28 GENERAL INTEGRALS OF PLANETARY MOTION. 



Let us represent the linear elements of the system by aj, a^, etc. Smce x, y, z, and 

 ^, 57, ^', are all linear co-ordinates, we have in the expressions (16) of the latter 



/. = [«">]. 

 Every time we differentiate these expressions with respect to the time, we 

 multiply the coefficients by b, a linear function of b^, 60, etc. Hence 



The form of the potential D. shows that 



a result which arises from the law of attraction proportional to the inverse square 

 of the distance. Whence 



di; 



cP^ SO. 

 In order that the differential equation — | = — may be satisfied identically we 



at- d^ 



must have 



or 



j(2) ^ [-„(-3)-[ oj, J ^ [a<-3)]. 



The expression (21) for g„ h being linear in a, is of the form 



Hence, when wfe express bi in terms of Cj, Co, etc., we must have 

 The fundamental property of homogeneous functions now gives 



Substituting in (46), we find 



^j^j^=-^h. 



which is the theorem enunciated. 



This theorem cannot be directly employed to obtain the values of 6;, for the 

 reason that V cannot be determined as a function of the canonical constants until 

 the equations of motion are completely integrated. 



§ 9. Siimmary of Results. 



The following is a brief summary of some of the results which follow from the 

 preceding investigation. 



We first suppose that we have found expressions for ^, r, and ^ of the form (12), 

 such as identically satisfy the differential equations (11). We also conceive the 



