40 ON PROBABILITY. 
equations, which serve to determine any unknown quantities, exceed in 
number that of the unknown quantities themselves. Our limits do not 
permit us to give the analysis upon which this proof is founded, we shall, n 
however, endeavour to explain shortly in what the method itself consists. 
The data furnished by observation lead in general to equations of this 
form @—bx+cy-+t &. = 0, 
equ. 
in which equation «, b, ¢, &c. are known quantities which vary from one i. 
equation to another. Each un” observation gives an equation - 
a, — b,x + c,y + &e. = 0. 
If the number of equations is equal to that of the unknown quantities 
*, Y, Zz, they may be determined by linear elimination, but generally the 
number of equations exceeds that of the unknown equations, and the ques- 
tion arises, of which.the solution is in scme measure arbitrary, what system 
of equations, equal in number to that of the unknown quantities a, Y, 2, 18 
most favourable for their determination. 
The method of least squares, which Laplace has proved to be the most 5 
advantageous, consists in determining the quantities a, Y, 2, so that the in 
quantity la, — bia + cy + &e. 125 cer 
+ {a —botcy + &}2 wy 
&ec. x 
+ {a,—- bo} c,y + &e 12 tho 
is a minimum. dif 
For this purpose it is only necessary to differentiate this sum separately, Bol 
with respect to each of the variables «, Y> 2, &c., and put the results sepa- Jo 
rately = 0. pen 
The equation, which is obtained by making x alone vary, is tw 
{a, —bie+ ay &ele, it 
T1®%—ba+ay4 &e.}s, i 
“+ &e. ie 
+ { a. —_ b, x + CY we &e 1b, = 0, 
or Sa, b,=23b2+y3h,c, + &c. = 0: 
each of the quantities z, y, 2, &e. furnishes a similar equation, and hence a 
system of equations results equal in number to those quantities, from whence 
they may be found by linear elimination. Tt js thus that many thousand 
observations may be made to concur in the determination of one element, 
Suppose, for example, the question consists in determining with accuracy 
the elements of the orbit of a planet, after having obtained them nearly by 
a first approximation. : 
Let) be the geocentric longitude observed, and suppose the uncertainty n 
is with respect to two only of the elements e and @, and let \ be the error 
of longitude, or the difference between the longitude of the planet observed 
at a given time, and that which is deduced by calculation from the approxi- 0p 
mate elements; Je, dw the errors of those elements; then by Taylor's 5 
theorem, neglecting the squares. &ec. of § e and d= 
dM dA; 
oN=(—)ée+(—)om 
de do