447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 223 
70. The foregoing result is represented thus [4, 1] = [3$0, 0, 1), which I proceed 
to explain. Consider in the first figure a line ; the symbol [3] denotes that in the 
second figure we have a conic with three points (a/). We are about to apply to this 
a quadric transformation ; (0, 0, 0) would denote that the three points of the principal 
system in the second figure were all of them arbitrary ; (0, 0, 1 ) that one of these 
points was a point a/ ; (0, 1, 1) that two of them were points a/ ; (1, 1, 1) that all 
three of them were points cq ; (0, 0, 2) would denote that one of the points was a 
point a a '; only in the present case we can have no such symbol, by reason that there 
are no points a/. Hence [3$001) denotes that the conic has applied to it a quadric 
transformation such that, in the transformation thereof, one point of the principal system 
coincides with one of the points (a/) on the conic. To [3], qua quadric transformation, 
belongs the number 2 ; and from 2, (001) we derive 3, (112), [in general k, (a, b, c) 
gives k', (a', b', c'), where k! = 2k — a — b — c, a' — k — b — c, b' = k—c — a, c' = k —a—b}. 
k — 2 corresponds to a symbol [3] of one number, k' = 3 to a symbol of two numbers ; 
viz., we change [3] into [30] ; we then, in the symbols (112) and (001), consider the 
frequencies of the several numbers 1, 2, ... taking those in the first symbol as positive, 
and those in the second symbol as negative ; or, what is the same thing, representing 
the frequency as an index, we have l 2 2 1 , l -1 ; or, combining, l 2-1 2 1 ; these indices 
are then added on to the numbers of [30] ; viz., the index of 1 to the first number, 
the index of 2 to the second number (and, in the case of more numbers, so on) : 
[30] is thus converted into [41], and we have the required equation 
[41] = [31001), 
where the rationale of this algorithmic process appears by the explanation, ante, No. 68. 
71. As another example take 
[8001] = [601 ([003). 
To [601], qua quartic transformation, belongs the number 4; and from 4, (003) we form 
5, (114); where the 5 indicates that [601] is to be changed into [6010]; then (114), 
(003), writing them in the form l 2 2°3 -1 4 a , show that to the numbers of [6010] we are 
to add 2, 0, — 1, 1 ; thus changing the symbol into [8001], so that we have the required 
relation. 
72. Mr Clifford calculated in this way the following table, showing how any trans 
formation of an order not exceeding 8 can be expressed by means of a series of quadric 
transformations ; the symbols Cr. 3, Cr. 4.1; 4.2, &c., refer to the order and number 
of Cremona’s tables, ante, No. 41. 
Cr. 3 . = [ 41] = [3$001), 
Cr. 4 . 1 = [601] = [41 ([002) = [3^001 $002), 
4.2 = [330] = [ 3$000) = [41$011) = [3$001$011),