340 Mr. BappacE on the Infiuence of Signs 
second stake by the result of the first, and that he makes it wu-7~ 
if he had won the first, but w+v if he had lost it; on this second 
bet he will either win or lose 
u+v, or U—v, : 
Supposing him to determine his third stake from his second, in 
the same manner as he fixed his second from his first, it is clear 
that according to the determination of his previous bets, he may 
stake on the third event either of the three sums 
U+2v, U, U— 2. 
And generally on the nth event, if he proceed according to this law, 
there are different bets which he may make according to the 
order in which the previous ones were decided. Now in any 
question in which such a mode of play entered, it would be ex- 
ceedingly tedious to consider separately all these cases, and to 
repeat the same or nearly the same reasoning for each individual 
case. This may be avoided by rendering the events indeterminate, 
for we then find his first profit may be denoted by 
u(—1)°, 
in which the letter @ represents any whole number whatever ; 
if it is an even number he wins, and if an odd one he loses; 
the same artifice applied to his second stake gives for it 
u—v(-1)% 
as a is still undetermined, this will represent that stake truly, 
whichever event has happened on the first. 
The result of this second stake may be represented by 
{2 SAGES) | 1a 
whether it is lost or gained, and this is still kept undecided by 
means of the letter D. 
The third stake will be 
u—v(—1)*- 2 - 1), 
