1862.] 183 



2ndly. If V be a rational and entire function of any quantities 

 a?!, a? 2 , . . . x n , involving, however, no powers of those quantities, and 

 all the coefficients being constant, and if in general V f represent the 

 sum of those terms in V which contain x { as a factor, and V^ the 

 sum of those terms in V which contain the product a? 4 x jt then the 

 determinant 



V V V V 



v 1 v 2 . . . v w 



V V V V 



1 r 11 12 * T 1 



V V V V 



Y 2 21 Y 22 * * V 2H 



V V V V 



T T ni nn 



will on development consist wholly of positive terms. 



3rdly. The definitions being as above, and the function V being in 

 form complete, i. e. containing all the terms which by definition it 

 can contain, the system of simultaneous equations 

 V V V 



y l =Pl> -y=A ' ' -^=P*> 



in which p lt p 2 , . . . p n represent positive proper fractions, will admit of 

 one, and only one, solution in positive integral values of x^ a? 2 , . . . x n . 

 4thly. The function V being incomplete in form, . e. wanting 

 some of the terms which it might by definition contain, the system 

 of equations 



V V V 



^-A. Y 2 =fe---f=^ 



will admit of one, and only one, solution in positive integral values of 

 #!, a? 2 , . . . X M provided that p^p^ . . .p*, beside being proper fractions, 

 satisfy certain conditions depending upon the actual form of V. 



These conditions are expressible by linear equations or inequations 

 of the general form, 



In the application to the theory of probabilities, the form of the 

 function V depends upon the explicitly determined logical connexion 

 of the events in the data; the equations or inequations of condition 

 correspond to the conditions of possible experience as a source of the 

 data. 



It appears, therefore, that, quite independently of any question of 

 the validity of the logical, and I ought perhaps to add philosophical, 



o2 



