TRANSACTIONS OF SECTION D. 



891 



Then, again, suppose we are given a number of results, and are not told how many 

 dice were used, how are we to find out the power to which we must raise (^ + 1), 

 since this depends on the number of dice P 



Before applying the law of chance to variations in which we cannot directly 

 measure the number of contributory causes (the analogue of the number of dice), 

 we must find some way out of these difficulties. 



The way is shown by the diagram (fig. 1). 



The rectangles in this diagram are proportional to the various terms of 

 (i + iy^ ) ^^^ ^^^J represent the most probable result of counting the number of 



Fig. 1. 



Diagram showing results of experiments with dice. 



I 



dice with more than three points in a series of trials with twelve dice. The heights 

 of these rectangles were determined by expanding (i + i)^^ ; but you notice the 

 dotted curve which is drawn through the tops of them. The general slope of this 

 curve is, yoa see, the same as the general slope of the series of rectangles ; and the 

 area of any strip of the curve which is bounded by the sides of a rectangle is very 

 nearly indeed tlie same as that of the rectangle itself. 



The constants upon which the shape of this curve depends are easily and 

 quickly obtained from any series of observations ; so that you can easily and 

 quickly see whether a set of observed phenomena obeys the symmetrical law of 

 chance or not. 



