156 



THEORY OF JUPITER'S SATELLITES. 



Thus we have determined our assumed expressions so that they satisfy 

 the differential equations to a specified degree of approximation ; and since 

 they contain four arbitrary constants, viz. c, e, /3, y, they are competent to 

 express any initial conditions, subject only to the proviso that e is small. 



[We ought now to proceed and discuss the third coordinate which 

 was eliminated from our equations at the beginning, and deduce the motions 

 of node and apse. Adams has left no indication of the method he would 

 have adopted. The following method may be indicated : 



We have 



hence 



xy yx = 0, 

 (r z-)({> = constant, 



where <f> = EM, 



ENM is the equator, 



NP the orbit, 



PM perpendicular to JEM, 



E a fixed point. 

 Hence we find EM. 



& 



Again - = sin MP = sin i sin NP, 



where i is the angle PNM. 



We may take smi = c. Hence we find NP, and 



tan NM = cos i tan NP ; 

 and EM being known, this gives the position of the node at any time. 



Again p is the mean rate of separation of P from A, where A is the 

 apse of the orbit; and the mean motion of P itself is the non-periodic 



part of -T. (NP) + cosi-j-(EN). Hence we find the motion of the apse.] 



