30 GENERAL INTEGRALS OF PLANETARY MOTION. 



to the centre of gravity of the system. The right-hand member of the last equa- 

 tion therefore becomes 



j = n 

 j=l 



Substituting for ^, 17, ^', and yj' their expressions (16), the constant term of this 

 expression becomes 



S'bh\ 



But if Ave add all the values of c, in (21), noting that by the form of the general 

 integrals we have 



*1 + *2 + *3 + + hn = 1 



j\+h-\-ji+ +y»3=o, 



we find, also, 

 and hence 



•^jCj = S'bh\ 



2(^)7' — ^'7)= 2c. 



Theorem V. — The constant part of the living force, which is itself equal to the 

 constant H in the integral of living forces, usually expressed in the form 



is represented by 



|(6iCi + 62C2 + + hn Oin\ 



as already shown in § 9. 



The constant part of II itself is therefore equal to 



The equality of H to the constant part of T may be shown by the preceding theory, 

 or it may be easily deduced directly from the theorem of living forces as shown by 

 Jacobi. {Voiiesungen ilber Dynamih, p. 29.) 



The conditions that the Lagrangian coefficients («„ IJ), the sum of the canonical 

 areolar velocities, and the difference between the potential and living force, are all 

 constant, give rise to a number of relations between the quantities ft. A;, and their 

 derivatives with respect to c, which I have not yet found of any use in the opera- 

 tions of integration. I therefore omit to cite them, especially as their complete 

 expressions are rather complex. 



The forms which we have been considering are those in which it would be 

 necessary to develop the expressions for co-ordinates of the planets, if we wished 

 these expressions to hold true for all time. The usual expressions are sufficiently 

 correct for a few centuries, but fail entirely when we extend the time beyond cer- 

 tain limits. But, in the case of the planetary system, we are obliged to adhere to 

 them for the reason that formulas developed in multiples of the 23* independent 

 arguments of that system would be unmanageable in practice. But, in the case 

 of the subsidiary systems, as the Tellurian and Jovian for instance, the secular 



* A linear relation of which we have not spoken must subsist between the quantities 61, 62, etc., 

 v/hich reduces the number of really independent arguments to 3)i^L 



