447] 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
207 
42. The system (a 2 , a 2 ... a n _j) geometrically determines completely the system 
(aj, aj... a! n _j); it ought therefore to determine it arithmetically; that is, given the 
one series of numbers, we ought to be able to determine, or at least to select from 
the table, the other series of numbers. Cremona has shown that the two series 
consist of the same numbers in the same or a different order. By examination of the 
tables, it appears that there are certain columns which are single (that is, no other 
column contains in a different order the same numbers), others that occur in pairs, 
the two columns of a pair containing the same numbers in a different order. Where 
the column is single, it is clear that this must give as well the values of (aj, aj ... a' n _j) 
as of (<*!, a 2 ...a n _j). Where there is a pair of columns, as far as Cremona has 
examined, if the one column is taken to be (a 1 , a 2 ... a^j) the other column is 
(aj, aj ...a' n _j)\ it appears, however, not to be shown that this is universally the case;, 
viz., it is not shown but that the two columns, instead of being reckoned as a pair, 
might be reckoned as two separate columns, each by itself representing the values as 
well of (a 1} a 2 ...a n _j) as of (aj, aj... a' n _i); neither is it shown that there are not, 
in any case, more than two columns having the same numbers in different orders. 
It seems, however, natural to suppose that the law, as exhibited in the tables, holds 
good generally; viz., that the tables contain only single columns, each, giving the values 
as well of (oq, a 2 ...a n _j) as of (a/, aj...a! n _j)\ or else pairs of columns, one giving 
the values of (a 1} a 2 ...a Mr _ 1 ), and the other those of (a/, a/ ... a' n _j); or, say, that the 
partitions are either sibi-reciprocal, or else conjugate. 
43. Assuming that the two systems (a 1} a 2 ...a il _ 1 ) and (aj, a/... a' n _j) are each 
known, there is still a question of grouping to be settled; viz., the Jacobian of the 
first plane consists of aj lines, aj conics, ... a' n _ 1 unicursal (n — l)-thics; each line, each 
conic, &c., passes a certain number of times through certain of the points ai, a 2 ... a n _ 1 : 
but through which of them? For instance, each of the aj lines will pass through two 
of the points a 1} a 2 , ... a n _x: will these be points a 1} or points a 2 , &c., or a point a x 
and a point a 2 , &c. ? The mere symmetry of the different groups of points determines 
certain conditions of the solution ( J ); for instance, if any particular one of the aj lines 
passes through two points a r , then each of the aj lines must pass through two 
points a r ; and since the points a r are symmetrical, we must in this way use all the 
pairs of points a r ; that is, if aj = \ a r (a r + 1), but not otherwise, it may be that each 
of the aj lines passes through two of the points a r . In the case of an equality 
a r = a s we could not hereby decide whether the line passed through two points a r or 
through two points a s . So, again, if any one of the aj lines pass through a point a r 
and a point a s , then each of the aj lines must do so likewise, and we must hereby 
exhaust the combinations of a point a r with a point a s ; viz., the assumed relation 
can only hold good if aj = a r a s . Similarly, each of the aj conics will pass through 
five of the points a lf a 2 ...a n _ 1 ; each of the aj nodal cubics will pass twice through 
one (have a double point there) and through six others of the points a A , a 2 ...a n _ 1 t 
which are the points so passed through? I do not know how a general solution is 
to be obtained, but most of the cases within the limits of the foregoing table have 
1 It is by such considerations of symmetry that Cremona has demonstrated the before mentioned theorem 
of the identity of the numbers (c^, a 2 ... a n _j) and (aj, aj ...