GENERAL EQUATIONS. 85 



and the corresponding quantity for Wj is 



w, p^ d cos i = 7 — ^, ^ ,, r^ ^ cos i 



Therefore the total moment of momentum about the 2-axis is 



M = — —-^"^d cos i-\-c,m,a,^o), cos i,-\-c^mna^o)^cos L (1) 



rrii + m^ i i i i i • 2 2 2 2 2 



The whole energy of the system is the kinetic energies of rotation, plus 

 the kinetic energies of revolution, plus the potential energy. 



The kinetic energy of rotation of m^ is ^c^m^a^coi, and there is a similar 

 expression for the kinetic energy of rotation of m^. 



The kinetic energy of revolution of m^ is 



\dt / ^ ^Xnii + m^/ ^Knii + m^/ \dt / 



and there is a similar expression for the kinetic energy of revolution Wz. 



The potential energy of the system is ^—^, where k^ is the gravita- 

 tion constant. '' 

 Therefore the total energy is 



_ . niiin- ,/,,,, m.m^ /drV k^m.m, , , , , , , .^^ 



^mj + Wj ^ m^ + m^Xdt J r '21111122222 \ / 



From the two-body problem we have 



r^^=±fc-^(mi + W2)a(l— e^) (3) 



the determination of the ambiguous sign depending upon the direction 

 of revolution, and 



■•"*'+ (i)'=*'('«'+'"')(7-i) (*) 



k-J irii + m^ 



(5) 



By means of (3), (4), and (5) equations (1) and (2) reduce to 



- +m,m2k^Ph ij—- 27zc{m,a,^ cos i^ 2Tzcim^a^ cos i^ 



(27r)Mm, + m3)4V^ e cost+ ^^ + -^^ (6) 



^ _ — m^i7ijc^ 27tc {in itti^ 27:02171^2^ 



7: ~ (27z)Hmi + m2)i P^ D^ 5? ^^ 



the sign of P' depending upon the direction of revolution, and the signs of 

 Dj and Dj upon the directions of rotation of m.^ and m^ respectively. 



