il8 jupiter and saturn. 



Periodic Inequalities. 



In the periodic part of the perturbing function,' the comparison 

 of the coefficients of (/' — A.) is as follows :* 



Le Verrier 

 a' R^,,^) = 8-876613 8-87727 



dR(n.^) 



= 9-420363 9-42109 



da 



a' R,,,„ = —0-4383544 —0-4378412 



a'a^;'-"^ = —0-6806245 -0-6804551 



aa 



The periodic perturbations of Jupiter which are due to the first 

 power of the mass of Saturn will be found on pp. 127-142 of VoL 

 X of Les Annates. The most interesting of these is the" great 

 inequality in longitude. We give on successive lines, the value 

 due to our and to Le Verrier's value of a. Le Verrier's other 

 figures are adopted unchanged. It should, however, be stated 

 that Le Verrier's figures have also been recomputed, and in some 

 cases, a quite trifling difference has been found ; hence in a few 

 cases Le Verrier's figures as given here, differ slightly from those 

 given in Les Annates. The symbols almost explain themselves, 

 (i and /3' serve to indicate the powers of the excentricities of 

 Jupiter and Saturn which enter into the numerical values. Terms of 

 the fifth order which involve the mutual inclination of the^two 

 planets' orbits have been omitted. 



— 135-61^^ +o-36/3-^ 4- 6-16/^^/32, X sin (5/'-2X-3Z£') 

 Le v.— 13600 +036 4- 6-36 



4- 78978 /3-/3, - 1 -98 ji^\^, -15-04 /3- /3^ X sin {$V -2\-{L' -2w). 

 LeV.+ 791-33 -1-99 -15-06 



-1521-42/3/32^ ^-3•58/3^/3^ +l5-I2/3/3\ X sin {$1' -2\-2G,' -w), 



LeV.-i524-5i +3-57 4-15-14 



+ 967-34 /3\ _I.84/32/3^- 5-59/3-\ xsin (5/'-2X-3a)')- 



Le V. 4- 968-90 —1-84 — 5 -36 



— 7-9i3/3;,2 x sin (5/'— 2X— w — 2 /). 

 Le v.- 7-939 



4- 17-728/3,772 X sin"(5/' — 2\— w'— 2/), 



Le V. + 17-780 



In computing these, Newcomb's development of the perturbing 

 function has been used and it will not be without interest to exhibit 

 the formulae. The first of the six parts of which the great in- 

 equality is made up involves the ratio of the semi-axes, the excen- 

 tricities and mutual inclination as follows : — ■ 



g3p|;; + g^F^;:]4-rr, 1^^+ etc. 



in which 



* Les Annales, X., p. 72. 



