360 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [46 



Another cause which aggravates the effect of the former is the near 

 approach to commensurability in the mean motions. 



Twice the mean motion of Jupiter differs very little from five times 

 that of Saturn. In other words, five periods of Jupiter occupy nearly the 

 same time as two of Saturn, so that if at a given time the planets were in 

 conjunction at certain points in their orbits, then after three synodic periods 

 they would be again in conjunction at points not far removed from their 

 positions at starting. Hence, whatever uncompensated perturbations may 

 have been produced in the motions of the two planets during these three 

 synodic periods will be very nearly repeated in the next three synodic periods, 

 and again in the next three, and so on. 



Hence the disturbances will go on accumulating in the same direction 

 during many revolutions of the two planets, and will become very important. 

 The inequalities of long period thus arising will affect all the elements of 

 the orbits of the two planets ; but the most important are those which affect 

 the mean longitudes of the bodies, since these are proportional to the square 

 of the period of the inequalities, whereas the inequalities affecting the other 

 elements are proportional to the period itself. 



The principal terms of the inequalities of mean longitude are of the 

 third order, if we consider the eccentricities of the orbits and their mutual 

 inclination to be small quantities of the first order. 



Terms of the same period, however, and those far more numerous and 

 more complicated in expression, occur among those of the fifth and of the 

 seventh order of small quantities, and M. Le Verrier has included these 

 terms also in his approximations. 



But the circumstance which contributes in the highest degree to cause 

 the superior complexity of the theories of the larger planets is the necessity, 

 in their case, of taking into account the terms which depend on the squares 

 and higher powers of the disturbing forces. 



I will endeavour to point out the nature of these terms and the manner 

 in which they arise. 



By the theory of the variation of elements we are able to express at 

 any given time the rate of variation of any one of the elements in terms 

 of the mean longitudes and the elements of the orbits of the disturbed and 

 the several disturbing bodies. If this rate of variation were given in terms 

 of the time and known quantities, we should at once find the value of the 





