OF THE LAWS OP THOUGHT, 172' 



standard form to wbich functions of every kind may be reduced^ 

 This form is not an arbitrary one, but ia determined by the circum- 

 stance that every conceivable object must rank under one or other 

 of the two contradictory classes x and 1 — x. Hence every con" 

 ceivable object is included in the expression, 



ux + V (l — as)', (8) 



proper values being given to u and v. For, if a given concept belong 

 to the class x, then, by making v = 0, the expression (8) becomes ux^ 

 which, by (1), means some x ; and if the given concept belong to the 

 class 1 — X, then, by making u = 0, the expression (8) becomes 

 V {1 — x), which, by (1) and (6), means some not x. Therefore^ 

 y (x) being any concept depending on x, we may put 



f{x) ~ ux + V (l — x) (9) 



It has been shown that one of the coefficients, u, v, must al- 

 ways be zero ; but the forms of these coefficients may be determined 

 more definitely. Eor, by making a? = in (9), the result i&v =/ (0) ^ 

 and by making a? = 1, there results % =/(l) ; by substituting which 

 values of u and v in (9), we get 



/(^) =/(l) ^ + /(O) (1 - cc) (10) 



This is the expansion or development of the function x. The ex- 

 pressions a;, 1 — X, are called the constituents of the expansion? 

 and /(I) and/(0) are termed the coefficients. The same phrase- 

 ology is employed when a function of two or more symbols is de- 

 veloped. 



Any one in the least degree acquainted with mathematical processes 

 will understand how the development of functions of two or more 

 symbols can be derived from equation (10). In fact, by (10), we 

 have 



/(^. y) =/(i. y) * +/(0, y) (1 - x). 



But again, by (10), 



/(i,y)=/(i. i)y + /(1, 0)(l --y), 

 and 



/(0,y)=/(0, l)y +/(0,0)(i -y). 

 :.f{x,y) =/(l, \)xy +/(1, 0)ar(l - y) 



+/(0,l)^(l-^)+/(0,0)(l-a;)(l~^) (11) 



The development of a function of three symbols may be written 

 down, as we ahall have occasion in the sequel to refer to it ; 



