192 REPORT—1862. 
ployed by Lagrange, ‘Mém, de Berlin, 1776), f can be expressed in terms of 
uw; viz. the result is 
fH=ut2r sin u+2d. 3 sin 2u+2)°. 3sindu+&e., 
where rest a Awad (=). Hence if u, sin uv, sin 2u, &c. are 
é 1+/1-é 
expressed in terms of the mean anomaly, f will be obtained in the form 
=g-+a series of multiple sines of g, the coefficients of the different terms 
being given in the first instance as functions of ¢ and \; and to complete tho 
development \ and its powers have to be developed in powers of e. The solu- 
tion is carried thus far in the ‘Mécanique Analytique’ (1788), and im the 
‘ Mécanique Céleste ’ (1799). 
30. We have next Bessel’s investigations in the Berlin Memoirs for 1816, 
1818, and 1824, and which are carried on mainly by means of the integral 
h 2r 
ont cos (hu—k sin u) du, 
20 
and various properties are there obtained and applications made of this im~ 
portant transcendant. 
31. Relating to this integral we have Jacobi’s memoir, “ Formule trans- 
formationis &c.” (1836), Liouville, ‘Sur l’intégrale “cos i (w—a sin uw) du” 
0 
(1841), and Hansen’s “ Ermittelung der absoluten Stérungen” (1843) ; the 
researches of Poisson in the ‘ Connaissance des Temps’ for 1825 and 1836 are 
closely connected with those of Bessel. 
32. A very elegant formula, giving the actual expression of the coefficients 
considered as functions of ¢ and X, is given by Greatheed in the paper “ Inyes- 
tigation of the General Term &c.” (1838) ; viz. this is 
fag teen {eer gn Teen sin "9 
r 
where, after developing in powers of A, the negative powers of must be 
rejected, and the term independent of A divided by 2. This result is ex- 
tended to other functions of f, Cayley “On certain Expansions &c.” (1842). 
33. An expression for the coefficient of the general term as a function of ¢ 
only is obtained, Lefort, ‘‘ Expression Numérique &c.” (1846). The expres- 
sion, which, from the nature of the case, is a very complicated one, is obtained 
by means of Bessel’s integral. This is an indirect process which really comes 
to the combination of the developments of f in terms of w, and w in terms of 
g; and an equivalent result is obtained directly in this manner, Creedy, 
‘General and Practical Solution &c.” (1855). 
34, We have also on the subject of these developments the very valuable 
and interesting researches of Hansen, contained in his ‘ Fundamenta Nova, 
&ec.’ (1838), in the memoir “Ermittelung der absoluten Storungen &e.” 
(1845), and in particular in the memoir “ Entwickelung des Products &e.” 
(1853). 
cos 
35. But the expression for the coefficient of the general term .- 79 in any 
of these expansions is so complicated that it was desirable to have for the 
coefficients corresponding to the values r=0, 1, 2,3, ... the finally reduced 
expressions in which the coefficient of each power of ¢ is given as a numerical 
