IN CASES OF LEGAL EVIDENCE. 149 



or as well as by them. In the particular case, antimony, the persalts of tin, and probably 

 some other substances, exhibit with some tests what to inferior skill would be such appearances. 

 Suppose there are n such causes of fallacious results, and let m several experiments be made, 

 Pi, g,, r, , be the respective probabilities that As, if it were present, and each of the other sub- 

 stances severally, if it were present, would produce the appearances witnessed by 

 the application of any one test. 

 <, that some substance, other than those known and enumerated, would do so if it were 

 present. 



Pa, q-i.-.t^ P„fU---tn the corresponding probabilities for the other tests. 



P the probability that As is present. 



Then P= P^-V^ P' 



]h--Pn + (If-9« + &C + t,.t^...t„ 



If this substance exist in moderate quantity, and even an ordinary degree of skill be employed, 

 the experiments may be varied so as to produce appearances which could not have been produced 

 by some one or more of the causes other than the presence of As, and therefore a factor will be 

 successively introduced into the terms ^i...?,, J'j...r„, &c. of the denominator, and the expres- 

 sion is reduced to 



I 

 P= 



1 + J.. .._=.. 



Px P„ 



It may be observed generallv, that it is the presence of this terra —...-2 in the denomi- 



Pi Pn 



nator representing possible hypotheses, yet unthought of, that distinguishes the proof of a physical 

 fact from a mathematical demonstration. 



The successive elimination of the known causes of error is precisely that which common sense 

 employs in arriving at a moral certainty ; when this cannot be effected, the previous expression 

 remains, and the probability of the fact alleged being true is arrived at by assigning numerical 

 values to its elementary probabilities. 



The evidence which is the subject of the following formulas is that, or nearly that which was 

 given in a case which occurred of a woman who was accused of having caused the death of her 

 husband, by administering As ; it is merely used as an illustration, and therefore no particular pains 

 is taken to state the evidence very accurately. The death and its cause were not disputed, the 

 probabilities therefore of the presence of As, and of its having caused the death, are taken in the 

 investigation as measured by 1. 



The first witness, whose evidence is here considered, alleged that she had seen the accused on 

 the morning of a particular day making some pills. 



Consider, therefore, first, the probability of a fact being true which depends for its evidence on 

 the testimony of a single witness. In such a case the allegation may have been made, either because 

 the event alleged took place, and the witness saw and believed it to do so ; or because the witness 

 believed it to have taken place, though it did not in fact do so ; or because the witness was actuated 

 by a wish to deceive, and made the allegation without believing in its truth. Call the event 

 alleged E^, and as a convenient mode of expressing the probabilities involved in the investiga- 

 tion, let there be w - 1 other events E.. £„, which include, first, all those by the belief in 



the occurrence of which the disposition, on the part of the witness, to make the jiarticular 

 allegation could be influenced, whetiier they might in fact have occurred or not: and secondly, 

 all those which the witness might be induced to allege on the particular subject, without believ- 

 ing them whether they could or could not have occurred. 



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