122 PROCEEDINGS OF THE AMERICAN ACADEMY 



which, in the solar system, unites with the former to carry the planet in 

 a continuous curve around its centre of motion. If the force of projec- 

 tion be ever so small, the planet will move in a curve, however elon- 

 gated, and change its direction gradually, though it may be with all 

 the rapidity of the comet shooting through its perihelion. When the 

 projectile force is nothing, the motion is rectilinear, and the direction 

 alters abruptly. Here, also, the case is made easy, and the authority 

 of the law of continuity vindicated. For in this instance, as in all 

 others where motion, and not simply directive power, is considered, the 

 velocity gradually diminishes, and prepares the way for a new motion 

 in the opposite direction. 



" It is well known, that sometimes the law ef the forces of nature 

 changes once or more in passing from one condition of nature to 

 another continuously connected with it. Thus the attraction of a solid 

 sphere is as the square of the distance from the centre inversely, so 

 long as the attracted body is on the outside. When the attracted body 

 comes within the surface, the attraction is directly as the distance from 

 the centre. In the case of a hollow shell, the law of its attraction 

 changes more than once. Within the shell, the attraction is con- 

 stant for all positions. Outside, it obeys the same law as in a solid 

 sphere. In the thickness itself, the attraction is subject to a third 

 law. The centre of gravity of the attracted body will pass abrupt- 

 ly from one to another of these three conditions ; but it is not always 

 safe to represent the whole body by its centre of gravity. As the 

 small body is passing through the surfaces of the large one, nei- 

 ther of the three laws stated above is applicable. Probably no sin- 

 gle law will follow the body through the various positions involved in 

 the entering of one of the bodies into the other. The law itself prob- 

 ably changes every instant, and thus the three partial laws, which are 

 so discontinuous, and which are derived from a consideration of only 

 the centre of gravity, will appear to be continuously connected when 

 those links which are neglected when we study the motions wholly 

 through the centre of gravity are restored. The mathematical function 

 itself, therefore, if made so general as to include all the conditions of 

 the experiment, might possibly be continuous from first to last. At 

 any rate, if we give full weight to this apparent breach of continuity in 

 the present mathematical expression of the law of attraction, it by no 

 means follows that the body which is attracted and passes into these 

 various exposures will change its velocity abruptly, as it comes under 



