112 



SECULAR VARIATIONS OF THE ELEMENTS OF 



Np L Np -.—.4- Np -^,4- N p '— - 



na na n a n a 



&c. 



| ^ 2 ^_ +N >2 JjLiiV" 2 -^,+^'" 2 ~+ &c. 1 sin (^+/3) 

 (. na na na n a ' J 



na na na n a 



in , ,-r/ ; »«' , -kth ii m" 



NqJ^J r N'4^+N"4 , -^ i 4 r N'"ci' 



&c. 



= { N* —+N' 2 ^ 1 +N" i 4 J r,+^'" 2 4Kr+ &c. 1 cos (gt+B) 

 (. «a ' na ma n a J 



(408) 



!• (409) 



And since, for the root a 4 , we have N i =Ni=Ni=N i "', &c, the system of equations 

 similar to equations (133) will be divisible by JV* 4 iV 4 ' &c, we shall have the follow- 

 ing equations of condition : — 



m . m . , r „ m rn' 



N \-N —r-:-\-N" -jr-j.A-N'" - ir - r . 



na ii a n a n a 



in, lit, „,. no in 



&i \-Ni—+W—„+Hist-h 



na na na n a 



m „, rn m . m 



na ' na ' na ' n a 

 &c. 



&c.=0 

 &c.=0 

 &c.=0 



(410) 



Now putting the first members of equations (408) and (409), equal to x and y 

 respectively, and the coefficient of sin or cos of (gt-\-B), in the same equations 

 equal to z, we shall have 



x=z sin (gt+B) ; y= cos (gt-\-B) ; 1 

 whence tan (gt-\-B)=x^-y. J 



(411) 



6. In order to find the values of x and y, which are to be used in equations (411), 

 we shall assume the following values of <p, q>, q>", &c, 6, 6', 6", &c, corresponding 

 to the beginning of the year 1850, at which epoch <=0. 



Mercury, 



<p -7° 



0' 



8".2, 



6 = 46° 



33' 



3".2 



Venus, 



$ =3 



23 



34.4, 



0' = 75 



20 



42.9 



The Earth, 



4>" =0 







0.0, 



6" = 











Mars, 



<?>'" =1 



51 



2.3, 



0"' = 48 



23 



36.8 



Jupiter, 



<^=1 



18 



40.3, 



/r = 98 



54 



20.5 



Saturn, 



<?/ =2 



29 



22.4, 



r =112 



19 



20.6 



Uranus, 



$"=0 



46 



29.9, 



e TI = 73 



14 



14.4 



Neptune, 



<p riI =l 



47 



0.9, 



rJJ =13O 



7 



45.3, 



Now, since 2=tan <p cos 6, and p=tan ty sin 6, we find, 



p =+0.0891690, log.jp =8.9502138; 



p' =+0.0573577, log./ =8.7585915; 



p" = 0. log. p" = — oo 



p'" =+0.0241597, log./" =8.3830911; 



