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SCIENCE 



[N. S. Vol. XXXIII. No. 838 



the most minute consequences of the laws 

 under which they run their courses. Their 

 orbits differ but little from circles, and for 

 practical purposes the series of terms 

 which represent their motions are easily 

 obtained, even if the process of so doing 

 may sometimes be a lengthy one. Far 

 otherwise is it with the asteroids. Many 

 of their orbits are highly eccentric and in- 

 clined at large angles to the plane near to 

 which the large planets circulate. But the 

 most interesting problems are those which 

 arise from the near presence of the second 

 largest member of the solar system, Jup- 

 iter, whose mass is but little less than one 

 thousandth that of the sun. 



Jupiter, great as it is compared with the 

 other planets, is yet small relatively to the 

 central body which mainly controls their 

 movements. In general its average effect 

 on a body which does not come very near 

 to it must be small. Under certain cir- 

 cumstances, it may cause very considerable 

 deviations for a time, even in planets which 

 do not approach it very closely. Astron- 

 omers have generally divided the disturb- 

 ances of the motion of one body produced 

 by another into two classes, periodic and 

 secular. The latter are properly those 

 which change the motion always in the 

 same sense so that they would ultimately 

 cause a change to another type of motion. 

 As a matter of fact, however, the division 

 is arbitrary and in a strict sense inaccu- 

 rate. So far as we know, all the disturb- 

 ances produced in our solar system by 

 gravitation are really periodic, but some of 

 the periods are so long, extending to thou- 

 sands and ten thoiisands of years, that it is 

 more convenient in the short space of time 

 during which we desire to know the motion 

 to treat certain of them as secular. 



These long period deviations are often 

 of very considerable extent. They may be 

 divided into two classes, "proper" and 



' ' accidental. ' ' In the former all the bodies 

 which have the same type of motion have 

 the long period terms, for instance, the 

 slow oscillations of the eccentricities of aU 

 the planets due to the attractions of one 

 on another are of this type and they arise 

 principally because the masses of the 

 planets are small , compared with the mass 

 of the sun. 



The accidental terms are those arising 

 from a synchronism of periods. They 

 should be regarded perhaps as a result of 

 the defects of our mode of representing 

 the motion in symbolic form. However 

 this may be, their presence causes a real 

 practical difficulty which must be solved. 

 We regard the motion of each body as 

 having a principal period of revolution 

 round the sun in a circle, with deviations 

 from circular motion due to the eccentri- 

 city of the orbit, these deviations having 

 periods which are multiples of the princi- 

 pal period. Thus there will always be 

 some period in the motion of one planet 

 very near to a period in the motion of 

 another planet. When the degree of ap- 

 proximation we require is settled there will 

 only be a limited number of accidental cor- 

 responding periods. If the difference be- 

 tween the two periods is small, a term of 

 long period arises, and the amplitude of 

 the oscillation is nearly proportional to the 

 long period (or to the period squared) and 

 it might seem that when the period became 

 infinitely great the amplitude would also 

 tend to infinity. In physical terms the mo- 

 tion would be unstable. It is not so in 

 general. When the amplitude begins to 

 approach very large values the motion may 

 still be stable. If it is, one of two things 

 has occurred. Either the difficulty is a 

 symbolic one, that is, our mode of repre- 

 senting the motion is defective for large 

 oscillations, and the difficulty can be 

 bridged by choosing some other analjrtical 



