490 
FORMATION OF EMPIRICAL LUNAR THEORY. 
pecting tlie formation of an empirical 
lunar theory. 
Mr Lubbock made some preliminary 
remarks, tending to prove, that al- 
though the astronomical tables con- 
nected with the lunar theory are suffi- 
ciently perfect for the more general 
purposes of navigation, yet astronomers 
are by no means satisfied with resting 
at this degree of exactness. It is an 
object much to be desired that they 
should reach, by calculation and theo- 
ry, a degree of accuracy far beyond this, 
and if possible construct lunar tables in 
which the fixed co-efficients should be 
as exact as could be obtained from the 
use of the very best instruments, erect- 
ed in fixed observatories. His object 
was to press the accomplishing of this 
by obtaining directly from observation 
the co-efficients and other quantities 
which would be necessary for the con- 
struction of tables, with as little reference 
to previous lunar theories as was practi- 
cable or proper. He observed, that the 
most important works upon the lunar 
theory were those of Messrs- Damoi- 
seau and Plana. The calculations of M. 
Damoiseau, however, are in such a very 
intricate form, that it is almost impossi- 
ble to verify them. The work of M. 
Plana constitutes an eatirely new era in 
the lunar theory, for in it the results are 
developed according to the powers of 
extentneities, inclination, and other ele- 
ments of the lunar and terrestrial orbits, 
as also the quantity M. denoting the 
ratio of the Sun’s mean motion to that 
of the Moon. In other respects, the cal- 
culations are similar to those of Damoi- 
seau ; and in both there finally meets us 
the almost insuperable difficulty, that 
the expressions for the co-efficients are 
series, having a very slow conveyance, 
and requiring, therefore, extreme labour 
to deduce them with any tolerable de- 
gree of accuracy. Now, it is principally 
these co-efficients that Mr. Lubbock 
wished to have the values of deduce em- 
pirically from the best observations ; 
and he even ventured to express a hope, 
that by this method tables might be con- 
structed of such minute exactness, as 
might serve to check the results obtain- 
ed from theory ; and his anticipations 
were the more sanguine, because nothing 
was wanting to complete success but 
the placing of a sufficient fund at the 
disposal of a committee, since abundant 
stores of the most minutely exact obser- 
vations were recorded ; and persons in 
every way competent to their reduction 
were to be had. 
Professor Sir William Hamilton then 
read his report on Mr. George B. Jer- 
rard’s mathematical researches, con- 
nected with the general solution of 
algebraic equations. 
Sir W. Hamilton wished, in the first 
place, to inform the Section, that no part 
of the grant SOI. had been expended, 
which the Association had so liberally 
placed at his disposal for the purpose of 
procuring the assistance of persons 
competent to verify, by numerical com- 
putations, the method of Mr. Jerrard. 
The reason that he had not deemed it 
necessary to resort to this expense was, 
that he had, at a very early period after 
the meeting of the 13ritish Association 
in Dublin, satisfied his own mind that 
the method of Mr. Jerrard entirely failed 
in accomplishing the solution of equa- 
tions of the fifth and sixth degree ; and 
he trusted that he should be able to lay 
before the Section, with as much clear- 
ness as the abstruse nature of the sub- 
ject would admit of, the principal steps 
of a demonstration, which, to the mind 
of the learned Professor himself, at least 
carried complete conviction, that the 
method of Mr. Jerrard was not appli- 
cable until the equation, as a minorlimit, 
had reached the seventh degree. In or- 
der that he might carry the Section fully 
along 'with him. Professor Hamilton 
stated, that it would be necessary to give 
again a rather detailed account of the pe- 
culic rities of the very i igenious notation, 
devised by Mr. Jerrard, for denoting 
certain algebraic jirocesses, or operations 
resorted to in the application of his 
method. The Professor then proceeded 
to detail to the Section the several steps 
of Mr. Jerrard’s method, clearly mark-, 
ing the steps previously known to an- 
alysts, and such as Mr. Jerrard had the 
merit of originating. 'I’he ])rincipal pe- 
culiarity oiformulce seemed to be, that 
in an equation, transferred in a ])articu- 
lar manner for the purpose of eliminat- 
ing the co-efficients of the original equa- 
tion, the co-efficients were so ingenious- 
ly obtained as to lie entirely independent 
of the degree of the original equation, 
and therefore to be of a similar form in 
all possible equations, the solutions of 
which were sought. As soon as he had 
prepared these formulae, the Professor 
proceeded to demonstrate to the Section^ 
::l 
