S EC T. 3.] ANY NUMBER OF OBSERVATIONS. 255 



D , will be given by the integral / (f J.dJ extended from J = D to J = I/. 

 This integral taken from the greatest negative value of J to the greatest positive 

 value, of more .generally from z/ = cc to // = -|- co must necessarily be equal 

 to unity. Supposing, therefore, any determinate system of the values of the 

 quantities p, q, r, s, etc., the probability that observation would give for V the 

 value M, will be expressed by y (M-- V), substituting in V for p, q, r, s, etc., 

 their values ; in the same manner 9 (M --V), (f (M&quot;--V&quot;\ etc. will express the . 

 probabilities that observation would give the values M , M&quot;, etc. of the func 

 tions V, V&quot;, etc. Wherefore, since we are authorized to regard all the observa 

 tions as events independent of each other, the product 



(f(MV) (f(M V) &amp;lt;f(M&quot;V&quot;) etc., =Sl 



will express the expectation or probability that all those values will result to 

 gether from observation. 



176. 



Now in the same manner as, when any determinate values whatever of the 

 unknown quantities being taken, a determinate probability corresponds, previ 

 ous to observation, to any system of values of the functions V, V, V&quot;, etc.; so, 

 inversely, after determinate values of the functions have resulted from observa 

 tion, a determinate probability will belong to every system of values of the un 

 known quantities, from which the values of the functions could possibly have 

 resulted : for, evidently, those systems will be regarded as the more probable in 

 which the greater expectation had existed of the event which actually occurred. 

 The estimation of this probability rests upon the following theorem : 



If, any hypothesis H being made, the probability of any determinate event E is h, and 

 if, another hypothesis H being made excluding the former and equally probable in itself, the 

 probability of the same event is h : then I say, wlien the event E has actually occurred, that 

 the probability that H was the true hypothesis, is to the probability that H was the true 

 hypothesis, as h to h . 



For demonstrating which let us suppose that, by a classification of all the cir 

 cumstances on which it depends whether, with II or II or some other hypothesis, 



