﻿A NEW METHOD FOR CORRECTING A PLANET's ORBIT. 463 



7. The first of these ■will be 8 w, such that if w° = 0, 



r = r c"-"° = r c* >", (6) 



c being the base of the Napierian system of logarithms. 



By its aid, and that of s, we can take account of all the variations of the elements 

 which give the dimensions, &c. of the orbit, but not of changes of the orbit-plane 

 itself; and s may be considered as the time when the planet would be in the same 

 heliocentric longitude (modified as before, like 7t), as it really is, if its elements were 

 M°, &c. The other variation, S w, may be defined as the change necessary to adapt 

 the natural logarithm of r, calculated by means of the approximate elements and for 

 the time z, to its true value for the time t. 



8. It will be noticed that we have so far adopted a similar course, in representing the 

 effect of changes in the elements, to that which Mr. Hansen has employed in his theory 

 of perturbations. Our mode of expressing it is taken, with the necessary changes, 

 from Prof. Zech's article on Hansen's Method, Vol. XL. of the Astronomische Nachrich- 

 ten, reproduced by Prof Encke, Mathematische Ahhaiidlungen der Berliner Akademie 

 for 1855, p. 39. 



9. The orbit-plane will be made to assume a new position by the change of SI and i ; 

 and the planet will be at a distance 8 Z from its position in the unchanged plane ; the 

 axis of Z being perpendicular to the orbit. 



10. Our first work will be to compute the partial difierentials of z, w, Z with respect 

 to the elements. It may be noticed, however, that these quantities, z, w, Z, enjoy but 

 a temporary existence as functions of the elements ; they correspond to nothing 

 actually existent. We shall, for convenience, omit the °. 



11. By the Theoria Motus, article 15, p. 15, 



, a a cos (I , ,,r , , , a a (»• + p) . _ , 

 dv = dCM-\- at) -\- - > ^" sin E 6? o) ; 



Hence, dA=^ aa os ^ j^ — ^_|_„A_| ^-rP sinE da, 4- dx; 



Letting now L — L, ^ tt — ;f , and L, — M ^ ;f , there will result 



, a a cos qo,,-r i \iQ1''-4"?'.t-i7 it /m\ 



dyi=—^d(L,—x + fit)-\--^^&mEdcp-\-dx. (7) 



From (5) we get dA = 8z~; (8) 



and as z° = f and 8 z is very small, we can assume -^- = -^— . So will 



' az at 



^ d A ,„. 



5J = 5z-j-^. (9) 



