82 THE ORBIT OP URANUS. 



In general, a series of terms of the form 



2a , sin (iA -\- s<j~) -\- 2 b t cos (iA -f- sg) 

 -\- 2a', sin (iA sg) -{- 2 b' t cos (i A sg), 



may be put in the form 



\ 2 ( t ',) cos iA 2 (b t J',) sin iA \ sin sg 

 + i 2 (, -(- a' t ) sin i^l -|- 2 (&, + &'<) sin iA \ cos #. 



All the periodic terms containing only g and 7 in the arguments may be put into 

 this form by taking 



Ag l, 



so that the coefficients of sin sy 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, 



(2 e " ~6~ e ') ^ x sin 2 ^ + (2 e ~ 

 etc - + etc. 



2 ~ F2 e3 ei7 X COS 



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

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



v = (w.e.O) -j- (tf.e.l) cos# -(- (w.c.2) cos 2g -J- etc. 



-f- (?7.s.l) sin^r -{- (t?.s.2) sin 2^ -f etc. 



= (p.c.O) -)- (p.c.l) cos gF + ( p.c.2) cos 2r/ -f- etc. 



4- (p.s.l) sin# -f- (p.s.2) sin 2j -f 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 0".04. 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 

 which 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 



