( 



t<2 THEORBITOFURANUS. 



In general, a series of terms of the form 



Xa i sin {iA -j- sij) -\-Xh^ cos {iA -\- sg) 

 -\- Sa'i sin {iA — sg) -\- "S, b\ cos (t J. — «(/), 



may be put in the form 



\ 2 (fl, — a',) cos I A — 2 (&i — b'i) sin iA \ sin sg 

 -\- 1 2 {('i -\- a,) sin iA -)- 2 (fe, + h',) sin ivl ^ cos sg. 



All the periodic terms containing only g and ? in the arguments may be put into 

 this form by taking 



A^g-I, 



so that the coefficients of sin s[/ and cos sg may all be expressed as a function of 

 the single variable argument A. 



The perturbations of the elements may be reduced to perturbations of the ( 

 co-ordinates expressed as the sum of several products of slowly varying functions : 

 into the sines and cosines of the multiples of g. We have, in fact, 



^v = H 



+ (2 — ^e = ),5eXsin<7 + ( 2 — ^e'') e^g X cos g 



-\- etc. -\- etc. 



It appears, therefore, that all the perturbations in which the arguments contain 1 

 the mean longitudes of only two planets may be put in the form 



8v = (v-cO) -\- (v.cA) cos^ -f- {v.c.2) cos 2g -j- etc, 



-f- (v.sA) sing -\- (v.s.2) sin 2g -\- etc. 



if5p = (p.c.O) 4- (p.^'.l)cos(7+(p.c.2)cos2(/4-etc. 



-\- (p.-s.l) sin^-[- (p-'''-2) sin 2j -\- etc. 



We have next to reduce to the same form those terms which contain the mean 

 longitudes of both Jupiter and Saturn, and which are given on page 78. We have 

 here twenty-four terms, each greater than O'.O-t. As most of these terms depend 

 on three independent arguments, they cannot be included in a double entry table, 

 while, if we include them as perturbations of the longitude in tables of single 

 entry, we shall have to enter twenty-two tables with as many different arguments. 

 But, by taking, for the argument A, the middle one in each series of arguments 

 Avhich depend on the same multiples of Jupiter and Saturn, and expressing the 

 terms above and below it in each series as coefficients of sin g, cos g, sin 2g, and 



