614 PROFESSOR BOOLE ON THE COMBINATION 
or, for simplicity, in the form 
U=0. 
The lower limit of w is, by (11) and (12), either 0 or — according as 
the latter quantity is positive or negative; the upper limit of w is the least of the 
quantities p, g, 7; suppose it p. First, let a be positive, then, making 
wu equal to this quantity, the value of U, as given in the first member of (14) 
becomes negative. Again, let w=p, then U becomes positive. Thus, as w varies 
from ? +1 sé up to p, U changes from negative to positive. Now 
CW =(Qu—p—q—rt 1) +4u(Qu—p—4 —r+1)+(p—u) (q—u) + (q—-u) (r—u) + (r7-u) 
(p—u) c : (15) 
which within the supposed limits is always positive. Hence U varies by continu- 
ous increase, and once only in its variation becomes equal to 0. 
ptq+r— 
ama 
Secondly, le 1 he negative, then uw, varying from 0 up to p, U as 
before will vary by continuous increase from a negative to a positive value. See 
the first member of (14). Whence U, changing by continuous increase from a 
negative to a positive value, will still only once become equal to 0. 
Wherefore, in either case, one root only of (10) will lie within the limits as- 
signed to win (11) and (12). And this one value substituted in (9) will give one 
set of values for s’, ¢, v’. 
20. The solutions which we have now obtained of the same problem on dif- 
ferent hypotheses with respect to the selection of the simple events, set in clear 
light the principles upon which the due selection of such hypotheses depends. 
The hypothesis which seems most readily to present itself utterly fails, while 
the other, based quite as much upon an apparently remote speculation on lan- 
guage, as upon the study of the laws of expectation as usually conceived, finds a 
support and confirmation within the realm of pure mathematics which is of the 
most remarkable kind. 
21. A practical simplification of the general method is suggested by that step 
of the preceding solution, which reduces (5) to the form (7). If we remove the 
traces (') from the letters in the latter system (and they do not at all affect the 
solution), we obtain what (5) would become if we replaced each of the symbols 
s, t, v, by unity. Practically, therefore, we may modify the general rule in the 
following manner :—Having obtained V, replace each of the symbols s, 1, &c., by 
unity, and proceed with the reduced value of V just as before, 7.¢., let V, 
represent that portion of V of which s is a factor, &c., then form the system of 
equations 
Vi aa wA+reC 
Bee re OE oi oat) 
