264 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



sented by the single symbol w, as a function or combination of 

 the events similarly denoted by the symbols s, t, &c., and it as- 

 signs by the laws of the Calculus of Logic the condition 



as connecting the events s, , &c., among themselves. We may, 

 therefore, by Principle vi., regard s, t, &c., as simple events, of 

 which the combination w, and the condition with which it is as- 

 sociated D 9 are definitely determined. 



Uniformity in the logical processes of reduction being de- 

 sirable, I shall here state the order which will generally be pur- 

 sued. 



12. By (VIII. 8), the primitive equations are reducible to 

 the forms 



(1) 



w(l-W)+W(l-w)=Q; 



under which they can be added together without impairing their 

 significance. We can then eliminate the symbols #, y, z, either 

 separately or together. If the latter course is chosen, it is ne- 

 cessary, after adding together the equations of the system, to 

 develop the result with reference to all the symbols to be elimi- 

 nated, and equate to the product of all the coefficients of the 

 constituents (VII. 9). 



As w is the symbol whose expression is sought, we may also, 

 by Prop. in. Chap, ix., express the result of elimination in the 

 form 



Ew + E'(\ - w) = 0. 



E and E' being successively determined by making in the 

 general system (1), w = 1 and w = 0, and eliminating the symbols 

 x, y, z, . . Thus the single equations from which E and E are 

 to be respectively determined become 



s (1 - S) + S(\ -s) + t(l - T) + T(l - f) . . + 1 -W= 0; 



From these it only remains to eliminate x, y, z, &c., and to de- 

 termine w by subsequent development^ 



