112 
CHRISTIE. 
of the ./-coefficients derivable from the equations of condition 
is zero, and the obvious principle that certainty is probability, 
namely, the highest degree of probability, a probability that 
excludes every alternative. Thus, wherever there is a system 
of n simultaneous linear equations in n unknowns, the 
method of least squares affords a valid transformation, which 
at times may facilitate their solution. Take, for example, 
the derivation of Fourier’s integral: The problem is, first, 
to determine the unique, the only possible, values of the 
coefficients in 
P= 00 
c Q + 2 (c p cos pj x -j- s p sin pj x) 
p = o 
so that the series may be the equivalent of <p (x) for all 
values of x between — % and -f <p (x) being subject to the 
3 3 
well-known limitations. Putting j x successively equal to 
to—1 to— 3 to—1 ,, . 
7T. — —— 7T, . . . ? -f-- 7 r. we obtain 
to 
to 
s. 
to 
'i °i 
1 — to . .1 — to 
1 —to 
1 4- cos -tt -j- sin -7r 4- cos 2- 7t -f- . . . — <p ( --= 0 
* m to 1 to 1 r \ m j J 
3 — to 
3 
I I ... /3 — to tt\ 
1 + COS - 7T+.. — (p\ - ^) = (J 
1 to 1 r \m j J 
. to — 1 
1 + COS ——— tt + 
to 
(m — 1 7T 
^ V to j 
Dl». 
whence by least squares, to obtain the unique values, which 
are therefore the most probable values of the coefficients, 
1 k = m — i /2 h + 1 — m ~\ 
'- = y.-o n—|— j) 
2 k -j- 1 — m iv\ 2 & -f- 1 — to 
m 3 J 1 m 
’2 h -h 1 —to tt\ . 2 & -f- 1— to 
--r ) sin p -7c • 
TO 3 J 1 TO 
Cp _ TO t J 
Jtv — ill - J. / 
^ H 
k = 0 ' 
9 k = m — 1 
- 2 
m i, o 
P 1 
