THE ORlilT OF URA NUS. 3 



In his researches on the motion of Uranus, tlic first thing done by Le Vcrrier 

 was to recompute the perturbations by Jupiter and Saturn. It will sufficiently 

 describe his method of doing this to say that in the developments he used the 

 purely mechanical method for the action of Saturn, and the algebraic development 

 of the perturbative function for the action of Jupiter, while in the integration he 

 used the method of the variation of elements. After completing the perturbations 

 of the first order he made the earliest attempt at a complete determination of those 

 of the second order. Beginning with the terms of this order which arise from the 

 secular variations of the elements, he determines them by recomputing the terms 

 of the first order for the epoch 2300, and assuming that the general term will then 

 be given by interpolating between the two terms thus found, supposing them to 

 increase uniformly with the time. Tliis proceeding has the sanction of such high 

 authority that it is wortli Avliile to call attention to its want of rigor. The dif- 

 ferential coefficient of each element being given in the form 



da , 1 . 



^,- =z /u cos of, 

 dt 



k being a function of the elements, the perturbation of the first order will be 



/.; . , 

 (a = Y sm lit. 

 b 



When we take into account the variation of /.•, and suppose it of the form A-^ -|- k't, 

 the process is equivalent to supposing that in this case 



/.•„ + A'7 . 



ta = ' — sm bt, 



b 



whereas it really contains the additional term, 



j^ cos bt, 



which appears to be neglected in the process in question. It will be seen that the 

 neglected coefficient is equal to the secular variation of the term during the time 

 tliat its argument requires to increase by an amount equal to the unit radius. It 

 is therefore the more important the longer the period of the inequality. 



To obtain the periodic terms of the second order Le Verricr begins by determin- 

 ing the ten principal terms of the perturbations of the elements of Saturn produced 

 by Jupiter. Next he takes up tlie terms in the mean longitude of Uranus which 

 depend on the square of the mass of Saturn. The only sensible terms he finds are 



— 1".17 sin (^*— 30 — 0".35 cos ((,' — 3^') 



+ 0".43 sin ((f" — 4-" + 4^ — 0".'2l cos {^" — 4;" + 4^, 



^, ^', and }^" being the mean anomalies of Uranus, Saturn, and Jupiter, respectively. 

 The terms depending on the product of the masses of Jupiter and Saturn arc then 

 taken up. Fifteen arguments are found the coefficients of which vary from a 

 small fraction of a second to one or two seconds, while a single one of long period 

 amounts to 32". 



When the method of variation of elements is used, it is necessary not only to 

 determine these variations to qviantities of the second order, but, in the transforma- 



