characteristic Quantities occurring in Central Motions. 7 



we understand the complete variation of the ergal. That is to 

 say, the complete variation must contain not merely the differ- 

 ence which is conditioned by the difference of the coordinates, 

 but also that which arises from the change of the form of the 

 function, consequently from the change (for example) of certain 

 constants occurring in the function. If we designate these con- 

 stants by c, c v &c, and their changed values by c + hc, c, + Sc,, 

 &c, and will employ the universal-variation symbol SU, my 

 equation must be written thus : — 



5XJ _ ^ Be- ~ 8c, - &c. = | Sv 2 + m^S log i. (5 b) 



In order to gain a more convenient way of writing the equa- 

 tion, it will perhaps be advisable to* introduce a special symbol 

 for that part of the variation which relates only to the alteration 

 of the coordinates, e. g. to employ a 8 with an index, putting 



My equation then reads, 



— fji 



6\U= — &v 2 -\-mv 2 8\ogi*. . . . (5 c) 



3. Having made these preliminary remarks, we can now pro- 

 ceed to the discussion of central motions. 



When we refer the motion of a point about a fixed centre of 

 attraction to polar coordinates whose middle point coincides with 

 that centre, two different processes present themselves for our 

 consideration — the angular motion of the radius vector, and the 

 motion of the point in the radius vector. The latter, so far as 

 the whole motion generally is stationary, consists in alternate 

 approach toward the centre and removal from it. 



When the time expended in an approach and recession is equal 

 to the period of revolution of the radius vector, after each revo- 

 lution the moving point comes again to the same place, and from 

 here commences a new revolution in the same path, and conse- 

 quently we have a perpetual motion in a closed path. It is the 

 same when during one revolution any whole number whatever of 



* In my first memoir relative to this subject I have, it is true, replaced 

 the sum on the left-hand side of my equation by the simple symbol 8\J ; 

 but I have there expressly attached to the letter U a different signification 

 from that here given to it — namely, by saying there, " let U denote the 

 ergal for the original motion. The change in the form of the ergal which 

 enters with the transition from one motion to the other I then expressed 

 thus — by designating the ergal for the altered motion by V +/aV, in which 

 V denotes a second function of the coordinates, and /i an infinitely small 

 constant. 



