264 DETERMINATION OF AN ORBIT FROM [BoOK 11. 



182. 



\ 



It will be necessary to preface the discussion concerning the degree of accu 

 racy to be assigned to the values of the unknown quantities, when there are sev 

 eral, with a more careful consideration of the function v v -j- v v -j- v&quot;v&quot; -f- etc., 

 which we will denote by W. 



I. Let us put 



, AW 



etc., 

 u f 



also 



(t 



and it is evident that we have p = P, and, since 



AW _ AW 2/d/ 



dp dp a dp 



that the function W is independent of p. The coefficient a = aa-\-a a -\-a&quot;a&quot;-\- 

 etc. will evidently always be a positive quantity. 

 II. In the same manner we will put 



also 



and we shall have 



, i AW p Ap B , , AW&quot; 



q 5 -, *--= Q *- n and - r - = 0, 



Aq a Aq a 1 Aq 



whence it is evident that the function W&quot; is independent both of p and q. 

 This would not be so if ft could become equal to zero. But it is evident 

 that W is derived from vv-\- v v -\- v&quot;v&quot; -\- etc., the quantity p being eliminated 

 from v, v , v&quot;, etc., by means of the equation p = ; hence, ft will be the sum of 

 the coefficients of qq in vv, v v , v&quot;v&quot;, etc., after the elimination; each of these 

 coefficients, in fact, is a square, nor can all vanish at once, except in the case 

 excluded above, in which the unknown quantities remain indeterminate. Thus 

 it is evident that ft must be a positive quantity. 



