388 PROBABILITY OF JUDGMENTS. [CHAP. XXI. 



included in the expression 



X m ti . t m -\ 



Now X. m may be resolved into two portions, viz., XX m and 

 (1 - X) X m , the former being the sum of those constituents of 

 X m which are found in X, the latter of those which are not found 

 in X. It is evident that in the developed expression of t, which 

 is equivalent to X, the coefficients of the constituents in the 

 former portion XX m will be 1, while those of the latter portion 

 (1 - X) X m will be 0. Hence the elements we have now con- 

 sidered will contribute to the development of t the terms 



XX m J, . . T m ., + (1 - X) X m -i, . . 7 m _! . 



Again, since Xi = t ly while X z t = t^t l = 0, &c., it is evident 

 that the only constituents involving t\ T 2 . . J m .\ as a factor which 



have coefficients of the form 1, 0, or - , will be included in the ex- 

 pression 



X 1 t\ t z . . t m _ i ; 



and reasoning as before, we see that this will contribute to the de- 

 velopment of t the terms 



XX, t^ . .J m -i + (Kl-X)X,* l ? 2 .. ?_!. 



Proceeding thus with the remaining terms of (6), we deduce 

 for the final expression of , 



t = XX m ti . . t m _i + XXrfi t z . t m _i . . + XX m - l t l . . OT _ 2 wt -i 

 + (1 - X) Xfi . . ?_! + (1 - X) X^Ta ..~f m ., + &c. (8) 



+ terms whose coefficients are . 



In this expression it is to be noted that XX m denotes the sum 

 of those constituents which are common to X and X m , that sum 

 being actually given by multiplying X and X m together, according 

 to the rules of the calculus of Logic. 



In passing from Logic to Algebra, we shall represent by 

 (XX m ) what the above product becomes, when, after effecting 

 the multiplication, or selecting the common constituents, we 

 give to the symbols a?,, . . #, a quantitative meaning. 



