METEORS OF AUGUST AND NOVEMBER. 137 



dg-' = cos6 . dg- — -sinisin (/ — Si) • dz — - sin 6 sin z cos (/ — ^)[drt + du + dw — dJJ] 



cos I 



,22; d/ = — dS^— tan6cos(/— S2) .di H — -[drt + du + dw — d Si] 



^ cos** 



d6 = sin (/ — Q,) . dii -f- sini cos (^ — S2) [drt -\- Av + dw — d Q,~\ 

 Expressions which I do not recollect to have met with before. They differ 

 from the formulae given by Gauss, Theoria Motus, p. 49, and by Santini, Ele- 

 menti di Astronomia, Vol. I., Cap. XVII., Prob. IX. and X., and by other 

 writers on the theory of elliptic motion, in containing tangential directions and 

 velocities instead of orbital positions and distances, and having in the bracket an 

 additional variable, u, which does not enter into the expressions for the varia- 

 tions of the latter class, and which here introduces new relations between the 

 remaining elements ^ and n, of which it is a function. 



In order to obtain d g',Al, and d h, in terms of the variations only of the ele- 

 ments proper, we must substitute the values of d ^, d v, and d u, in terms of 

 those of the elements. From the Theoria Motus, p. 15, with small modifica- 

 tions, we have 



J « ^ ^ rj TT . ., J J n . f 2 4- e cos y) sin V J , 



av =. — cos Id if + ?, d /J — d 7t] + ^ — — . d <?> 



rr ^ ^ -* cos 4) ^ 



(23) 



dr=; -.d«-f- tan <^ sin ?; [d if -f- ^/d w — d 7t] — - cos cos w . d ^ 

 a G) 6> 



and from the equations 

 n = <2~' 



(24) 



p — a cos^ ^ . = g^ r^ sin^ '^ — ir — a) ^^ ^^^^ ^ 

 by means of differentiation, making', for conciseness, N' = , and put' 



ting for tan u, its value = tan (a — {n + v)\ 



dw = — -.-.da 

 2 a 



(25) dg=i- .da .dr 



2 aag rrg 



du = tan (^n — {n + v)^ ^^N'. — + N'ci.— — tB.n^.d^~\ 



expressions which, substituted in (22), would, after reduction, furnish the gene« 

 ral solution of the problem for all relations of an orbit to the ecliptic. 

 VIII. — 2 K 



