MATHEMATICAL SECTION. 105 
Titan and Hyperion, a and n are the mean values of r and w, and 
a,, n,, etc., are constants to be determined. In the differential equa- 
tions of motion for r and w, those terms not containing explicitly the 
mass of Titan were expanded by Taylor’s theorem into series of 
sums of terms containing cosines of multiples of @ affected with con- 
stant coefficients; while the terms containing the mass of Titan ex- 
plicitly were expanded mechanically into a series of cosines of mul- 
tiples of @ by means of assumed values of the coefficients. Equating 
the coefficients of the cosines of equal multiples of 0, a number of 
equations were obtained from which to derive a corresponding num- 
ber of the quantities a,, ,, etc. With these new values were ob- 
tained and the process repeated. Instead, however, of considering 
m’ (the mass of Titan) as known, a, was assumed to be given and m’ 
was considered as one of the unknowns. 
[This paper appeared in full in the Annals of Mathematics. 4°. Char- 
lottesville, Va., 1887. Vol. 8, No. 6, p. 161.] 
Brief remarks on. Mr. Stonn’s communication were made by the 
Chairman and by Messrs. Bakrr, H1Lu, and KuMMELL. 
Mr. E. B. Exxiorr presented a paper on 
THE QUOTIENTS OF SPACE-DIRECTED LINES. 
He wrote down some of the fundamental relations of this analy- 
sis and explained the nature and properties of the special symbols 
employed. He called attention to the lectures of Hamilton on 
quaternions, and to his “letters” on the same subject, as they ap- 
pear in Nichol’s Cyclopedia ;* and commented on the transition 
from Hamilton’s primary conception of a quarternion as a quotient 
of two directed right lines in space to his secondary conception of 
a quaternion as the sum of a directed right line and a number. 
The presentation of this paper was followed by a discussion, in 
which Messrs. BAKER, H1Lu, Sronr, Woopwarb, and the Chair- 
man participated. 
* See A cyclopedia of the physical sciences, ete., by J. P. Nichol. * * 
8°. London and Glasgow, 1857. pp. 625-628. 
