GENERAL INTEGRALS OF PLANETARY MOTION. 19 



§ 6. Development of £1, £Lj, and n'j. 



We have next to find the forms of the expressions D.j and Q.'j which enter into 

 the equations (23) and (24). In the first place' we have 



V(x, — xjf + (y, - yjf + {z, - zjf 



We now substitute for x, y, and z their expressions (9) as linear functions of ^, 

 >7, and ^ respectively. By this substitution we shall introduce no terms of the form 

 ^y;, yj^, or ^^. Hence, when we substitute for ^, yj, and ^, their expressions in infinite 

 periodic series, the reduced expressions will contain cosines only. In fact, using 

 the forms 



^j = SJCi cos N 

 Yii = SJCi sin N 

 ^i = Sk\ sin N', 



we shall have from (12) when we put for brevity 



/an _ a^\^,^ j^(^_^ \^.^ ^ etc. . . . =Jcu, 



Xi — ccj = Sliy cos iV ; 



y^ — %. = Shi sin N; (32) 



Zj — 2,- = S'Uij sin N'. 



Each denominator in D, will therefore assume the form 



i/ (M^coTi^T^P^M^lvT+XM'surFp. 



When we form these three squares we find that every term of the form li cos 

 (iV^-|-iVv) in the first square is destroyed by a corresponding term — h cos (iV^-|- N^ 

 in the,second square. Hence the sum of these two squares will only contain terms 

 of the form 



h cos {N^ — iVv). 



Since in each value (15) of iVwe have 



\ + 'k + % + + izn = 1, 



we shall have in iV^ — N, 



%i = 0. 



Also, since in N' the sum of these coefficients is zero, it follows that the same 

 thing will hold true of the third of the preceding squares. The denominator in 

 question may therefore be expressed in the form 



VSTc cos, N, 

 in which each N is of the form 



where 



