50 THE ORBIT OF NEPTUNE. 



place of the planet, this hypothesis would be sufficiently near the truth for an 

 entire year or more. But the error of geocentric place would be subject to an 

 annual period though the errors of the heliocentric place should be invariable. 

 Let us estimate the error of the hypothesis in question. Put 



r =z radius vector of Neptune. 



D = difference of longitude of Sun and Neptune. 



Sv, h; errors of heliocentric longitude and radius vector. 



Then the errors of geocentric longitude will be, approximately, 



. /, , cosZ>\ Sr . 



Of this expression the part 



^^ n 1 ^'' ■ n 

 — cos JJ-}- -^ smlJ 

 r T' 



will not be regularly progressive, but will change with the sine and cosine of D, 

 the period of which is about 368 days. 



The integral of this expression gives for the mean value of the error, while D 

 is iuci'easing from Do to D^, 



8v sinZ>i — sinZ>o Sr cosD^ — cosZ>o 



T A — A »^ D, — D, 



By putting 



D= ^'-^% 8 = D,-D = D-D„ 



and developing according to powers of 8, this expression becomes 



i%o.z,(i_f)+^:si„i,(i-f). 



This, plus the error of heliocentric longitude, is the mean error which will be 

 given by a series of observations equally scattered through a period + 5 on each 

 side of the mean epoch D. But what we really want is the error at the mean 

 epoch itself J that is, 



8v 4- — cos D4-—, sin D; 

 so that we must correct the mean error actually found by the quantity 



I — cos £>-{-— sm JJ], 

 \r r- / 



or, since S is generally about 11', and r about 30, 



.027(|cosD + |oSini)) 



S^/8v 

 6 



