442 Prof. Boole on a General Method in 



the ratios s, t, &c. being found from (9), must be substituted in 

 (10). These ratios being positive fractions, we must employ a 

 set of values of 5, /, &c., which consists solely of positive frac- 

 tions. It will hereafter be shown, that when the problem is a 

 real one, the system (9) furnishes one, and only one set of values 

 answering the required description : that set must therefore be 

 taken. This is the only addition required to the general rule as 

 given in the Laws of Thought. 



The combined systems (9) and (10) may be elegantly deduced 

 by the following method, originally communicated to me by 

 Professor Donkin. 



The probabilities, before the nexus, of the events V, V„ V<. . 

 and A are the corresponding quantitative functions V, V„ V< . . 

 and A. The probabilities of the same events after the nexus are 

 1, jo, 9 . . and u respectively. Now the only effect of the nexus 

 is to exclude a number of hypotheses unfavourable to the hap- 

 pening of the above events, without affecting the cases favourable 

 to their happening. Hence the several probabilities have to each 

 other the same ratio before the nexus as after, and therefore 



V:V.:V,..:A=l:j9:^..;t^; 

 or 



p q ' ' u ' 



a system equivalent to the system (9) and (10). 



The investigation is conducted in the same manner when the 

 function C presents itself in the final logical development (5), 

 and the general rule thus established is the following : — 



Rule.—Yoxm the symbolical expressions of the events whose 

 probabilities are given or sought, and equate such of them as 

 relate to compound events to a new set of logical symbols, s, t, 

 &c. Express also any absolute conditions which may be given 

 in the data. From the combined system determine by the Cal- 

 culus of Logic, Wj the event whose probability is sought in terms 

 of all the events s, /, &c. whose probabilities are given, and let 

 the result be 



«.;=A+0BH-5c+iD. 



Then representing the aggregate A -f B + C by V, and the sum 

 of those constituents in V of which 5 is a factor by V„ and so 

 on, form the algebraic system of equations 



^=^..=V, (L) 



p q 



Prob. u;= :^-t£^, (IL) 



wherein p, q, &c. are the given probabilities of s, t, &c. 



