K. Pearson 
295 
Now either of the curves in lUustrations I. and II. is a good example of the 
impossibility of using the method of least squares for systematic curve-fitting. 
The reader need only attempt to write down the type-equations, which must be 
solved to find the constants, and he will realise the simply appalling amount of 
lengthy approximations which must be carried out even after rough values of these 
constants have been guessed by some one or other means. 
But it is not only algebraic and exponential curves for which the method of 
least squares fails ; it fails also for trigonometrical curves. I will now illustrate 
this in the very simplest case possible. 
(7) Illustration III. Let it be required to fit the simplest sine-curve 
y = a sin {nx + a) 
to the aneroid barometer observations in the Illustration in § 3. Let us write the 
equation in the form 
y = A sin nx + B cos nx (xvii). 
Then the three type-equations to find n, A and B, arising from applying the 
method of least squares, are the following : 
AS (cos^ nx) + ^BS (sin 2nx) = S {y cos nx), 
^A8 (sin 2nx) + BS (sin- nx) = S {y sin nx), 
AS {yx sin nx) - BS {yx cos = \ {A- — B-) S {x sin 2nx) - ABS (x cos 2nx). 
Here S denotes a summation with regard to the eleven values of x and y given 
on p. 279 ; after these have been substituted in the summations, we must eliminate 
A and B, and we shall then have an equation to determine n. Afterwards the 
values of A and B must be found by substituting the value of n in the two first 
equations. We may leave this as an exercise to those readers who have faith in 
the method of least squares applied to curve-fitting ! * 
Now let us turn to the method of moments. There are three constants to 
be found, so we must find the area and the first two moments of the observa- 
tions and of the theoretical curve. 
Taking the origin at the middle of the range 21, and writing Mq, M^, 
for the area and first two moments, we have 
2Bsmnl ^ , /—I cos nl sm nl 
Mo = , il/i = 2A -I- — „— 
n \ n n- 
^/^ nn (^^ ^^^^ ^''^ COS nl 2 sin nl 
= LB \ 1 ~ — 
* The equation to find n is intractable even if we place the origin at the centre of the range and 
evaluate by trigonometry the summations not involving ;/ or x outside the trigonometrical terms. 
