12 
REPORT— 1847. 
but the former is far more convergent. The factors of vhich the quantity 
{l+P+9}^ 
is formed, each constitute generally a highly and regularly coo^rpng series ftm^ 
at once by the table. Their product must also generally convene rapidly, uidilsi 
therefore the product of {1 by 
If / is the time reckoned from the time of the perihelion passage of «, r<r', 
n/rst)— esintf; 
and if 360°—nV is the mean anomaly of m' at the time of the perihelion pisssprf 
— o—nV* 
■n, 
cos*C^=co8 j In—%smw|, 
sini^'’*8ini |n— — esini;} : 
n’ I » J 
and os — is a fraction, cost and sin tj' can be developed in a series rapidly'®''' 
explicitly only tlio variable qunntitv a. Mr. Farley b»ol' 
^lat^ for me u table of the values of cos sin cos 2 i sin 2 1 &c. for J«P'» 
terms of the eccpnlric anomaly of those comets for which 
anilti'willirr.l'lfl* of tables presents wherever they 
emr hi.f o"ly the march of the tigures affords securitya?^ 
taken whit ® ^c^iuircs facility in aucli c^culations srstematically 
1* are more easy than they would be ‘if the quaat*®,"; 
3- LnirtXn''? by a common migiu. or L troublesome as 
The u*e^S individuals, or by the same individual a: different ti^- 
on the ramrnrr- V* queatinn in a literal or algebraic developn^’ol: ^ 
can thua be rnTt^Luy fecStTd“"^ P^perty of the numerical development ma 
into three classes, which I 
numcriral vali.^Q Literal or algebraical are those which result *1“ 
plrte OH/uti-liioJ constants are inserted last, and after the development u« ' 
« part onlv of the 
aeveral diSe symbols, or when the development is iu«J'‘ 
of these but nnt ‘"'i'cn the numerical values are inserted after a p«^ 
accomplished, finally, orithnetic or ^ 
inserted in nl^e nt result when the numerical values of the coostaftO 
attem^d. ^ symbols before any step of the de^lopoir®' » 
formed generally preferable, for this reason, that if it can 
Sned to Ae serve* for every possible value whichm^ 
function due to \T instance, is the development of tbediswriB®? 
Ww c33ld be acLn^ “ development in terms of the lequirite 
plished and carried out to a .«<ufficient extent, and K be«? 
piTferablc“r?anv^I!“'®M constants could be easily introduced, it 
Iwtiuns of Kneke’s ComPt^i* s.developrnent, in his ' Memoir on die Ped ' 
tion onlv of ilm nr Saturn, is n ywan-/]/n*aldevelopment, beeauM*!® 
explicit fii t- general. The conversion of the quantities P^; 
''•hiJe the caJeuktim* multiples of / is an anlhmsiicel pr^j 
or algebraical. ^ quantities . in p. 29 of M. Hansen’s paper, is 11161*1 
