CASE XII. 



117 



XII. CASE iz^O i^d^O a2 = e = S = 0. 



If i is to be taken different from zero and subject to change it is neces- 

 sary to suppose that i^ may vary also, for all the interactions are mutual. 

 In this case equations (9') become 



M' =P'i sin i sin Q, -f 



mj sin ii sin Qi 



D 



_ ... „i • • o . ^^1 sin i. cos ^i 

 M" = P^ sin I cos Q +^ jz 



(114) 



We may choose the xy-plane so that it shall coincide with the invariable 

 plane of the system. Then M' = M" = 0, and equations (114) give 



Q>i = Q-{-n7: (n=Oorl) 



With this relation equations (114) become 



[P^sint± 

 whence 



m, sin i.l . ^ _ r„. . . Wj sin in ^^ 



— ^-yr — ^ sin 86=0 P*sini±— i-yc — ^ cos 8^=0 



P* sm I ± — ^-y^ — * =0 



(115) 



In this discussion we shall take D as essentially positive and let i and t^ 

 vary from to z. Then we must take the lower of the two signs in (115), 

 or Q>i=Q-\-7t. This equation and equations (9) and (10), under the 

 hypotheses of this section, give us 



= P* sin I ^-yr — - 



n^ Til • , ^1 cos t, 

 M =Pi cos * + ~-v^ — - 



7t ~ Pi '^D^ 



(116) 



Eliminating D and tj from the last of these equations by means of the 

 first two, we have 



^ = -^ + P^-2McostP^+M2 



TT P' 



The conditions for a maximum or minimum of E considered as a func- 

 tion of i and P are 



-^ -— =2MP* sm t =0 



TT 01 



rn^dE _2^^ , 2 J^_2 M cos i _ 

 li'dP~ZP^^^P^~^ Pi 



