620 PROFESSOR ROOLE ON THE COMBINATION 



the appearance of the symbol ^ in (14). I have thought it better to complete 



the investigation, especially as it will serve as a model for the one which follows. 



24. Before proceeding to the second solution of the problem, I will endeavour 

 to explain the principle on which it will be founded. It is involved in the fol- 

 lowing definition. 



Definition. The mean strength of any probabilities of an event which are 

 founded upon different judgments or observations is to be measured by that sup- 

 posed probability of the event a priori which those judgments or observations fol- 

 lowing thereupon would not tend to alter. 



Thus, suppose we were considering the question of the suitableness of a newly 

 discovered island for the growth of a particular plant, and that the probability 

 of its suitableness, as dependent upon general impressions of the climate were r ; 

 but that added special observations, — such as analysis of the soil, determination of 

 allied species growing in the locality, &c, had some of them the effect of raising, 

 others that of depressing, the general expectation before entertained. Now we 

 might suppose that expectation to have had such a measure, that the added obser- 

 vations should, when united, leave the mind in the same state as before. I call 

 that measure the mean value of the testimonies — the value about which, to adopt 

 (for illustration, not for argument) a mechanical analogy, they balance each other. 

 I conceive that in thus doing, I am only giving a scientific meaning to a term which 

 has been hitherto used in a vague sense. I shall show that the formula of the arith- 

 metical mean is a special determination applicable only to particular problems, of 

 the more general mean of which I here speak, and that other determinations of 

 it exist, applicable to other problems, but possessing, in common, certain definite 

 characteristics. 



To apply this principle to the problem under consideration, we must add to 

 the data a new element, viz., the a priori value of Prob. z, i.e., the value which 

 the mind is supposed to attach to it before the evidence furnished by the obser- 

 vations. We will suppose this value r. We must then seek, as before, the a pos- 

 teriori value of Prob. z, i.e., its value after the observations, and, equating the two 

 expressions, determine thence the value of r. 



I shall, in referring to the above principle, speak of it as the " principle of the 



mean." 



Special solution of Problem I. founded upon the principle of the mean. 

 Assigning to z the a priori probability r, our data are the following : 

 Prob. oc — a x Prob. y — a 2 Prob. z = r 



Prob. w = a 1 c l Prob. v = a 2 c 2 



Prob. vjz—a x c x p x Prob. vz = a 2 c 2 p 2 . 



with the conditions wx = vy=0 wv=0 



