G THE OEBIT OF NEPTUNE. 



than the unknown quantities. We may then substitute these linear functions 

 themselves in place of an equal number of the unknown quantities. If the 

 equations are not absolutely identical, the coefficients of the other unknown qiian- 

 tities will not entirely vanish by the substitution, and thus we shall still have 

 the whole number of unknown quantities, only the coefficients of certain of them 

 will be very small. The solution by least squares can then be performed without 

 trouble, because the extra decimals will be necessary only in multiplying by the 

 very small coefficients, when they can be introduced with ease. Afterward the 

 values of the original unknown quantities can be deduced from those of the linear 

 functions, and the unknown quantities which have been retained. 

 Suppose, for example, that the equations of condition are 



a^x + l>iy + Ci2 := 5ii 

 a.\x + h.{ij + C2Z = 7^2 

 O'i^ + % + C32 — n^ 

 a^^x + 6,?/ + cji — n^ 



A simple inspection, or, at least,- an attempt to solve three of the most diverse 

 of the equations, will show if the given n equations are really equivalent to only 

 one or two. Then we should put 



X=i ax Ar ^y + yz 

 Y= a'x + [3'y + y'z 



the coefficients a, (3, y, being entirely arbitrary, and so taken that when X and F 

 were substituted for x and y the coefficients of z should be as small as possible. 

 It would conduce to simplicity if a and /3', or a' and (3, could each be made zero, 

 which could always be done. 



If we attempt to correct the elements of a planet's orbit by observations extending 

 over only a few degrees, the equations of condition will necessarily be of the kind 

 referred to. Hence a transformation of this kind will be advisable. An example 

 wdll be given in the correction of the orbit of Neptune from observation. 



§ 5. Ten years have elapsed since the publication of Kowalski's theory, and 

 no general revision of the orbit has been published by any astronomer, so far as 

 the writer is aware. The observations which have accumulated in the mean time 

 would seem sufficient to fix the elements exactly enough to give the place of the 

 planet within 5" during the remainder of the present century. It is, therefore, 

 proposed, 



1. To determine the elements of the orbit of Neptune with as much exactness 

 as a series of observations extending through an arc of forty degrees will admit of. 



2. To inquire whether the mass of Uranus can be concluded from the motions 

 of Neptune. 



