84 OBSERVATIONS ON THE ORBITS 



of the earth, and I shall regard it as exactly directed towards this centre; in 

 which case, the only effect of the earth's attraction is to increase the' relative 

 velocity without changing the relative direction. The only change which is, 

 therefore, required in Erman's paper is to increase the relative velocity v' by 

 this increase of velocity. The increased velocity is determined by the formula 



in which 



R = the earth's radius, 



^ = the attractive force of the earth at the unit of distance, 

 Va = the increased velocity, 

 Hence 



?1 — ^ = 0-13932, v,"" = v'^ + 0-13932 



R r, 



and the five values of v^', corresponding to the five values of v', calculated by 

 Erman, are 



0-91118, 1-41818, 1-69177, 193830, 217363. 



Nothing farther seems needed upon this point, and I therefore leave it to notice 

 an omission by Professor Erman. 



He has neglected the negative sign of the radical in the equation 

 v' = — e COS. u + \J {v^ — e^ sin. \), 

 and this sign may be used as long as the resulting value of v' is not negative. 

 Thus, for the value of v 



= 0-77382 

 we should find 



v' = 0-29426 

 velocity in perihelion = 2-235 

 perihelion distance = 0-4186 

 T = 0-60973 

 (0 = 12° 3' 

 which shows that Erman's conclusions, regarding the relative velocity and the 

 inclination of the orbit, are unsound. 



In reviewing some of his numerical results, I differ a little from him, but 

 the difference is of no practical importance. 



Another most important point for consideration is the difference of direction 



