136 RESEARCHES CONCERNING THE PERIODICAL 



d y, and d z, for variations of y, /I, and /?, may be useful, and that is in enabling 

 us to estimate the probable errors of any system of elements for a group or 

 cluster of these meteors derived from assumed values of y, /I, and /?. The 

 formulae for computing variations of elements for a change of radial positions 

 and distances in an orbit, or of geocentric positions and distances in the heavens, 

 are stated at length by writers on the laws of elliptic motion, Gauss, Littrow, 

 Santini, and others. As I do not recollect to have met with similar expres- 

 sions for the variations of tangential directions and velocities, or relative direc- 

 rections and velocities, I shall here point out the method employed in preparing 

 the requisite formulae. 



In the formulae (S), making y' = y cos ^, g' = g' cos b, and differentiating, 

 we obtain, after making the requisite reductions, 



d^' = + cos (;i — /) cos/3, dy — ^ sin {?i — l).d?u — ^ cos (2.-1) tan (S.d^ 



d/= - sin (a — Z)cos/?. dy + ^cosa — /). d;i — ?^sin (;i-— /)tan/3.d/3 

 9 9 9' 



20) d J = " cos^ b [tan /3 — tan b cos {% — /)] cos /?. dy 



+ '^ cos b sin 5 sin (/I — /) . d ^ 

 9' 



+ y_ cos^ 5 fl + tan S-tan /3 cos {T^. — /)1 . d/3 

 9' 



Expressions equivalent to those given by the authors above quoted for the 

 variations of positions and distances, as might be expected from the symmetri- 

 cal form of the fundamental equations, from which they are derived. Also, as 

 before, ^ =■ n -\- v -\- u, denoting the longitude on the orbit of the point [/ J] 

 towards which the true motion of the asteroid is directed, and n — Q, =. n + v 

 + u — 9,, being the argument of latitude, the equations 



tan (^ — Si) = cos i ism {n + V + u — Si) 

 Sin 5 = sin « sin [n + v + u — Si) 

 tan b = sin i tan (n + v + u — Si) cos (I — Si) 



(21) . / V 



= tanz sin {I — si) 

 cos {n + V + u — SI ) = cosb cos (Z — si) 

 g' = g cos b 



give by differentiation, substitution, and reduction. 



