GENERAL INTEGRALS OF PLANETARY MOTION. 27 



of the derivative of the constant term of the living force loith respect to its correspond- 

 ing canonical element. That is to say, if we represent the constant term of the living 

 force by 7, and suppose V to be expressed in terms of the canonical elements, we 

 shall have 



dV 



dV 



O^n = T, 



From the expressions (9) for x, and the corresponding expressions for y and z, it 

 will be seen that the expression for the relative living force is 





-\- etc. etc. etc. 



-\- corresponding terms in -/y' and ^'. 



Here the coefficients of ^', etc., are those which we have shown to form an ortho- 

 gonal system, and, by the properties of such a system, the expression reduces to 



Substituting for ^', )7', and ^'' their periodic expressions 



^ = — Sbk sin N 

 y( = Sbli cos ^V 

 ^' = SBli cos iV', 



the constant term of the living force is found to be 



the sign /S' having the signification given on page 12. Compare this expression 

 with that of C; in (21). Multiply each c^ by its corresponding 6^, and add all the 

 products, remembering that 



b =r ^\?;^ -[- i.^Jj -|- etc. for ^ and >7, and 

 * =./i^i + J2&2 + etc. for Z,. 

 We thus find, from the expression for Fjust given, 



2 Y= h,G^ + 6,c, + J3C3 + + h,,fi^, 



db- dh- 

 Diff"erentiating this expression with respect to C; and substituting — '- for — i 



dcj dCi 



we have 



nSy T , Sb: , dh , . dhi ,^r.N 



^Wr''-^'^ec+'-^d+ -^'-act- ^''^ 



We have now to show that b is a homogeneous tunction of the degree — 3 in 

 (ci, Co, C3,,). Let us represent such a function of the ntli degree by [c'"']. 



