439 



6thli/, Where testimonies are in a high degree cumulative, e.g. 

 where their concurrence was d priori very improbable, the formula 

 for the strength of the united testimonies p^, p^, . . . pn, tends to 

 assume the following expression as its limits, viz. : — 



Pi Pi • • • P" 



Pi Pa • • • P" + (^ -Pi) ^1 -Pa) • • • (1 -P") 



This is commonly assumed to be the general solution. The au- 

 thor shows that the proof of it, as usually given, involves the neglect 

 of a real and important consideration, — viz., that it is to the same 

 fact that the testimonies relate. To the true general solution it 

 stands in the position of a limiting value, applicable only on the 

 hypothesis of the testimonies being of a very unexpected kind, or 

 of their concurrence being very unexpected. 



Sthly, When the probabilities are not cumulative, but some of 

 them are felt to be too great, others too small, and a kind of mean 

 between them is required, a definite result is again obtained, which 

 may be thus stated. 



Let pj, po» . . pnj represent the separate probabilities, then is the 



mean pi'obability represented by the formula 

 i_ 



(PlPa, . . . . pn)n 1. 



IpiPa-.Pn}" + 1(1 -Pi) (1-Pa) ■ • (1-P..)} ^^'■' 



This formula is the counterpart of (2). It expresses the average 

 of the probabilities furnished by differing judgments, even as the 

 formula of the arithmetical mean expresses the average among dif- 

 fering measures of a numerical magnitude. But it differs in cha- 

 racter and in the consequences which it involves from the latter 

 formula. 



Thus if the testimonies are two in number, and the probabilities 

 which they furnish in favour of an event are p and q, the formula be- 

 comes , and furnishes a value which does not 



Vp'y + V U-pJ(i-^) 

 lie half-way between the values p and q, but which, as may easily 

 be shown, lies nearer to the one of those values which is the nearer 

 either to 1 or to 0. This indicates, that if we have to take an 

 average between two judgments, one of which partakes more of the 

 character of certainty than the other, the former will have greater 

 weight in determining the final state of expectation. This, the author 

 observes, is accordant with our instinctive feelings. 



The formula, it is to be observed, is not applicable to cases in 



