THE ORBIT OF NETTUNE. 



i + % + CjZ + etc ..... = !, 

 2 x + % + c. 2 z + etc ..... rr n.,, 

 etc* etc. etc. etc. etc.; 



each unknown quantity is given in the form 



etc., 



in which R represents the determinant formed from all the coefficients a, b, etc. 

 in the given equations, and A 1} A. 2 , etc. the partial determinants, obtained by 

 omitting column a, row 1, column a, row 2, etc. 



If, now, the number of equations is greater than that of the unknown quantities, 

 and they are solved by the method of least squares, the form of the solution will 

 be the same as the above, except that for R will be substituted the sum of the 

 squares of all the determinants R, formed by solving separately every combination 

 of such number of the given equations as is equal to the number of unknown 

 quantities, and for A l} A. 2 , etc., certain powers and products of the partial deter- 

 minants which enter into the separate solutions. Hence, if these determinants 

 are very small, the corresponding determinants in the solution by least squares 

 will be very small quantities of the second order. But the determinants will all 

 be very small if the equations are nearly equivalent to a number less than that 

 of the unknown quantities ; that is; if they can be put into the form 



rp 



aX+ (3Y+yZ+ etc. + p = n 4 , 

 a'X+ @'Y+ yZ+ztc. + p' = 6 , 

 etc. etc. etc. etc. etc. etc.; 



the quantities X, Y, Z, etc. being less in number than the unknown quantities, 

 and p being a very small linear function of the unknown quantities. If the p's 

 vanish, all the determinants will vanish with it ; whence, if they are very small, 

 the determinants will be very small likewise. Calling a system of equations 

 identical when they really give fewer independent relations than there are un- 

 known quantities, the theorem sought may be expressed as follows : 



If a system of equations differ from identity by a very small quantity, the normal 

 equations derived from them ivill be identical to small quantities of tlie second order. 



Hence, if such a system of equations is to be solved by least squares, it will be 

 necessary to carry the solution to nearly twice as many decimals as are necessary 

 in the original coefficients. Thus, in the case under consideration, as Professor 

 Kowalski considered it necessary to retain four places of decimals in the coefficients 

 of the unknown quantities, it would have been necessary to include at least six 

 or seven decimals in the normal equations, instead of only four. 



But the necessity for so long a numerical calculation can be avoided by a suitable 

 transformation of the equations of condition. If the equations are identical, 

 they really give certain linear functions of the unknown quantities less in number 



