Miscellanea 
297 
Now suppose we had to determine our agH from (i) by the method of least squares, each array 
being weighted by its frequency. We should have to make 
. 21 
a minnuum. 
or a,= Ks/'Ks, 
in agreement with the result obtained in (iii) bis above as a result of the -v/aj's being orthogonal 
functions. Now the fit by least squares to the means of the arrays is precisely the same as the fit 
by least squares to the whole swarm of points. In other words if we fit our regression line as 
above indicated to the whole population it will be "the best" fitting curve, if we make the 
assumption that least squares does give the best fit*. Anyhow it is likely to be a good fit, and 
that sufl&ces for our present purposes. 
We will now write Z=(.r-.r)/o-^ and yx={]/x — y)l(^y^ whence it follows that qit = S(n^ V^X')jJV. 
We shall further write : 
and fe = M28 + 2/o-/'^^ = 2^-^28+2 = 2^ ' 
P28-1— ^28 + 4 • 
We may rewrite (i) Y=a„\j/„ f + +u„\j/n (i) bis. 
Multiply both sides by ?i^\//-g/iV and sum : 
ao ~y" yo -«o, 
if we assume ■4^0= 1 as we are at liberty to do. But the left-hand side is zero. Therefore «u— 0. 
Now assume = A' - xj/'o and multiply both sides by n^c^f^jN and sum : 
^K^o^/'O^O by hypothesis =^^^^^^#^^ -c,, ^J!^ . 
But ^^^^^^^ = 0, .-. ci„ = 0, and >/,i=A'. 
Multiply both sides by XjN and sum : 
S{n,YX) -SKA'fj S{n,X'^) ^ 
Hence aj = r. 
Now assume X- — c-n^i — c-^yi-^o. Multiply by n^-^^/N and sum : 
Thus C2o=l- 
Multiply by w_c\|^i/A^ and sum : 
S(n,^^X^) SjnM 
0- c,, ^ , 
= V/3l = C2I -j^ =C21 • 
Thus C2i = \/^i. 
* In order that this statement should be absolutely true our arrays would have to follow the normal 
or Gaussian distribution. This produces, however, au unnecessary Umitation. It is known, but possibly 
not well known, that expansions in orthogonal functions as defined by (ii) give least square fits. I am 
rather inclined to think that this is an argument in favour of least square fits, rather than a justification 
of the expansion, i.e. that the method of least squares has a wider validity than Gauss' proof provides. 
