AND IMAGINARY ROOTS OR EQUATIONS. 
607 
NOTES TO SECTION II. 
Received May 7, 1864. 
On the probability of the specific superior limit to the number of real roots in a 
superlinear equation equalling any assigned integer. 
(21) The question comes to that of determining the probability of a per-rotatory or 
trans-rotatory pencil with a definite number of rays of each kind possessing a given 
variation-index. 
probability in question in general terms, as follows. 
1. For a per-rotatory pencil of (Jj positive and v negative rays. Let [p, v, g~] be the 
probability of the rays being so disposed as to give rise to 2 g variations of sign in 
making a complete revolution. Then there will be g distinct groups of positive, and g 
of negative rays. The number of partitions with permutations of the parcels inter se 
of gj elements in g parcels is (^~ 1 ) (^ ~ + 1 ) ? and of v elements into g parcels is 
(y-l)(y-2)..(y-ff + l) 
If we combine each parcel with each in every possible way, and then imagine the 
combined parcels let into a circle containing m-j-n places and shifted round in the circle 
through a complete revolution, we shall obtain 
i „K/ ^ — — — <7+1)1 (v — l)(y — 2)..(y— 0+1) 
^ + )x lXR 1.2..(g 1) 
arrangements ; but on examination it will be found that every arrangement so produced 
will be repeated g times ; moreover it is obvious that no other arrangement giving rise 
to g groups of each sort can be found. Hence the true number of distinct groupings 
of the sort in question is 
±JH fc-l)(|*-2)-(, *-</ + !) (v 1) (y — 2) .. (y — g -f- 1) 
9 \.2..{g-\) 
It seems hardly worth while to pursue this subject in greater detail. I will only notice that when m is even 
the chance of the specific maximum attaining the absolute maximum, i. e. becoming 2oj, will depend on the pro- 
portion of the ways in which in a cycle of n elements w of them may be marked with a distinctive sign in such 
a way that no two of such signs shall come together. Accordingly I find by a computation of no great difficulty 
(understanding i xx to mean 1.2.3. ..x), 
n-(n — ii>— 1) 
“ 7rui7r(ri. — 2w ) 
and hence, since the total number of combinations of n elements m and w together is 
r (n) 
r(n-w)7r(n-w-l) 
7 r(n — l)7r(n — 2w) 
— oi) 
, I deduce 
Thus when n has its minimum value, viz. 
p 2oi = 7rw ~^~ — f-, and becomes very small as u increases. When 
Jr ( w 1) 
again n increases towards infinity approaches indefinitely near to unity, and the chance approaches near to 
certainty of the specific not beoming less than the absolute maximum of real roots. 
