334 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 



planets it is only in special cases that terms depending on the square of 

 the disturbing force need be taken into account, and it is scarcely ever 

 necessary to consider terms of the next order of approximation. 



In the case of the Moon, however, it would be necessary to repeat the 

 process of approximation at least four or five times, in order to obtain 

 results of the accuracy required in the present state of the theory. If we 

 consider that the disturbing function consists of a great number of terms, 

 and that each term gives rise to a corresponding term in the value of 

 each of the disturbed elements, while powers and products of the corrections 

 of all the elements in every possible combination, up to a certain order, 

 have to be taken into account, it may be readily imagined how impracticable 

 it would be by such a process to carry on the approximation to a greater 

 extent than has been already done by Plana. Every process in which the 

 approximations require to be repeated several times, is subject to the 

 inconveniences that have been described, and these inconveniences are much 

 greater when, as in the present case, we have to make successive approxi- 

 mations to the values of the six elements of the orbit, instead of to the 

 values of the three coordinates of the Moon. 



It was with the view of avoiding this excessive complication of the 

 method of successive approximations that M. Delaunay devised his method 

 of integrating the differential equations of the Moon's motion. The funda- 

 mental idea of this method consists in attacking the difficulty by small 

 portions at a time, and in replacing these extremely complicated successive 

 approximations by a much greater number of distinct operations, each of 

 which is comparatively simple, so that it may be carried out to any degree 

 of exactness that may be desirable, while the mind is relieved by being able 

 readily to embrace the whole of each operation in one view. 



It is difficult, without the use of algebraical symbols to give an idea 

 of M. Delaunay's beautiful method, but I must endeavour, in some measure, 

 to fulfil this task, and I must crave your indulgence should I fail in the 

 attempt. 



The theory of the variation of the arbitrary constants gives, as is well 

 known, the differential coefficients of the elliptic elements with respect to the 

 time, in terms of the elements themselves and the partial differential 

 coefficients of a certain function, called the Disturbing Function, taken with 

 respect to those elements. By a proper choice of elements, the differential 

 equations may be reduced to their simplest, or to what is called their 

 canonical form. In this form the six elements are divided into three pairs, 



