388 
note on Lambert’s theorem for elliptic motion. 
[472 
there is no longer any real ambiguity; and it must be possible to decide between 
the two values of x '• the criterion is, in fact, a very simple one, viz. drawing a chord 
from A through the other focus H of the ellipse, this either separates, or it does not 
separate, B from the force-focus S', and I say that in the expression of Time AB, 
in the former case (viz. when chord A is a separator) we have % < 180 ; in the latter 
case (viz. when chord A is not a separator) we have x > 180°. 
It of course follows that, in the case of transition, when the line AB passes 
through H, we must have % = 180°: this is at once seen to be so; for x = 180° gives 
the condition 4<x = p + p + c; but if cr, a, are the distances of AB from the focus H, 
then 2a = p + &, 2a = p' + a-', and the condition becomes cr -f- cr' = c; that is AB must 
pass through H. 
As a verification of the new criterion, I consider the point A as having a fixed 
position on the orbit, but the point B as having successively different positions; and 
writing down the two formulae 
Time AB = X ~ X ~ sin x + sin x, 
Time BA = co — co' — sin co -f- sin co', 
(where for simplicity the constant factor P 4- 2tt is omitted) I proceed to compare 
these for different positions of the point B. We have, in every case, cos co — cos x, and 
cos co' = cos x ; whence (%, co being each positive and less than 360°) co — x or else 
60 + % = 360°, viz. the former equation subsists if co, x> are eac ^ ^ ess or eac ^ greater 
than 180°, the latter if the one is greater, the other less than 180°. And again 
(x, & being each less than +180°) we have co' = x', or else co'=~x', according as 
co', x have the same or opposite signs. 
A 
Now in the figure, suppose that B occupies successively the different positions 
Pi, B 2 ,...B 5 , the criteria for x> X ( or "> as follows. 
Ch. A. 
Ch. B. 
therefore 
z AS 
z BSA 
1 
sep. 
not 
X< 
180° « > 180° 
or CO + X = 
2vr 
<180° 
> 180° X ' 
= + 
co' = — 
or 
/ 
co — 
-x> 
2 
sep. 
sep. 
< 
< 
» a) = X 
< 
< 
+ 
+ 
/ 
CO = 
x> 
3 
not 
not 
> 
> 
» Ù) = X 
> 
> 
+ 
+ 
>> 
/ 
CO = 
x> 
4 
not 
not 
> 
> 
» 6> = X 
> 
> 
- 
- 
}) 
/ 
CO = 
x> 
5 
not 
sep. 
> 
< 
» co + x= : 
27r 
> 
< 
- 
- 
/ 
CO = 
~x- 
o