Classification of modes of motion in “families” 545 
hereafter be density) which determines the motion exactly. In the 
particular case of the elliptic motion used for illustration the motion 
was stable, but other cases of motion might be adduced in which the 
motion would be unstable, and it would be found that classification 
in a family and specification by some measurable quantity would be 
equally applicable. 
A complex mechanical system may be capable of motion in several 
distinct modes or types, and the motions corresponding to each such 
type may be arranged as before in families. For the sake of simpli- 
city I will suppose that only two types are possible, so that there will 
ca 
Planet 
Fig. 1. 
A “family” of elliptic orbits with constant rotational momentum. 
only be two families ; and the rotational momentum is to be constant. 
The two types of motion will have certain features in common which 
we denote in a sort of shorthand by the letter A. Similarly the two 
types may be described as 4 +a and A +6, so that a and 6 denote 
the specific differences which discriminate the families from one 
another. Now following in imagination the family of the type 4 +a, 
let us begin with the case where the specific difference a is well 
marked. As we cast our eyes along the series forming the family, we 
find the difference a becoming less conspicuous. It gradually dwindles 
until it disappears ; beyond this point it either becomes reversed, or 
else the type has ceased to be a possible one. In our shorthand we 
D. 35 
