318 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



Representing the first member of the equation by V, we have 



which, when X = 0, assumes the form - p - p 2 . . - p n + 1 ? and 

 is negative in value. 

 Again, we have 



consisting of a series of terms which, under the given restrictions 

 with reference to the value of X, are positive. 



Lastly, when X = , we have 





 Pi 



which is positive. 



From all this it appears, that if we construct a curve, the or- 

 dinates of which shall represent the value of V corresponding to 

 the abscissa X, that curve will pass through the origin, and will 

 for small values of X lie beneath the abscissa. Its convexity will, 



between the limits X = and X = be downwards, and at the 



Pi 



extreme limit the curve will be above the abscissa, its ordinate 



PI 



being positive. It follows from this description, that it will in- 

 tersect the abscissa once, and only once, within the limits speci- 



fied, viz., between the values X = 0, and X = . 



Pi 



The solution of the problem is, therefore, expressed by (11), 

 the value of X being that root of the equation (10), which lies 



within the limits and , , . . . 

 Pi p* Pn 



The constant c is obviously the probability, that if the events 

 #i 5 #2> #n> all happen, or all fail, they will all happen. 



This determination of the value of X suffices for all problems 

 in which the data are the same as in the one just considered. It 

 is, as from previous discussions we are prepared to expect, a de- 

 termination independent of the form of the function 0. 



