126 THE QUANTITATIVE METHOD IN BIOLOGY 



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

 (face) is |. 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 | 

 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+f)^. (See 

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

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



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

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



{a + b + c + d + e+f)^ = 



Table a 



a^ + 2ab + 2ac + 2ad + 2ae + zaf + 2bf + 2cf + 2df + 2ef +P 

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

 + c^ + 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 = 1, b = 2, c =2, d = ^, e = ^, f = 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^ —a + a = 'L + x = 2 



fl6=i+2=3 



flc = I + 3 = 4 



h^ =b + b =2 + 2=4 



/2=/+/ =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. 5. 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.) 



' 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.) 



