438 



theory they must be, infinitesimal, the values of W^ Wg . . . W^, 

 which depend upon ratios will be finite. If any one of the quan- 

 tities Cj, Cg . . . Cn, is equal to unity, the formula verifies itself; 

 for if it is certain that if an observation be correct, the result which 

 it furnishes must be adopted. 



C 1 



If C. =1, then ]^_Q =Q = infinity. Therefore, the expression 



becomes ^ — » 



-— =pi 



1-0, 



If C^ Cg . . . Cn are equal, the formula degenerates into 



P1+P2 + ' • •P"' (2.) 



n 



and expresses the principle of the arithmetical mean. 



The author conceives this investigation to be of value, not on ac- 

 count of the result to which it conducts us, but on account of the con- 

 nexion which it establishes between the doctrine of the arithmetical 

 mean and the logical theory of probabilities. 



In the second special problem, viz., the problem of the combina- 

 tion of testimonies to a fact or hypothesis, results are obtained 

 which may thus be described : — 



1st, The complete solution involves arbitrary constants, and is 

 therefore indefinite. It admits, however, in various cases of a de- 

 finite value, and leads to many general conclusions of considerable 

 interest. 



2dli/f The arbitrary constants relate, not to the probability of the 

 fact or hypothesis as dependent upon the testimonies or evidences, but 

 to the probabilities of the testimonies themselves. Thus, if two 

 symptoms are observed, each of which gives a certain probability of 

 the existence of a disease, the strength of the joint presumption does 

 not altogether depend upon that of the separate presumption, but is 

 affected, for instance, by the ct priori likelihood of concurrence of 

 the symptoms. 



ddl^, In general, combined presumptions would be strengthened 

 by the d, pi-iori unlikelihood of their combination. 



4thli/y In many cases the arbitrary elements disappear. Thus, 

 if one of the presumptions amount to certainty, the combination 

 will always indicate certainty, however unfavourable the opposite 

 presumption may be. 



