588 
NOTES AND REFERENCES. 
48. The expressions f 2 x — X(a — b) 2 (x — c)(x — d)... for the Sturmian functions in 
terms of the roots, or (to use Sylvester’s term), say the endoscopic expressions of these 
functions, were obtained by him, Phil. Mag. vol. xv. (1839). 
It was interesting to express these in terms of the sums of powers 8 ly S 2 , &c., 
that is in terms of symmetrical functions of the coefficients, but for the actual 
expression of the Sturmian functions in terms of the coefficients the process is a very 
circuitous one, and the proper course is to start directly from the exoscopic expressions 
as linear functions of fx, f'x also due to Sylvester (see his Memoir “ On a theory of 
the syzygetic relations of two rational integral functions &c.,” Phil. Trans, t. CXLiil. (1853), 
pp. 407—548), which is what is done in the subsequent paper 65. 
49. I attach some value to this paper as a contribution to the theory of the 
Gamma function. 
50. 55, 70, 95, 98. The general theorem of § I. was given in a very different and 
less suggestive notation, by an anonymous writer, “ Théorèmes appartenant à la géométrie 
de la règle,” Gergonne t. ix. (1818-19), pp. 289—291 ; viz. the statement is in effect 
as follows: considering in a plane or in space any n points 1, 2, 3 ...%; then joining 
these in order, take 12 any point in the line 1, 2 ; 23 any point in the line 2, 3, 
and so on to n — l.n. Take then 123 the intersection of the lines 1, 23 and 12, 3; 
234 the intersection of the lines 2, 34 and 23, 4 ; and so on to n — 2. n — 1. n. Take 
then 1234 the intersection of the lines 1, 234; 12, 34; and 123, 4 (viz. these three 
lines will meet in a point) : 2345 the intersection of the lines 2, 345 ; 23, 45 ; 234, 5 
(viz. these three lines will meet in a point)... and so on to n — 3 . n — 2 . n — 1. n. And 
so on for 12345, &c. up to 123...%; in the successive constructions we have four, 
five, ... and finally n lines which in each case meet in a point. A proof is given by 
Gergonne, t. xi. (1820-21). 
A large part of this paper relates to the theory of the relations to each other of 
the 60 Pascalian lines derived from the hexagons which can be formed with the same 
six points upon a conic : the literature of the question is very extensive, and I hope to 
refer to it again in the Notes to another volume. 
54. See for an addition which should have been printed with this paper, the 
last paragraph of 92. 
63. Boole’s theorem of integration which is here demonstrated is a very remarkable 
one, and it would be very interesting to investigate the general forms, or a larger 
number of particular forms, for the functions P, Q satisfying the condition mentioned 
in the theorem. It may be remarked that the demonstration is very closely connected 
with Lejeune-Dirichlet’s method for the determination of certain definite integrals 
referred to in 2 and 3, we have a triple integral the real part of which is = fP h- Qi n+( i 
or 0, according as P is or is not comprised between the limits 0 and 1. It includes 
Boole’s formula mentioned in 44 and in 64. 
65. See 48. 
66. The method here employed of establishing the theory of Laplace’s coefficients 
in %-dimensional space by means of rectangular coordinates, has I think some advantage