OF TESTIMONIES OR JUDGMENTS. 651 



the conditions of limitation being 



w > u> c' + cp — 1 M>c + c'g — 1 



u <cp u <c'q . . • • (18) 

 Now, since u is greater then c + c'q — 1 it is a fortiori greater then cp + c'q-l. Thus 

 within the limits assigned to u, all the factors of each term of (17) will be positive. 



If then we give to u the value which belongs to the highest of its inferior 

 limits, the first member of (17) will be reduced to its second term, and will be 

 negative. If we give to u the value which belongs to the lowest of its superior 

 limits, the first member of (17) will be reduced to its first term, and will be 

 positive. Moreover, that member is a quadratic function of u. Hence there is 

 one root, and only one, within the limits specified. 



We must now express %, i/, s, and t, in terms of u. Their values determined 

 from the system (16) are as follows, viz. : — 



x _ c(l-p) C(l-q) 



u — ( + c'q—l) y u — (c' + cp — 1) 



u— (c + c'q— 1) u 



c(l — p) c'q — u 



t _ u-(c' + cp-l) v u 



c'(l — q) cp — u 



All these expressions become pdsitive when u is determined in accordance 

 with the conditions (18). 



It would seem from the above, as well as from reasonings analogous to those 

 of Proposition I., that when the algebraic system belonging to a problem in the 

 theory of probabilities is placed in the form 



V V 



z * y 



the limits of variation of the first member of any equation subject to the condi- 

 tion, that the variables shall all be positive, and shall vary in subjection to all 

 the other equations of the system, will not in general be and 1, as in the case 

 contemplated in Prop. I., but will correspond with the limits of value of the 

 second member of the same equation as determined by the conditions of possible 

 experience. 



This conclusion I have in various cases independently verified. The analytical 

 theory still, however, demands a more thorough investigation. 



APPENDIX B. 



A note to Archbishop Whately's Logic, Book III., sec. 14, contains a rule 

 for computing the joint force of two probabilities in favour of a conclusion which, 

 as actually applied, is at variance with the preceding results. For this reason, 



