262 DETERMINATION OF AN ORBIT PROM [BoOK II. 



omitted one or inconsistent with it, or, which amounts to the same thing, when 

 it is possible to assign a linear function 



aP 4- @ _j_ y # -{- $S 4- etc., 



which is identically either equal to zero, or, at least, free from all the unknown 

 quantities p, q, r, s, etc. Let us assume, therefore, 



P _|_ 4. yj{ _j_ #8 _|_ etc. = x. 



We at once have the identical equation 



(i, _|_ m) v 4. ( v r 4. w /) v 4- ( v &quot; _)_ m &quot;) v &quot; -f etc. = pP + q Q -\- rR -j- sS + etc. 

 If, accordingly, by the substitutions 



p = a x, q = (9 x, r = y x, s = d x, etc. 

 we suppose the functions v, v, v&quot;, to become respectively, 



m -j- &quot;L x, m -\- \ x, m&quot; - \- H x, etc., 

 we shall evidently have the identical equation 



(I I _|_ XT _|_ x&quot;X&quot; + etc.) xx (\m 4- I m -f l&quot;m&quot; etc.) * = x*, 

 that is, 



1 1 _|_ XT -f X T -f etc. = 0, x + X m 4- XV + X&quot; WZ &quot; + etc. = : 



hence it must follow that X = 0, X = 0, X&quot; = 0, etc. and also x = 0. Then it is 

 evident, that all the functions V, V V&quot;, are such that their values are not 

 changed, even if the quantities p, q, r, s, etc. receive any increments or decre 

 ments whatever, proportional to the numbers a, ft, y, d, etc. : but we have already 

 mentioned before, that cases of this kind, in which evidently the determination 

 of the unknown quantities would not be possible, even if the true values of the 

 functions V, V, V&quot;, etc., should be given, do not belong to this subject. 



Finally, we can easily reduce to the case here considered, all the others in 

 which the functions V, V, V&quot;, etc. are not linear. Letting, for instance, n, x, (, &amp;gt; 

 o, etc., denote approximate values of the unknown quantities jo, q, r, s, etc., (which 

 we shall easily obtain if at first we only use v of the p, equations V=M, V = M\ 

 V&quot; M&quot;, etc.), we will introduce in place of the unknown quantities the others, 

 /&amp;gt; q , r , s , etc., putting p = n -\-p, q = % -(- /, r = (&amp;gt; -f- r , s a -)- * , etc. : tin- 

 values of these new unknown quantities will evidently be so small that their 



