156 AN EXAMINATION OF SOME QUESTIONS 



successive powers of 2. In order to determine these, let us sup- 

 pose that at some point of the game, the last event has proved 

 favourable ; then, by the conditions, the next stake is 2 m ; and 

 whatever be the course of succeeding events, 2m will always form 

 part of the stake ; therefore it need not be multiplied by any 

 function of a, b, c, &c. the letters which determine the winning 

 or losing of the subsequent events. We may therefore assume 

 2 M as the constant part of every stake, without reference to 

 any particular order in their occurrence. If the failure or hap- 

 pening of this event is represented by ( — 1)", a being an odd 

 number in the first, and an even number in the second case, 

 we must multiply the next power of 2, or 2', by some function 

 of a which shall vanish when a is an even number, and be- 

 come unity when it is an odd one. Such a function is easily 



found, and one of the simplest is ^ — -^ • The next stake 



is therefore m (2+ ^'^f^" 2) , whatever be the form of «. 



The failing or happening of this event may be represented 

 by (_ 1)\ and the profit of the player by this event is then 



represented by m (2 + a / ^ ' ' 



The third stake must comprehend the second power of 2, 

 and will be of the form 



If b is an even number, or the second event is favourable, in 

 that case, the new stake would be 2 « ; and therefore both 

 f(a, b) and/, (a, b) must vanish. This will take place if each 



has a factor of the form ^^^^^=^, and the new stake in con- 



sequence 



