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 



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 A + a and A + b, so that a and b denote 

 the specific differences which discriminate the families from one 

 another. Now following in imagination the family of the type A + 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 



