CHAP. V.] PRINCIPLES OF SYMBOLICAL REASONING. 75 



-z)+f(l 9 9 l)x(l-y)z 

 + /(!, 0, 0) (1 -y) (I-*) +/(0, 1, 1) (1 - tf)yz 

 + /(0, 1,0) (l-a)y (l-*)+/(0,0,l) (l-*)(l-y)z 



in which /(I, 1,1) represents what the function /*(#, ?/, 2:) be- 

 comes when we make therein x - 1, y = 1, 2=1, and so on for 

 the rest. 



11. It is now easy to see the general law which determines 

 the expansion of any proposed function, and to reduce the me- 

 thod of effecting the expansion to a rule. But before proceeding 

 to the expression of such a rule, it will be convenient to premise 

 the following observations : 



Each form of expansion that we have obtained consists of cer- 

 tain terms, into which the symbols x, y, &c. enter, multiplied by 

 coefficients, into which those symbols do not enter. Thus the 

 expansion of f(x) consists of two terms, as and 1 - x, multiplied 

 by the coefficients f(l) and/(0) respectively. And the expan- 

 sion of /(#, y) consists of the four terms xy, x (1 - y), (1 - x) y, 

 and (1 - #), (1 - y), multiplied by the coefficients /(I, !),/(!, 0), 

 /(O, 1),/(0, 0), respectively. The terms #, 1 -#, in the former 

 case, and the terms xy^ x(\ -y), &c., in the latter, we shall call 

 the constituents of the expansion. It is evident that they are in 

 form independent of the form of the function to be expanded. 

 Of the constituent xy, x and y are termed the factors. 



The general rule of development will therefore consist of two 

 parts, the first of which will relate to the formation of the consti- 

 tuents of the expansion, the second to the determination of their 

 respective coefficients. It is as follows : 



1st. To expand any function of the symbols x, y, z. Form a 

 series of constituents in the following manner : Let the first con- 

 stituent be the product of the symbols ; change in this product 

 any symbol z into I z, for the second constituent. Then in 

 both these change any other symbol y into 1 - y, for two more 

 constituents. Then in the four constituents thus obtained change 

 any other symbol X into 1 - x, for four new constituents, and so 

 on until the number of possible changes is exhausted. 



2ndly . To find the coefficient of any constituent. If that con- 



