2 -l, 
a/ + 3a/ + 6a/ ... + \n (n - 1) a! n _ x = \n (n + 3) — 2, 
a' 2 + 3a/ ... + \ (n - l)(n - 2) a' n _ a =^(n- 1) (n - 2), 
a/ + 2a 2 + 3a 3 ' ... + (n — 1) a n_ x = 3n — 3. 
32. To any line ax' + by' +cY = 0 in the second plane there corresponds in the 
first plane a curve aX + bY + cZ of the order n ; and to any line a'x + b'y + dz = 0 in 
the first plane there corresponds in the second plane a curve a'X'+ b'Y'+c'Z'= 0 of 
the same order n; the curves aX+bY + cZ — 0 in the first plane are, it is clear, a 
system, and the entire system, of curves each satisfying the conditions which have 
been stated in regard to the individual curves X = 0, F = 0, Z — 0, and being as 
already mentioned unicursal ; and similarly the curves a'X' + b'Y + c'Z' = 0 in the second 
plane are a system, and the entire system, of curves each satisfying the conditions 
which have been stated in regard to the individual curves X' = 0, Y = 0, Z = 0 ; and 
being also unicursal. We may say that to the lines of the second plane there 
corresponds in the first plane the réseau of curves aX + bY+cZ = 0; and to the lines 
of the first plane there corresponds in the second plane the réseau of curves 
a'X'+ b'Y'+ c'Z'= 0; these réseau being systems satisfying respectively the conditions 
just referred to. 
33. We have next to enquire what are the curves in the second plane which 
correspond to the a x + a 2 ... + a n _ x points of the first plane. I remark that the 
<x x + a. 2 ... + a n _ x points are termed by Cremona the principal points of the first plane, 
and the corresponding curves the principal curves of the second plane. But it will be