126 THE QUANTITATIVE METHOD IN BIOLOGY 



faces of the die becomes visible. The frequency of each event 

 (face) is J. If we represent the faces by six letters, a,b,c,d,e,f, 

 and ascribe to each letter the value of its frequency, which is J 

 for each, the possibilities are expressed by the polynomial 



a + b + c + d + e+f = ^ = i (certitude)^ 



Two dice being cast successively, the simple events are com- 

 bined two by two, and thirty-six compound events are equally 

 possible, according to the expression {a + b + c + d + e +/)^. (See 

 § 100.) These thirty-six events are represented by aa, ab . . . 

 ba, bb . . . etc. The frequency is J^- ^^^ each. 



If the order of succession is neglected, the thirty-six events 

 are reduced to twenty-one (see § lOo) — viz. 



[a-\-b-vc + d + e-^fY== 



Table « 



a^ + 2ab + 2ac + 2ad + 2ae + 2af + 2bf + 2cf + 2df + 2ef +p 

 + b^ + 2bc + 2bd + 2be + 2ce + 2ed + e^ 

 4- c2 + 2cd + d^ 



It is possible to go further. Since each letter represents one 

 of the faces of a die, we may ascribe to each letter two distinct 

 values : (i) the value of the frequency of the corresponding 

 simple event, which is J for each letter ; (2) the value of the 

 corresponding face of the die (a = i, 6 = 2, c = 3, ^ = 4, ^ = 5, / = 6) . 

 This facial value is quite independent of the frequency. 



In Table a each monomial is the expression of a compound 

 event. The arithmetical sum of the letters of each monomial is 

 the facial value of the corresponding compound event ; for 

 instance : 



a& = I + 2 = 3 



^c = 1 + 3 = 4 



52 =^4.^=2 + 2=4 



>=/+/ =6 + 6 = 12 



When only the facial values are taken into account, the twenty- 

 one events in Table a are reduced to eleven, characterized by 

 the values 



2, 3. 4» S> 6, 7, 8, 9, 10, II, 12 



(In Table a the monomials are arranged in such a way that 

 they have the same facial value in each vertical column.) 



1 This polynomial is also the expression of the six possible events when one 

 ball is taken from an urn containing six sorts of balls in equal number, (See 

 § 100.) 



