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SCIENCE 



[N. S. Vol. XXXIII. No. 8S8 



become smaller as the mean angular veloc- 

 ity increases negatively. It is true that we 

 are accustomed to express the small oscilla- 

 tions of a pendulum by means of angles 

 which increase uniformly with the time, 

 but it is the angle of oscillation which is so 

 expressed and this angle varies between 

 finite limits. 



I must apologize for insisting at such 

 length on an elementary illustration with 

 which you are all familiar. Though it 

 serves to make clear the types of devia- 

 tions which Jupiter may induce in the 

 motion of an asteroid, the analogy must 

 not be pushed too far. It fails to illustrate 

 what may happen to a planet when the 

 stage which corresponds to the rod at rest 

 near the lowest point of the circle has been 

 passed. And it takes no account of the 

 numerous short period changes which the 

 planet experiences and which can not be 

 altogether neglected. Moreover, forces 

 may be present which may tend to increase 

 the oscillatory motion, so that after a 

 period of hesitation the planet may be 

 compelled to settle down into a motion 

 which corresponds to a rotation of the rod 

 in the negative direction, and this inde- 

 pendently of the minute forces which may 

 determine its motion just after the velocity 

 has first become zero. 



These periodic changes, in which the 

 variable is constrained to remain between 

 finite limits, are generally known as libra- 

 tions. The familiar example furnished by 

 the motion of the moon relative to its 

 center of mass is of course a result of the 

 synchronism of its period of rotation about 

 its axis with that about the earth. The 

 delicate balancing of the conditions neces- 

 sary for a libration makes the term un- 

 usually appropriate in the cases of the mo- 

 tions of the asteroids. 



All the earlier discoveries indicated that 

 the smaller members of the solar system, 



which were not satellites of some planet, 

 occupied a more or less well-defined region 

 between the orbits of Mars and Jupiter. 

 But the distribution was by no means con- 

 tinuous. It was soon seen that if the mean 

 periods as then determined were arranged 

 in order of magnitude there were gaps in 

 the list which could not be explained by 

 the law of averages. It was, I believe, 

 Professor Kirkwood, of Indiana, who first 

 pointed out that these gaps existed at or 

 near the places where the periods are com- 

 mensurable with the period of Jupiter. 

 "With the increasing number of planets dis- 

 covered the discontinuities have become 

 more accentuated. In general, the smaller 

 the two whole numbers which represent 

 the ratio, the wider is the gap. It has been 

 known that such cases of commensurabil- 

 ity produced great difficulties in computa- 

 tion and that the ordinary methods failed. 

 The idea was this — near commensurability 

 of a forced period and a natural period 

 produces large deviations, growing larger 

 the more closely the whole number ratio 

 was approached. It was thought that when 

 the ratio became exact the deviation would 

 be infinite, or in physical language the 

 orbit was imstable. But the deviation, at 

 any rate so far as at present known, does 

 not tend to an infinite limit. There ap- 

 pears to be a maximum value which can 

 not be exceeded, and if the planet is so 

 started that this maximum is passed, the 

 ratio of the two periods becomes exact and 

 oscillations about this exact ratio occur. 

 The natural period becomes a forced 

 period. The analogy to the case of the rod 

 is evident. 



As early as 1812 Bessel had pointed out 

 that the periods of Jupiter and Pallas are 

 very nearly in the ratio of 7 to 18 and that 

 the attraction of Jupiter must maintain 

 this exactly, that is, the deviations from it 

 would be oscillatory and not secular. New- 



