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 2w; and 

 whatever be the course of succeeding events, 2u 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 u 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 ( — l) a , 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 1 , 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 u (2 -\ 2 %) > whatever be the form of a. 



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 u (2+ l ~~ { ~ lY 21 ) ( — 1)\ 



The third stake must comprehend the second power of 2, 

 and will be of the form 



u(2 + 2 f{a,b) + ^f x (a,b)y 



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

 that case, the new stake would be 2w; 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 



