138 RESEARCHES CONCERNING THE PERIODICAL 



In the present case, the asteroid being near the observer, or (neglecting the 

 small quantities already mentioned) near the earth's centre, and consequently 

 near its node, recollecting that 



tan u = tan (n — {n + v)j = tan {a — ®) = tan (n — S2) 



sin (/ — ^) = ± sin {I — ©) 

 cos (/ — Q,) z= ± cos {I — e) 



(26) 



tan b = tan i sin {I — Si) 

 d ^ = 

 d7t + d?; = d© = 



and using the upper sign for a convergent point north of the ecliptic, and ob- 

 serving that d H may be neglected, since it could only be introduced into these 

 formulae by making part of the value of d v, which, in this instance, disappears, 



and calling N — -. , and substituting and reducing, there result for 



the case of a meteoric asteroid, the following expressions of the meteor's true cur- 

 tate velocity and direction, in terms of those of the three independent elements, 



d e-' = ± ^ tan 6 sin (/ — ©) d i + ^ iV \ —-^ : — tan^ b\.dn + ^ tan'' 6 tan ^ . d ? 



° w w L2 (a — r) J u 



(27) d/=q:tan5cos(^-e).d^• + iV■'-^^-^.dn-^^"(^-®^^^"^d, 



COS^ b CDS'* 



db = ± s\n{l — ©) . d i + iVtan 6 . d n — tan 6 tan ^ . d 9 

 More simple expressions for the variations of some of the elements may be de- 

 rived from the equations (9) and from the expression 



(28) v/ a cos ^ sin i = g- r sin 6 = r Z -\- r y sin j3 

 thus respectively 



dn = — 3n^w(y+Gcos4<).dy — 3n^Gy'sin(Z — x).dx+3ni Gy'\cos{L — x)tan^ — sin^B l.dji 



(29) ^ . , 



... cota , rn^^ to sin ^ , mi y' 



d A = cot * cot z . d z — .an : — .—. . d > : — -. — -. . d 



3 n sin I sin 9 sm i sin ^ 



These expressions establish the principle stated in Section IV., No. 6, that 

 the independent elements may be reduced to three. They would, if the data 

 of Tables I., II., and III. possessed the requisite precision to warrant the pre- 

 sumption that the elements of Table VI. are approximations towards the real 

 values, enable us to estimate the effect of the uncertainty of the assumed values 

 of y, /I, and ^. At present this cannot be considered the case. It is worthy 



