168 SECULAR VARIATIONS OF THE ELEMENTS OF 



Equations (408) and (409) will also become 



f AT m < AT' / m ' , AT" " m " i * 1 



=» ! {^+ j ^+M sinw+ < 3m); 



(539) 



(at m , ,t, , m' , , T „ , m" , } 



1 Nq \-N'qJ—+N"qJ— t +&c. \ 



{ na na ' na ' ) 



| JV 8 — +JV a -?^ 7 +&c. I cos (#H-/3 (0) ); 



(540) 



If we substitute in these equations the values of p , po\ & c > 2o? <7o'j & c -> given 



by equations (53 



), they will become 



(541) 



AT m , AT/ / 1^ , AT „ „ Wl" . p ) O 



Np \-Np i-Np U&c. Vcos« 4 



na na na ) 



AT m , A7/ / Ifl , AT// // Wl" , p ) • o 



Nq \-N'q UNq U&c. > sm « 4 



na na na ) 



N 2 —MN' 2 - 7 ^ T -\-N" 2 ^-„-\-&c. 1 am(gt4-0«»); 

 na na na j 



at vn , at/ i vn' , ato // m" , o 1 o 



Nq \-Nq \-Nd — -U&c. >cos/? 4 



na na no J 



JVp— +JV>^+iyy'^,-|-&c. 1 sin 5 

 na ' r«a ' w a j 



na na ' ' n a J 



j^ 2 — +i\T' 2 ^+i\r'' 2 — +&c. \ cos(gt4-3 m ). 

 na na' na ' J 



Now according to equations (410), the coefficient of iV 4 in this equation is equal 

 to nothing; and if we substitute the values of the coefficients of acos/3 4 , and a sin /3 4 , 

 which are given by equations (408) and (409), both members of equations (541) 

 and (542) will be divisible by the coefficients of a sin (gt-\-P m ), and a cos (gt-\-p {0) ), 

 and we shall find 



(542) 



Whence we get 

 Therefore 



sin (gt+p—pj=a sin (gt+p™); 1 

 cos (gt+P—pi)=a cos (gt+p m ). J 



tan (fft+p-p^tea (^+/3<°>) 



P^=p—p ii anda=l. 



(543) 



(544) 



It therefore follows that in order to apply our numbers to the invariable plane, 

 we have only to diminish the constants p, /3 1? /5 2 , &c, by the longitude of the 

 ascending node of that plane, on the fixed ecliptic of 1850, and neglect the con- 

 stant term. 



