344 Mr. Baspace on the Influence of Signs 
which the choice is to be made, is always finite, and the second 
player can only make his selection out of that number di- 
minished by unity. The first player, at his second stroke; 
can only choose out of the same number diminished by two, 
and so on. Now it is evident, that if the individual actually 
chosen at each step, were fixed permanently, the reasoning must 
diverge into an immense multitude of cases; whereas, if any one 
indifferently is chosen by the first, and again, any one indiffer- 
ently out of the k-1 remaining ones by the second, and so on, 
the things chosen by the two parties would all be comprehended 
in two expressions. 
It is this power of representing any one of k—p things, where 
p have already been taken out of k, (subject only to the condition 
that the expression for it can never represent any one of those 
already chosen,) which enables us to delay the decision of the 
individuals actually selected until the conclusion; and thus by 
their means, to satisfy the other conditions of the problem. Seve- 
ral instances of such questions, will be noticed in a future paper: 
The only means that have hitherto been employed for this purpose, 
are the roots of unity, and the sines, and other similar functions 
of submultiples of 7, and from the great length of the formule 
in which they occur, I am by no means sanguine in my ex- 
pectation of much success in those enquiries, until some more 
condensed method of indicating and to a certain extent also of 
executing such operations, shall have been contrived. The 
principle in discussion, appears to me to be the only one, by which 
any general and complete solutions can be arrived at: more partial 
views may doubtless be taken, more adapted to the present state 
of symbolic language, and even these become extremely valuable, 
in such difficult enquiries, not merely from the advances they 
themselves introduce, but as the ground-work of generalization 
to more perfect methods. 
