228 RESULTS OF RECENT RESEARCHES ON THE 
Accordingly we haye the system of fundamental equations : 
HH = y => ke => 2, plane of momentum, | 
1 : ! 
Pye pe at AA ee 
= y¥ + kz — —, surface of energy, r Me ed (4). 
9 9 a . 878 1 
oy = laws = il, quae OF mechiny. J 
These equations represent all possible interactions of the system, but in their present 
form are very difficult to interpret. The general problem to which they give rise 
seems to be insoluble, but we can solve and interpret them fully for one particular 
case which is in close accord with the conditions existing in nature; and it is possible to 
show by analogy that all other cases will be essentially similar to the one of which we 
shall treat. 
By taking the case of two equal stars rotating in the same direction with equal 
angular velocities, or substituting (3) of (4) in (1) of (4), we reduce the plaiie of momentum 
to a particular line of that plane : 
ae — He + (14 k) = a — Ae’ + 2 = 0, since & = 1. 
The equation of the energy surface passes into the form 
fo) 
see al 
Go 
The curve of rigidity becomes 
n = 24, wheren = Vy 4 #2 
ve 
Every point in the plane of momentum represents one configuration of the system, 2. é., 
one distance apart, one velocity of axial rotation, one moment of momentum of orbital 
motion. This point therefore determines the dynamic condition of the system, and by the 
motion of this point we may discover the changes which are taking place in any case that 
may be imagined. As we have restricted the plane of momentum to one line, the guiding 
point representing the configuration of the system will simply glide back and forth along 
this line. In the same manner the surface of energy is now restricted to a curve formed 
by cutting that surface by a certain plane; the guiding point that would slide along 
the energy surface is thus restricted to one line of the surface given by the transformed 
equation. [The reader who may desire to examine this question exhaustively must be 
referred to my Inaugural Dissertation, Die Hntwickelung des Doppelsternsysteme, Berlin, 
1893, R. Friedliinder & Sohn. | 
As the tides raised in the stars are subjected to frictional resistance, energy 1s 
