_— 
§ 2. Suppose, we have measured some series of the quantities 
L,M,N...., between which the following relation exists: 
NP Ze ye Dn 
where X, Y, Z.... are unknown quantities which we want to calculate. 
We assume that the number of equations between the observed 
quantities is larger than the number of unknown quantities, so that 
we want to calculate the most probable values of X, Y,Z.... by 
means of the method of least squares. 
Let L,, M,, N,.... be a set of values belonging together, as yielded 
by the observations, 
l,,m,,n,.... the errors made in these observations, 
DU > Mn, > M2n,.... the mean errors in those measurements ZL, 
M,, N,...., which we assume to be known 
before hand, 
A,, Y,,4,-~-- & set of approximate values for X, Y,Z.... 
DE the corrections to be calculated, which must 
be applied to those approximate values. 
Each measurement gives then according to (1) an equation: 
¢ oD 
Bt Mm Nn). SPH Xe Krey Bee... = 4, (2) 
where: 
(37), 
OL di, = Lr MS 
x=X,,Y=Y... 
my = aaa: 
= ED ee 
diel Me Eset a) TEAR 
0 
N 
| 
Fo=F(L 
Yet zw, y,2.... must be chosen so that ') 
Ee m,? n,? 
1 dj | 1 . . . © 
= ( — dt rig ge ) Is a minimum. . . (3) 
Ml Mm? Mn, 
If now the coefficients Y, ,Z...., are known, what errors, /,, mn, 7 
en 
correspond to the observed quantities ,, .1/,,.V,...? It is evident 
that various sets of quantities L, J/, V.... tay have given rise to 
the same sets of quantities ce M,, N,...., and that those values of 
L, M,N and hence of /,,m,,n, are the most probable for which 
ba a n,? eel 
GE | Gear ee INE 
eS Wii ta Ny, 
1 Koti auscu, Lehrbuch der praktischen Physik, p. 16 considers the equations: 
Ub =f (AS BC nat Sraa) 
“where r,s...., and often « are instrumental readings” and yet he determines 
(see p. 11) A, B, C so that the sum of the squares of errors in u is as small as possible. 
16% 
