689] BY A BEAL CORRESPONDENCE OF TWO PLANES. 321 
point (W'), and this may be distinguished accordingly as a counter-barrier 12; and 
in like manner the cross-point (TP) through which it passes will be called a cross- 
point (TTV); and the barrier corresponding hereto, and the branch-point V at which 
it terminates, will in like manner be called a barrier 12, and a branch-point V v ,. 
Each barrier and branch-point will thus have a pair of suffixes; and the corresponding 
counter-barrier and cross-point will have the same pair of suffixes. It is to be observed 
that two or more of these corresponding figures may very well have the same pair 
of suffixes; but that such corresponding figures must be distinguished from each 
other; thus, if there are two branch-points V 12 , these may be distinguished as the 
branch-points aF 12 and ¡3 V 12 , and the barriers, counter-barriers, and cross-points by 
means of these same letters a and /3, (or otherwise), as may be convenient. It would 
seem that not only the number of the points V must be even, but the number of 
each set of points V 12 must also be even (see post, No. 15). 
13. It is also to be noticed that the determination of the suffixes of the several 
points V, &c., depends first upon the arbitrary choice of the suffixes of the points 
Q', and next on the choice of the system of barriers; but that, these being assumed, 
the suffixes of the several points V, &c., are completely determinate. 
14. Taking now any point S whatever, and supposing that P moves from Q 
continuously to S by a path which does not meet a barrier, the points P' will move 
from the several points Q' to the several points S' by paths not meeting the counter 
barriers ; viz. to each point S' there will be a path from some point Q; and giving 
to such point S' the suffix of the point Q', the suffixes of the several points S' 
which correspond to any point whatever, S, will be completely determined. The 
determination depends of course on the assumptions referred to No. 13, but not in 
anywise on the position of the point S. 
It will be noticed that, as all the points V are connected together by the 
barriers, the only closed paths from a point to itself are paths not including any, 
or including all, of the points V; and that between such paths there is no real 
distinction. 
15. Consider a point P moving continuously in any manner. The several corre 
sponding points Pi, P 2 ,.., P n ' will each of them move continuously, but the suffixes 
interchange; viz. when P arrives at and then passes over a barrier aft, the corre 
sponding points P^ and Pfj' will each arrive at the corresponding counter-barrier a/3, 
and, on passing over this, P a ' will be changed into P/ and Pp into P a ', the other 
points P' remaining unchanged; and the like in other cases. This in fact includes 
the whole or the greater part of the foregoing theory. Thus, if P describe a closed 
curve not cutting any barrier, there will be no change of suffix; and when P returns 
to its original position each of the corresponding points Pi, Pi,.., P n ' will describe 
a closed curve, returning to its original position. But suppose that P describes a 
closed curve, cutting once only 7 a barrier 12; suppose that the path is from P to 
Q, and then crossing the barrier to P, and thence again to P; Pi passes to Qi, 
and then crossing the counter-barrier it passes from P 2 ' to Pi; while at the same 
time Pi passes to Qi, and then crossing the counter-barrier it passes from P/ to P/; 
C. X. 41