GENERAL INTEGRALS OF PLANETARY MOTION. 29 



quantities h and 6 as expressed in terms of 3« canonical constants Ci, c.^,c.^ 



C3„, so chosen that the expression 



(c;, 4)= 2 < — ^ — Jil^^^A L'_!J — etc. > 



^ '' '' «= 1 1 (9c^. dh <5c, o7,. ^ ^c,. (54 i 



shall reduce to unity when Ic =y, and shall vanish whenever any other of the 6?i 

 quantities q Cim ^\ hn is substituted for 4. Then : — 



Theorem I. — If, taking the entire series of ^n co-ordinates represented by 



^1 ^«5 >7i yim Kx ^7n we multiply the square of each coefficient 



fc by the coefficient of the time in the corresponding angle i^^ -f i^.^ -f etc. (that 

 is, by the corresponding quantity ^>l -|- {362 + etc., o\ j-p^ ^jih + etc.), and by the 

 coefficient ij or _;} of any one of the ;i's, as X^-, which 1 is to be the same throughout, 

 then all tlie constants c, except c,-, will identically disappear from the sum of all 

 these products, which sum will reduce identically to 2cj. This theorem is expressed 

 in equation (21). 



Theorem 11. — The 3n coefficients of the time, 61, h.^, etc., considered as fimctions 



of q, C2, etc., fulfil the o conditions expressed by 



where i andy may have any values at pleasure from 1 to 3tc. They are therefore 

 all the partial derivatives of some one function of Cj, Cj c^^. 



Theorem TIL — This function is the negative of the constant term of the expres- 

 sion for the living force in terms of Cj, Cj, etc., as shown in the last section. 



Theorem IV. — The sum of the canonical elements q, Cj c^^ is equal to the 



" constant of areas," this constant being either the sum of the canonical areolar 

 velocities on the plane of XY, or, which is the same, the sum of the products ob- 

 tained by multiplying the actual areolar velocity of each body around any point, 

 fixed with reference to the centre of gravity of the system, by the mass of the body. 



This theorem is demonstrated as follows : The sum 



i = n 

 1 = 



is known to be a constant by tlie principle of conservation of areas. From the ex- 

 pression (9) for Xj, and the corresponding expression for y^, introducing the quantity 

 ttoi as in (8), we have 



(^y, - x',2/,) ='i"'i"^'(,v. - r.%) ; 



j = ok=o m^i 

 multiplying by m,-, and then summing with respect to i, we have 



j = /l = I i = m^ ) 



By the condition of the orthogonal system (8) the sum in brackets vanishes when- 

 ever y is different from k, and becomes unity when these indices are equal. More- 

 over in (5) ^'0 and r^ vanish whenever the origin of co-ordinates is fixed relatively 



