OF TESTIMONIES OR JUDGMENTS. 651 
the conditions of limitation being 
u> 0 u>e+ep—1 u>ct+egq—1 
u<cp u<cg : ? : s (18) 
Now, since w is greater then c+¢q—1 itis @ fortiori greater then cp+c’g—1. Thus 
within the limits assigned to ~, all the factors of each term of (17) will be positive. 
If then we give to w 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 wu 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 2, y, s, and ¢, in terms of uw. Their values determined 
from the system (16) are as follows, viz. :— 
c(1—p) : e(1—q) 
a u—(¢e+ep—1) 
~u—(+eg—1) 
— Ur(eteg—T) x 
e(1—p) cq-—u 
_ u-(¢+ep~1) x 
e(1—q) cp—u 
All these expressions become pdsitive when w 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 
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 0 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 WuatEy’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, 
