608 
PROFESSOR SYLVESTER ON THE REAL 
And the total number of arrangements, which is the number of ways in which [Jj things 
can be distributed over (p-\-v) places, is Hence we obtain 
r r(i*-l)(f*-2)»0*-^ + l)x(v-l)(v-2).. (v-^ + l ) 1 
^-^ + v _i)| 1.2,(^-l)(1.2..y) j 
7TfA7r(|X — l)7TV7r(v — l) 
~irgz(g — l)z{g.—g)w{v—g)z{^ + v—l) 
[Throughout these investigations ir(x) is used in the same sense as n.r, to signify the 
factorial 1.2.3...#.] 
If there should appear any obscurity in the statement of the method by which has 
been obtained the number of distinct distributions of the p, v elements into g groups 
of each, the reader is referred to the equation in differences obtained further on in this 
Note, by which all doubt of the correctness of the result will be removed. 
(22) For a trans-rotatory pencil of rays, to ascertain the probability of the variation- 
index being 2 <7+1. 
Imagine a circular arrangement of (x positive elements and v negative elements con- 
taining 2y variations. 
Let this circle be supposed opened out at any point and the variations of the open 
pencil so formed to be reckoned according to the trans-rotatory law, which is that in 
passing from one extremity to the other a change is to be seen as a variation, and a 
variation as a change. If the break is made between two negative or between two 
positive elements, the number of variations obviously becomes increased by one unit; but 
if between a positive and a negative element, that number becomes decreased by one 
unit. The number of these latter intervals is 2y, and of the former 2y. 
Hence the probability of the index becoming 2y + l is and of its becoming 
ft + K 
2rv 
If, then, we denote the probability to be calculated by [g,, v, g-\-\\ it is obvious that 
we shall have 
h 9+¥\ = ' + [X , +V 9 9\ + ~uT+ v ^ S'+l]- 
JU. + V 
But by the formula previously obtained it will easily be seen that 
Hence 
v ’ 5 , + 1 ]— ^ g(g + 1)^ tf 6 ’ 9]’ 
2 ip-g^-g) 
=(^- 1 ) &s ^ ] (,) 
9 (zg.) 2 (zv) z z[x.Tr(g. — l)Trvz(v—l) 
*{9— l)v{g+l)n(p + v)n{g,— g)z{y— g) z(g— \)zgz{g. + v— g)z{v— g) 
