158 T H E R B I T O F URANUS. 



CHAPTER VII. 



FORMATION AND SOLUTION OF THE EQUATIONS OP CONDITION RESULTING 

 FROM THE PRECEDING COMPARISONS. 



In tlie preceding chapter we liave obtainid from observations a scries of cor- 

 rections to the geocentric positions of Uranus resulting from the provisional 

 theory. The further operations are as follows: — 



1. To reduce all the corrections in right ascension and declination to correc- 

 tions in geocentric longitude and latitude. IMost of the corrections are already 

 so expressed, so that tliis reduction is necessary in only a few cases. 



2. To find the mean value of the correction in geocentric longitude during each 

 opposition, and to express this mean value in terms of the correction to the helio- 

 centric co-ordinates. 



3. To express these corrections to the heliocentric co-ordinates in terms of cor- 

 rections to the elements of Uranus and the mass of Neptune. 



4. To solve the equations of condition thus formed. 



The first of these processes is too simple to make it necessary to present any 

 details of it. With regard to the second I have sought, not the simple correction 

 to the geocentric longitude, but this correction multiplied by such a factor as it 

 was supposed would make the probable error of the correction 0".5. The equations 

 for expressing the error of geocentric longitude in terms of errors of heliocentric 

 longitude and radius vector have been given on page 129. The first observation 

 of Flamstead, p. 107, gives the equation 



+ 22"=:1.04§X + .027,5p 



hi being the correction to the heliocentric longitude, and i5p that to the Ncperian 

 logarithm of the radius vector. From the discordance of Flarastead's clock errors 

 it may be estimated that the probable error of the first member of this equation is 

 10". Therefore we divide the equation by 20, which gives 



^V? = l".l = .0525;i + .OOl^p. 



In the opposition of 1715 we have four observations. The best were those of 

 March 4 and 10, of which we may estimate the probable error at 10", and the 

 worst that of March 5, of which the probable error may be estimated at 20", 

 while that of April 29 is intermediate in certainty. The separate observations 

 give the equations 



March 4, (V=.+ 28" =1.06(^;i; Weight, 4 



March 5,hl = -\-U = 1.06^;i; Weight, 1 



March 10, ^ = + 36 = l.OG^A; AVeight, 4 



April 29, ;/=-)- 2 = 1.04U -f .04^p; Weight, 2. 

 Mean hi ^ -\- 27.6 = 1.056,^;i +.003,^ ; probable error = ± 6". 



