and on the Distribution of the Fixed Stars. 527 



obtain two definite limits, within which the probability sought 

 must lie independently of all further results of experience. The 

 extreme case is that in which the proposition, whose probability 

 is sought, has no logical connexion whatever with the premises. 

 Even then the method does not fail. The final expression con- 

 sists of a series of terms, each multiplied by an arbitrary con- 

 stant, and the nature of the experience by which the constants 

 are to be determined is assigned ; but in the absence of that ex- 

 perience, the limiting values of the series are and 1. As a 

 general example of the method I will take the following case. 

 I suppose that, from obsei-vations made upon the health of a 

 district during a period of sickness, it appeared that a portion of 

 the houses represented by the fraction p were vdsited by fever ; 

 a portion represented by the fraction cj by cbolera ; and a por- 

 tion represented by the fraction r were free from both these 

 diseases, and at tbe same time in a proper sanitary condition as 

 respects cleanliness and ventilation : required the probability 

 that any house taken at random was in a defective sanitary 



condition. ^ • i ■ i vf 



The solution of this problem would be sufficiently simple it 

 we were permitted to assume that the three events represented 

 by the presence of fever in a house, the presence of cholera, and 

 the prevalence of a defective sanitary condition, were independent 

 of each other. This, however, we have no right to assume, ihe 

 solution must be sought for on other principles. The foUomng 

 is the residt to which the general method conducts me. 

 Probability sought 



1-r ^ 1-^ 



c and J being arbitrary constants. Assigning to them the ex- 

 treme values and 1, we get 



{l- p-r)il-q-r) 



r^- ' 



and 1 -/• for the Umits of its value. The interpretation of the 

 arbitrary constants afforded by the final logical equation is, that 

 e is thc'ui'obability that if a house is visited by fever its sanitary 

 condition IS defective, c' the pro])ability that if a house is visited 

 by cholera without fever its sanitary condition is detective. 

 These elements must be determined by further experience. It 

 may be; further remarked, that for the application of the general 

 method. It is perfectly indilFerent whether the number of our 

 data is equal to the number of simple events or not, and that the 

 mode of their combination is equally luumportant. Ihc case 

 above given is a very simple one; but for tins reason its verifi- 



