888 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
In whatever manner the two bodies may interact on one another, the resultant 
moment of momentum h must remain constant, and therefore (a) will always give one 
relation between n, x, and 77 ; a second relation would be given by a knowledge of the 
nature of the interaction between the two bodies. 
The equation (a) might be illustrated by taking n, x, 77 as the three rectangular 
co-ordinates of a point, and the resulting surface might be called the surface of 
momentum, in analogy with the “line of momentum” in the above paper. 
This surface is obviously a hyperboloid, which cuts the plane of nx in the straight 
line n-\-x=h; the planes of n-q and 77 = 1 in the straight line determined by n = h ; and 
the plane of xrj in the rectangular hyperbola x(l —r))=h. 
The contour lines of this surface for various values of n are a family of rectangular 
hyperbolas with common asymptotes, viz. : 77=1 and r= 0 . It does not however 
seem worth while to give a figure of them. 
If the satellite raises frictional tides of any kind in the planet, the system is non¬ 
conservative of energy, and therefore in equation (J3) x and 77 must so vary that z 
may always diminish. 
Suppose that equation (/3) be represented by a surface the points .on which have 
co-ordinates x, rj, z, and suppose that the axis of 2 be vertical. Then each point on 
the surface represents by the co-ordinates x and 77 one configuration of the system, 
with given moment of momentum h. Then since the energy must diminish, it follows 
that the point which represents the configuration of the system must always move 
down hill. To determine the exact path pursued by the point it would be necessary 
to take into consideration the nature of the frictional tides which are being raised by 
the satellite. 
I will now consider the nature of the surface of energy. 
It is clear that it is only necessary to consider positive values of 77 lying between 
zero and unity, because values of 77 greater than unity correspond to a hyperbolic 
orbit; and the more interesting part of the surface is that for which 77 is a pretty 
small fraction. 
The curves, formed on the surface by the intersection of vertical planes parallel to x, 
have maxima and minima points determined by clz/dx=0. 
This condition gives by differentiation of (fi) 
x * 
x 
I-77 (I-7?) 2 
(7) 
From the considerations adduced in previous papers, namely, those in the Proc. Roy. 
Soc., No. 197, 1879, and No. 202 , 1880, it follows that this equation has either two 
real roots or no real roots. 
When 77=0 the equation has real roots provided h be greater than 4/3% and since 
this case corresponds to that of all but one of the satellites of the solar system, I shall 
