188 REPORT—1862. 
APPENDIX. 
On the Solution of the Linear Equation of Finite Differences in its most 
General Form. By Prof. Sytvester, F.R.S. 
The author exhibited (and illustrated with examples) a simple and readily ap- 
plied method of obtaining the general term (and consequently the complete solution) 
of an equation of finite differences with any number of independent variables, a 
question which, although touched upon by Libri and laboriously investigated by 
Binet, had hitherto, to the best of his knowledge, remained unsolved even in the case 
of an equation with but one independent variable with non-constant coefficients ; 
when the coefficients are supposed constant, the well-known solution flows as 
an immediate corollary from the author’s general form. Essentially the method 
depends upon the adoption of a natural principle of notation for the given coeffi- 
cients, according to which each coefficient is to be denoted by a twofold group 
of indices, the number of the double indices in a group being equal to the num- 
ber of independent variables in the given equation. Thus, supposing tm,n,p... to 
be expressible by means of the given general equation, as a sum of w’s with infe- 
rior indices, the coefficient of wu, ,,,--- in that sum must be denoted by the double 
index group tog i = ." i: The process for obtaining the general term in tz, y, z+++ 
pi Ale lare 
is then shown to be reducible virtually to the problem of effecting the simulta- 
neous decomposition of the integer variables 2, y, s... into parts in every possible 
manner and order of relative arrangement, the magnitudes of such parts being 
limited by the degree or degrees of the given equation in respect of these variables. 
The collective value of the terms thus obtained constituting the complete solution 
may be termed, in the author's nomenclature, a hyper-cumulant, whose properties 
and their applications remain to be studied out as those of the elementary kinds of 
common cumulants have been to a considerable extent in the ordinary theory of 
continued fractions. The first stage in the process of constructing the terms of a 
general cumulant or general hyper-cumulant is almost identical with that of finding 
the coefficients in the expansion of a power of a polynomial function of one or 
several variables, differing from it indeed only in the circumstance that permutations 
which lead to repetitions in the latter case, represent distinct values in the former. 
On Aérolites. By Professor N. 8. Masketyye. 
Professor Maskelyne prefaced a series of notices of meteorites lately added to the 
collection in the British Museum by some observations on the phenomena that 
accompany the fall of such bodies to the earth. Loud reports and the develop- 
ment of brilliant light in the sky are among the most generally observed of these 
phenomena. The fallen mass or its fragments, besides the marked characters they 
constantly present, as well in composition as in the mode of aggregation of their 
component minerals, exhibit also invariably a superficial enamelling or incrustation. 
The meteorite which fell at Butsura, in India, on May 12, 1861, accompanied by 
successive reports and a luminosity in the sky visible in the daytime, presented 
some new and very interesting facts bearing on the cause of this incrustation. The 
whole of the fragments found, though they fell in four places, at distances of three 
or four miles apart, formed the parts of a large piece of an aérolite, fitting to one 
another with great exactness, with the exception of two of them, between which 
an intermediate fragment had been lost. Some of the fragments were found to be 
entirely coated with crust, yet capable of being adjusted to each other with 
unmistakeable accuracy; others again exhibited no such incrustation at the parts 
where they fitted to each other, and were yet, like the former, found several miles 
asunder. It was obvious from this that some of these fragments had become 
coated with crust after they had been severed, while others had been so severed 
without becoming subsequently incrusted. 
That the incrustation was the result of superficial fusion seems the best explana- 
tion of its presence on the meteorite, as well as of the partiality with which 1t was 
distributed. Such asuperficial fusion, however, could only result from the develop- 
ment of heat of enormous temperature very instantaneously ; and the best if not 
