яШШШИШШ 
48 
JAMES JOSEPH SYLVESTER. 
[900 
901] 
three terms, to a class in the University, with a concluding so-called lecture 34, 
which is due to Hammond. The subject, as is well known, is that of the functions 
of a dependent variable, y, and its differential coefficients, y, y", ..., in regard to x 
(or, rather, the functions of y', y", ...), which remain unaltered by the interchange 
of the variables x and y: this is a less stringent condition than that imposed by 
Halphen (“These,” 1878) on his differential invariants, and the theory is accordingly 
a more extensive one. A passage may be quoted:—“One is surprised to reflect on 
the change which is come over Algebra in the last quarter of a century. It is now 
possible to enlarge to an almost unlimited extent on any branch of it. These thirty 
lectures, embracing only a fragment of the theory of reciprocants, might be compared 
to an unfinished epic in thirty cantos. Does it not seem as if Algebra had attained 
to the dignity of a fine art, in which the workman has a free hand' to develop his 
conceptions, as in a musical theme or a subject for painting ? Formerly, it consisted 
in detached theorems, but nowadays it has reached a point in which every properly - 
developed algebraical composition, like a skilful landscape, is expected to suggest the 
notion of an infinite distance lying beyond the limits of the canvas.” And, indeed, 
the theory has already spread itself out far and wide, not only in these lectures by 
its founder, but in various papers by auditors of them, and others,—Elliott, Hammond, 
Leudesdorf, Rogers, Macmahon, Berry, Forsyth. 
I 
Sylvester’s latest important investigations relate to the Hamiltonian numbers: 
there is a memoir, Crelle, t. c. (1887), and, by Sylvester and Hammond jointly, two 
memoirs in the Philosophical Transactions. The subject is that of the series of 
and 
numbers 2, 3, 5, 11, 47, 923, calculated thus far by Sir W. R. Hamilton in his 
well-known Report to the British Association, on Jerrard’s method. A formula for 
the independent calculation of any term of the series was obtained by Sylvester, but 
the remarkable law by means of a generating function was discovered by Hammond, 
viz. E 0) E ly E 2 , ..., being the series 3, 4, 6, ... of the foregoing numbers, each 
increased by unity; then these are calculated by the formula 
(1 - t'fo+t (1 - t) E > + t 2 (1 - + ... = 1 - 21, 
equating the powers of t on the two sides respectively: observe the paradox, t = \, 
then the formula gives 0 = sum of a series of positive powers of 
Enough has been said to call to mind some of Sylvester’s achievements in 
mathematical science. Nothing further has been attempted in the foregoing very 
imperfect sketch. 
where 
then 
whence 
powers 
c