1010 



of the planet's coordinates are found at once by the integration of 

 the proper differential equations. But this method, though perhaps 

 preferable to the former for calculating the inequalities depending 

 on the first power of the disturbing force, is inapplicable, or at any 

 rate seems as yet not to have been applied to the calculation of those 

 which depend on the square and higher powers of that force, so 

 that it cannot be regarded as affording a complete solution of the 

 problem. 



In Hansen's method, the perturbations are applied to the mean 

 longitude and to the logarithm of the radius vector. The disturbed 

 mean longitude, combined with invariahle elements, gives the true 

 longitude in the orbit, and the logarithm of the true radius vector 

 is found by adding its perturbations to that value which corre- 

 sponds with the disturbed mean longitude and the same invariable 

 elements as before. The perturbations of latitude are determined 

 in a manner more analogous to the ordinary method of variation of 

 elements. Thus, with respect to the longitude and radius vector, 

 Hansen avoids the inconvenience of having to calculate the pertur- 

 bations of twice as many quantities as are finally wanted, while at 

 the same time his formulae take into account all powers of the 

 disturbing force. 



Hansen finds also that the series which expresses the perturba- 

 tions of the mean longitude is more convergent than that in ordinary 

 use which gives the perturbations of the true longitude ; and Mr. 

 Adams found this to be the case in his investigation of the disturb- 

 ances of Uranus, the use of the perturbations of the mean longitude, 

 instead of those of the true, having been attended with considerable 

 advantage. 



Astronomers have long seen the convenience of applying the in- 

 equalities of long period to the mean longitude of the planet, and 

 similar advantages, though not of so marked a character, follow from 

 applying all the inequalities in the same manner. 



Only a part of Hansen's investigations respecting the perturba- 

 tions of bodies moving in orbits of great excentricity and inclination 

 has yet appeared. In this part he treats of the case where the di- 

 stance of the disturbed body from the sun is always less than that 

 of the disturbing. This is perhaps the most important case, as it 

 includes the disturbances of the minor planets and of Encke's comet 

 produced by Jupiter, and the planets exterior to it. 



An example is given of the application of the formulas to the 

 disturbances of Encke's comet caused by Saturn. In this case the 

 method succeeds perfectly, and there is no doubt of its applicability 

 when the distance of the disturbed and disturbing bodies from the 

 sun never become very nearly equal to each other. If these distances 

 ever approach very closely to equality, or if one of them is sometimes 

 greater and sometimes less than the other, the calculations become 

 much more complicated. Hansen's lunar theory, " Fundamenta 

 nova investigationis orbitae lunag," contains the most complete view 

 of the principles of the method first adverted to above ; but the 

 numerical results of his formulae have not yet been published. 



