634 PROFESSOR BOOLE ON THE COMBINATION 



we have, from the data, the equations, 



U+t + fA + V = C 

 U + t + g + <T = C' 

 u + fl = cp 



u + Q = c'q, 

 to which must be added the inequations 



w>0 *>0 /u>0 j/>0 ()>0 <r>0 

 u + t + fJ. + v + (> + o~<l 



Proceeding as in Art. 14, we find ultimately as the conditions of possible ex- 

 perience 



C p^l-c'(l-q) c'q^l-c(l-p) .... (11) 



together with the usual condition that c, d, p and q must be positive proper frac- 

 tions, or at any rate must not transcend the values and 1. We find, too, as 

 the conditions limiting u and t, 



u < cp u<cq 



u>c + c'q — l u>c' + cp — 1 m>0 



i<cl— p <<c'l 



(12) 



i>c + c'l — q — 1 t>c' + c —p — 1 t > 



The solution of the problem assumes, therefore, the following form and cha- 

 racter : — 



1st, It involves two constants c and c, which are arbitrary, except in that 

 they are subject to the conditions (11). 



2ndly, The values of u and t, determined from (8) and (9), in subjection to the 

 conditions (12), are to be substituted in the formula (10). 



3dly, In the absence of any means of determining c and c 7 , the value obtained 

 will be indeterminate, except for particular values of p and q. Some general 

 conclusions may nevertheless be deduced from its expression indicating the man- 

 ner in which expectation is influenced by circumstances insufficient of themselves 

 to give to it a definite amount of strength. This will appear from the following 

 analysis. 



Analysis of the Solution. 



35. The solution is contained in the numbered results, from (8) to (12) inclu- 

 sive, of the preceding Article. Of these, (11) expresses the conditions of possible 

 experience, (12) the conditions limiting u and t From (8) and (9), these quan- 

 ties are to be determined in accordance with (12), and the resulting values sub- 

 stituted in (10). 



By a proper reduction of (8) and (9), the solution may also be put in the fol- 

 lowing form : — 



du? + (cc'm — aa')u — acc'pq = (1) 



at 2 - (cc'm + aa')t-a'cc'(l-p)(l-q) = ... (2) 

 where a = cp + c'q — l a' = c(l—p) + c'(l-q) — l m=p + q—l. 



