AND IMAGINARY EOOTS OF EQUATIONS. 
653 
effectually the character of the roots in an equation of the fifth degree ( 64 ) ; whereas in 
the symmetrical and invariantive method which I have employed three have been seen 
to s uffi ce. 
In an equation of the seventh degree the case of 0 or 4 will be distinguishable from 
that of 2 or 6 imaginary roots by the sign of the discriminant, and then again the case 
of 0 from that of 4, and of 2 from that of 6, by other invariantive criterion-systems. So 
for an equation of the ninth degree, the first separation will be that of the 0, 4, or 8 
case from that of 2 or 6 ; then it may be conjectured the 2 case will be invariantively 
separated from the 6, and the 0 or 8 from that of 4, and, finally, 0 and 8 from each 
other — the reduction of cases apparently depending upon the relation of the index of 
the equation to the powers of the number 2. This much we know (from art. 49) as 
matter of certainty, that no single criterion other than the discriminant can ever serve 
to distinguish one form of roots -from another so that all other criteria must accom- 
pany each other in groups ; and accordingly the scheme of criteria, established in the 
foregoing investigation is in kind the very simplest a ’priori conceivable. 
(M) ^ p or an e q Ua tion of the nih. degree there are n — 1 variable criteria, each capable of being + or — , and 
thus giving rise to 2 n ~ l conceivable diversities of combination. The actual number possible, however, is consider- 
ably less than this ; and I find by an easy method that this number, when n is odd, is 2 n ~‘ 1 -\ — 2^— — ~- 9 , and 
when n is even, is 2 n ~ 2 -f- 
n(n—l) 
„ n fn , \ 
n 2 n (r-\) 
( b ) Not quite foreign to this subject is the inquiry as to the comparative probability of each different succes- 
sion or each different family of successions possessing equivalent characters ; and, as connected therewith, the 
comparative probability of a certain specified number of the roots of an equation of a given degree being real 
and the remainder imaginary. In the simplest case of a quadratic equation of which the coefficients are real 
but otherwise arbitrary, I find that upon the particular hypothesis of the squares of the three coefficients being 
limited by one and the same quantity, the probability of the roots being imaginary is ~j|~ ’ or "3727932, 
a little less than -§■ , this being the value of the integral da (\ — ; but we are not at liberty to infer 
from this the value of the probability in question when the coefficients are left absolutely unlimited. A case 
in point, as illustrating the effect of imposing a limit in questions of this kind, occurs in the problem (which I 
raised in my lectures on Partitions) of finding the probability that four points placed at hazard in a plane will 
form the angles of a reentrant quadrilateral, which Professor Cayley has shown is exactly l in the absence of 
any limit. For if ABCD be the four points, and ABC the greatest of the four triangles of which they may be 
regarded as the angular points, and if through A, B, C be drawn lines parallel to BC, CA, AB respectively, the 
triangle a jSy so formed will be four times as great as ABC, and the point D must be somewhere within a/3y, 
otherwise ABC would not be less than each of the three other triangles ABD, BCD, CAD ; and consequently, 
since D must lie within ABC when the quadrilateral is reentrant, the probability in question is or 
ccpy 
Now it is easy to see, by using the very same construction, that if any contour whatever be imposed as a limit 
upon the positions of the four points, the probability referred to will exceed i by a finite quantity — a result 
somewhat paradoxical, since d 'priori one would have supposed that the value of it for the case of no limit would 
be the mean of the values corresponding to the respective suppositions of every possible form of limit. 
4 S 2 
