Method of Least Squares. 369 
of the square of the selected abscissa. It is zero. Yet it is 
infinitely probable that the real value is greater than zero. 
It appears from the subexponential form* incidental to this 
problem, that if, after taking the mean of a group of observa- 
tions of the type I. A. 3, we take the mean of a second group 
of the same type, the mean of those two means (or of any 
number of such means) is not likely to be as much nearer the 
true value as might have been expected. 
A variant of Prob. (3) is obtained by the distribution of the 
observations into groups each of the type (3). This variant 
might be obtained from the variant of Prob. I. Thus, suppo- 
sing the limits of the groups still given, let our knowledge of 
the weight corresponding to each group be degraded through 
all degrees of conjectural knowledge to absolute uncertainty. 
Now let the barriers which separate the groups be unfixed, 
and finally become absolutely uncertain, and we shall have 
reached a fourth case, 
I. A 4, in which we do not know the weight of any of the 
observations ; and they may all, for all we know, have differ- 
ent weights. The solution may be reserved till we reach the 
more general case of I. B. 4. 
I. B. Facility-curves which are symmetricalf , but not pro- 
bability-curves, present similar cases for the exact solution of 
which it would be necessary to assume as given some particular 
1 / 2/3 \ ~ 
form, e. g. .— (a+- : ^-)e ' 2 , where a + /3 = 1 ; unless indeed 
V ire \ c l 
as Donkin has observed^, it were possible by an unimaginable 
perfection of the calculus of functions to take, as it were, a 
mean of all admissible functions. 
Approximate solutions of cases under this heading are given 
by the Method of Least Squares, as discovered by Laplace. 
Concerning these it may be observed that, if the subexponential 
form is an appreciable ingredient of the elemental facility- 
curves, then the factors assigned by the method of least squares 
are not in general the most advantageous. It is obvious to 
object that the number of the observations in the case to which 
the Method of Least Squares is applicable is supposed infinite; 
and therefore that the principle mentioned in the Postscript 
of the previous paper will apply. But then infinite is to be 
interpreted here as very large. Or, rather, not so very large; 
as we are told in recommendation of this very theorem whose 
* See previous paper. 
t Some of the following inquiries can only by courtesy be included 
under tlie term Least Squares. 
\ Phil. Mag. [4] vol. ii. p. 56. 
Phil. Mag. S. 5. Vol. 16. No. 101. Nov. 1883. 2 D 
