SECT. 3.] PLACES IN ORBIT. 109 



latter by t. Now it is evident that if the approximate value of any one of the 

 quantities p, e, IT, is known, the two remaining ones can be determined from them, 

 and afterwards, by the methods explained in the first section, the time corre 

 sponding to the motion from the first place to the second. If this proves to be 

 equal to the given time t, the assumed value of p, e, or 77, is the true one, and the 

 orbit is found ; but if not, the calculation repeated with another value differing a 

 little from the first, will show how great a change in the value of the time corre 

 sponds to a small change in the values of p, e-, U; whence the correct value will 

 be discovered by simple interpolation. And if the calculation is repeated anew 

 with this, the resulting time will either agree exactly with that given, or at least 

 differ very little from it, so that, by applying new corrections, as perfect an agree 

 ment can be attained as our logarithmic and trigonometrical tables allow. 



The problem, therefore, is reduced to this, for the case in which the orbit is 

 still wholly unknown, to determine an approximate value of any one of the quan 

 tities p, e, U. We will now give a method by which the value of p is obtained 

 with such accuracy that for small values of // it will require no further correc 

 tion ; and thus the whole orbit will be determined by the first computation with 

 all the accuracy the common tables allow. This method, however, can hardly 

 ever be used, except for moderate values of z/, because the determination of 

 an orbit wholly unknown, on account of the very intricate complexity of the 

 problem, can only be undertaken with observations not very distant from each 

 other, or rather with such as do not involve very considerable heliocentric 

 motion. 



86. 



Denoting the indefinite or variable radius vector corresponding to the true 

 anomaly v U by (&amp;gt;, the area of the sector described by the heavenly body in 

 the time t will be %f() y d v, this integral being extended from v = JY to v = N , 

 and thus, (k being taken in the meaning of article 6), kt\/p=/i)()dv. Now it 

 is evident from the fomulas developed by COTES, that if (f x expresses any 

 function whatever of #, the continually approximating value of the integral 

 ftpx.dx taken from as = utoz=.u-{-Jis given by the formulas 



