Method of Least Squares. 373 
Then the general equation for x becomes 
nx n ~ l — (n— l)1,x r .x n ~ 2 + {n — 2)Xx r x s x n ~ 3 , 
which reduces to a quadratic. And in the general case the 
coefficients of the equation of the (n — l)th degree, consisting 
of combinations of the observations, might possibly be eva- 
luated by a calculating machine. Whereas the alternative 
methods do not seem equally to lend themselves to mechanical 
treatment. It may be questioned, therefore, whether even in 
respect of convenience the DeMorgan-Glaisher method has the 
advantage. 
II. The difficulties thicken as the hypotheses become more 
rarefied. For unsymmetrical facility-curves the most pro- 
bable value is not in general, even presumably, coincident 
with the most advantageous value. The most probable 
value may, of course, be determined if we assume a particular 
form of facility-curve, and determine by inverse calculations 
analogous to those already given the value of the constants. 
It should be observed that we must here expressly assume, 
what before was taken for granted, that is, not only what the 
form of each generating facility-curve is, but also how it is 
disposed with regard to the real point. Whereas in unsym- 
metrical curves the longest ordinate, the centre of gravity, 
the bisection of the area, may all correspond to different ab- 
scissae, which of them, if any of them, corresponds to the real 
point ? 
The scruple just raised affects also that approximative me- 
thod of determining the most advantageous value, which is 
afforded by Poisson's extension of what may be called Laplace's 
law of errors ; I mean Poisson's proof that, even in the case 
of unsymmetrical curves, the several values of a linear func- 
tion of observation, -^-^ — ' 2 2 (x x , x 2 , &c. being nume- 
7X + 72 + &C. v 7 " _ & 
rous), may be regarded as ranged under a probability-curve, 
whose centre is the mean of the mean errors of the elemental 
curves. If now each facility-curve be so disposed about the 
"real point" (as, waving metaphysical difficulties, I have 
called it), then indeed the weighted mean* will be the most 
advantageous value ; but not in general. Poisson himself, as 
I understand him, points out this cause of failure (Connaissance 
des Temps, 1832 " Suite," § 9 beginning). 
A fresh difficulty, connected with a fresh quassitum, now 
also makes its appearance. We have all along described the 
method of least squares (taken in a large sense) as an inverse 
* Assuming of course that the utility-functions are symmetrical, and 
that the influence of suhexponential elements may be neglected. 
