502 
MR. G. H. DARWIN ON A PLANET 
would however be hardly repaid by the results, because most of the interesting 
general conclusions may be drawn from the case of two satellites, where we have only 
to deal with curves. 
If the case of a single satellite be considered, we see that the energy locus is a curve, 
and the transit along the steepest path degenerates merely into travelling down hill. 
Now as the slopes of the energy curve are not altered in direction, but merely in 
steepness, by taking the abscissas of points on the curve as any power of £ the solution 
may still be obtained if we take x (or ^) as the abscissa instead of £ This reduces 
the solution to exactly that which was given in a previous paper, where the graphical 
method was applied to the case of a single satellite.* 
r 
§ 5. The graphical method in the case of two satellites. 
I now return to the special case in which there are only two satellites. The equation 
to the surface of energy is given in (21). The maxima and minima values of z (if any) 
are given by equating bz/bx and bz/by to zero. This gives 
fl — K]X 7 — K.f = 
h—Kpd—Kgf= 
K i & l 
\ 1 j 
K 2 f J 
(24) 
By (15) and (19) we see that these equations may be written 
They also lead to the equations 
(ad) 
it — 12 , 
n—il u 
1\4 ^ 1 i A _a ! 
(25) 
-(ad) 3 -p -1=0 
!- 
(2/ i ) 1 -k~(r) 3 +k=° ! 
. . . (26) 
Kr, 
Now an equation of the form Y 4 -'—a H 3 -{-/3=0 may be written 
(Y/3~f —a/3 - *(T/3 -l ) 3 +l = 0- And I have proved in a previous papert that an 
equation ad— hx 5j r 1 = 0 has two real roots, if h be greater than 4/3*, but has no real 
roots if h be less than 4/3*. Hence it follows that this equation in Y has two real 
roots, if a be greater than 4/T/3*, but no real roots if it be less. 
Then if we consider the two equations (26) as biquadratics for ad and f respectively, 
we see that the first has, or has not, a pair of real roots, according as 
* Pi’oc. Roy. Soc., No. 197, 1879. 
t Ibid., No. 202, 1880, p. 260-263 
