MEASUREMENT OF VARIABLE PROPERTIES 159 



has been observed ; see § 98) ^ all the possible compound events 

 and their frequencies may be calculated a priori by expanding 

 (a + bY\ in which a = b = \%% = \. 



The prisms being extracted, for instance, in series of 10, and 

 this operation being repeated, for instance, 102,400 times,^ all 

 the events (combinations of 10 prisms) are represented by — 



(rt + 6)1" = fli" + lOfl'ft + 45a8j2 + i2oa'63 + zioa^^ + 252a^b^ + 

 2ioa^b^ + I20fl36' + 45a^b^ + loab^ + 6" = i 



The denominator of all the terms is 2i° = i024. The fre- 

 quency of each term is Yihr multiplied by its coefficient. 



Example : The frequency of the combination a^b^ (2 prisms 

 a + 8 prisms b) is ^wr ; since we have extracted 1024 x 100 

 series, a^b^ occurs approximately 45 x 100 times. 



I ascribe to the letters a and b two values : 



(i) The value |, which is the frequency of the corresponding 

 simple events. 



(2) A second value (facial value), which is quite independent 

 of the frequency and represents the measure of the correspond- 

 ing simple events (§ loi, p. 126) — i.e. a = 2 cm. and 5 = 3 cm. 

 In this way the sum of the letters of each term is the total 

 height of the 10 prisms of the corresponding combination 

 (resultant of the simple causes). 



Example : The term xoab^ represents i prism « + 9 prisms b, 

 the total height of which is (2 x i) + (3 x 9) = 29 cm. 



By a very simple calculation one finds that the total heights 

 increase regularly from the first to the last term, the values 

 being : 



20, 21 . . . 29, 30 cm. 



We construct now a large number of new prisms of eleven 

 sorts, the respective heights of which are given by the above 

 values (20 to 30 cm.), the number of each sort being propor- 

 tional to the frequency (numerical value) of the corresponding 

 term. 



(The base of all the prisms is i cm^) 



Examples: The term 0'-° = ^^^ represents the frequency 

 of the combination the total height of which is 20 cm. 

 We construct, for instance, 1000 prisms of 20 cm. The term 

 ioa'&=i-^fx represents the frequency of the prisms of 21 cm. 

 According to this, we construct 10,000 prisms of this sort, etc. 

 The total number of prisms is 1,024,000. 



All the prisms (each within a case) are mingled and put 



^ Each prism is supposed to be enclosed within a case, in such a. way that 

 it is impossible to distmguish an a from a 6 without opening the case. 



^ 102,400 = 2'° X 100. I take this figure in order to simplify the calculations 

 below. 



