222 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. [447 
Special Cases—Reduction of the General Rational Transformation to a Series of Quadric 
Transformations. 
68. It was remarked by Mr Clifford that any Cremona-transformation whatever 
may be obtained by this method of repeated quadric transformations, if only the 
principal systems, instead of being completely arbitrary, are properly related to each 
other. To take the simplest instance; suppose that we have 
First figure. Second figure. Third figure. 
A, B, C A', B', C' B", C" 
II 
E, F D', E', F' D", E", F" 
viz., in the transformation between the first and second figures, we have the principal 
systems ABC and A'B'C' (arbitrary as before); but in the transformation between the 
second and third figures, the principal systems are D'E'F' and D"E"F", where I)\ 
instead of being arbitrary, coincides with A'. And we then have B", C" in the third 
figure corresponding to B', C’ in the second figure, and E, F in the first figure corre 
sponding to E', F 1 in the second figure. This being so, to a line in the first figure 
corresponds in the second figure a conic through A', B', C'. But A' — D'; viz., this 
conic passes through a point D' of the principal system of the second figure, in regard 
to the transformation between the second and third figures. That is, (k, a, b, c referring 
to the second figure, and k\ a', b', c' to the third figure, k — 2, a, = 1, b = 0, c = 0, and 
therefore k' = 3, a' = 2, b' = 1, d = 1,) corresponding to the conic we have in the third 
figure a curve, order 3 (cubic curve), passing twice through D", but once through E" 
and F" respectively; this cubic curve passes also through the points B", C" which 
correspond to B', C' respectively; that is, 
cubic passes through E", F", B", C" each 1 time 
„ „ D" 2 times; 
or, corresponding to a line in the first figure, we have in the third figure a curve, 
order 3, passing through four fixed points each 1 time, and through one fixed point 
2 times. That is, we have n = 3, a/ = 4, a 2 '=l. And in the same manner, to a line 
in the third figure there corresponds in the first figure a cubic through four fixed 
points (viz., B, C, E, F) each 1 time, and through one fixed point, A, 2 times; so 
that also a x — 4, a 2 = 1. The transformation is thus of the order 3, and the form 
4j 1 2 and 4j 1 2 (this is in fact the only cubic transformation; see the Tables, ante, No. 41). 
69. Mr Clifford has also devised a very convenient algorithm for this decom 
position of a transformation of any order into quadric transformations. The quadric 
transformation is denoted by [3], the cubic transformation by [41], the quartic trans 
formations by [601], [330], the quintic ones by [8001], [3310], [0600], and so on ; see 
the Tables just referred to. (This is substantially the same as a notation employed 
above, the zeros enabling the omission of the suffixes; viz., [8001] = 8 4 1 4 ; and so in other 
cases.)