108. (3) De Forest^{[1]} has given a proof unencumbered by imaginaries
of what is the fundamental proposition in Laplace's theory
that, if a polynomial of the form

A_{0} + A_{1}*z* + A_{2}*z*^{2} + . . . + A_{m}*z*^{m}

be raised to the *n*th power and expanded in the form

B_{0} + B_{1}*z* + B_{2}*z*^{2} + . . . + B_{mn}*z*^{mn}

then the magnitudes of the B's in the neighbourhood of their maximum
(say B_{1}) will be disposed in accordance with a “probability-curve,”
or normal law of error.

109. (4) Professor Morgan Crofton's original proof of the law of
error is based on a datum obtained by observing the effect which the
introduction of a new element produces on the frequency-locus
for the aggregate of elements. It seems to be assumed, very properly,
that the sought function involves as constants some at least of the
mean powers of the aggregate, in particular the mean second power,
say *k*. We may without loss of generality refer each of the elements
(and accordingly the aggregate) to its respective centre of gravity.
Then if *y*, = *f*(*x*), is the ordinate of the frequency-locus for the aggregate
before taking in a new element, and *y* = ∂*y* the ordinate after that
operation, by a well-known principle,^{[2]} *y* + ∂*y* = [Sφ_{m}(ξ)*f*(*x* − ξ)∆ξ],
where η, = φ_{m}(ξ), is the frequency-locus for the new element, and the
square brackets indicate that the summation is to extend over the
whole range of values assumed by that element. Expanding in
ascending powers of (each value of) ξ and neglecting powers above the
second, as is found to be legitimate under the conditions specified,
we have (since the first mean power of the element vanishes)

∂*y* = ½[Sξ^{2}φ^{m}(ξ)∆ξ]*d*^{2}*f**dx*^{2}.

From the fundamental proposition that the mean square for the
aggregate equals the sum of mean squares for the elements it follows
that [Sξ^{2}φ_{m}(ξ)∆ξ] the mean second power of deviation for the *m*th
element is equal to ∂*k*, the addition to *k* the mean second power of
deviation for the aggregate. There is thus obtained a partial
differential equation of the second order

*dy*

*dk*= ½

*d*

^{2}

*y*

*dk*

^{2}

A subsidiary equation is (in effect) obtained by Professor Crofton
from the property that if the unit according to which the axis of
*x* is graduated is altered in any assigned ratio, there must be a corresponding
alteration both of the ordinate expressing the frequency:
of the aggregate and of the mean square of deviation for the aggregation.
By supposing the alteration indefinitely small he obtains a
second partial differential equation, viz. (in the notation here adopted)

*y*+

*x*

*dy*

*dx*+ 2

*k*

*dy*

*dk*= 0.

From these two equations, regard being had to certai~~a~~n other conditions
of the problem,^{[3]} it is deducible that *y* = C*e*^{−x2/2k}, where C is a
constant of which the value is determined by the condition that

.

110. (5) The condition on which Professor Crofton's proof is based
may be called differential, as obtained from the introduction of a
single new element. There is also an integral condition obtained
from the introduction of a whole set of new elements. For let A
be the sum of *m*_{1} elements, fluctuating according to the sought law
of error. Let B be the sum of another set of elements *m*_{2} in number
(*m*_{1} and *m*_{2} both large). Then Q a quantity formed by adding
together each pair of concurrent values presented by A and B must also
conform to the law of error, since Q is the sum of *m*_{1} + *m*_{2} elements.
The general form which satisfies this condition of *reproductivity* is
limited by other conditions to the normal law of error.^{[4]}

111. The list of variant proofs is not yet exhausted,^{[5]} but enough
has been said to establish the proposition that a sum of numerous
elements of the kind described will fluctuate approximately according
to the normal law of error.

112. As the number of elements is increased, the constant above
designated *k* continually increases; so that the curve representing
Varieties of Linear Function.
the frequency of the compound magnitude spreads out
from its centre. It is otherwise if instead of the simple
sum we consider the linear function formed by adding the
*m* elements each multiplied by 1/*m*. The “spread” of
the average thus constituted will continually diminish as the
number of the elements is increased; the sides closing in as the
vertex rises up. The change in “spread” produced by the accession
of new elements is illustrated by the transition from the high to the
low curve, in fig. 10, in the case of a sum; in the case of an *average*
(arithmetic mean) by the reverse relation.

113. The proposition which has been proved for linear functions
may be extended to any other function of numerous variables,
Extension to Non-linear Functions.
each representing the value assumed by an independently
fluctuating element; if the function may be expanded
in ascending powers of the variables, according to
Taylor's theorem, and all the powers after the first
may be neglected. The matter is not so simple as it is
often represented, when the variable elements may assume large,
perhaps infinite, values; but with the aid of the postulate above
enunciated the difficulty can be overcome.^{[6]}

114. All the proofs which have been noticed have been extended
to errors in two (or more) dimensions.^{[7]} Let Q be the sum of a
Extension to two or more Dimensions.
number of elements, each of which, being a function
of two variables, *x* and *y*, assumes different pairs of
values according to a law of frequency *z*_{r} = *f*_{r}(*x*, *y*), the
functions being in general different for different elements.
The frequency with which Q assumes values of the
variables between *x* and +∆*x* and between *y* and *y* + ∆*y* is *z*∆*x*∆*y*, if

;

where, as in the simpler case, *a* = ∑*a*_{r}, *a*_{r} being the arithmetic mean
of the values of *x* assumed in the long run by one of the elements,
*b* is the corresponding sum for values of *y*, and

*k* = ∑[∬(*x* − *a*_{r})^{2}*f*_{r}(*x*, *y*)*dxdy*]

*m* = ∑[∬(*y* − *b*_{r})^{2}*f*_{r}(*x*, *y*)*dxdy*]

*l* = ∑[∬(*x* − *a*_{r})(*y* − *b*_{r})*f*_{r}(*x*, *y*)*dxdy*];

the summation extending over all the elements, and the integration
between the extreme limits of each; supposing that the law of
frequency for each element is continuous,
otherwise summation is to be
substituted for integration. For
example, let each element be constituted
as follows: Three coins having been
tossed, the number of heads presented
by the first and second coins together
is put for *x*, the number of heads
presented by the second and third coins
together is put for *y*. The law of
frequency for the element is represented
in fig. 11, the integers outside denoting
the values of *x* or *y*, the fractions
inside probabilities of particular values
of *x* and *y* concurring.

2 | |||||||||||||||||||||||||||

1 |
| ||||||||||||||||||||||||||

0 | 1 | 2 |

Fig. 11.

If *i* is the distance from 0 to 1 and
from 1 to 2 on the abscissa, and *i*′ the corresponding distance on the
ordinate, the mean of the values of *x* for the element—∆*a*, as we may
say,—is *i*, and the corresponding mean square of horizontal deviations
is ½*i*^{2}. Likewise ∆*b* = *i*′; ∆*m* = ½*i*′^{2}; and ∆*l* = ⅛(+*i* × +*i*′ − *i* × −*i*′) = ¼*ii*′.
Accordingly, if *n* such elements are put together (if *n* steps
of the kind which the diagram represents are taken), the frequency
with which a particular pair of aggregates *x* and *y* will concur, with
which a particular point on the plane of *xy*, namely, *x* = *ri* and
*y* = *ri*, will be reached, is given by the equation

^{[8]}.

115. A verification is afforded by a set of statistics obtained with
dice by Weldon, and here reproduced by his permission. A *success*
is in this experiment defined, not by obtaining a head when a coin
is tossed, but by obtaining a face with more than three points on it
when a die is tossed; the probabilities of the two events are the same,
or rather would be if coins and dice were perfectly symmetrical.^{[9]}
Professor Weldon virtually took six *steps* of the sort above described
when, six painted dice having been thrown, he added the number of
successes in that painted batch to the number of successes in another
batch of six to form his *x*, and to the number of successes in a *third*
batch of six to form his *y*. The result is represented in the annexed
table, where each degree on the axis of *x* and *y* respectively corresponds
to the *i* and *i*′ of the preceding paragraphs, and *i* = *i*′.
The observed frequencies being represented by numerals, a general
correspondence between the facts and the formula is apparent.

- ↑ The
*Analyst*(Iowa), vols. v., vi., vii.*passim*; and especially vi. 142 seq., vii. 172 seq. - ↑ Morgan Crofton,
*loc. cit.*p. 781, col.*a*. The principle has been used by the present writer in the*Phil. Mag.*(1883). xvi. 301. - ↑ For a criticism and extension of Crofton's proof see the already
cited paper on “The Law of Error,”
*Camb. Phil. Trans.*(1905), pt. i. § 2. Space does not permit the reproduction of Crofton's proof as given in the 9th ed. of the*Ency. Brit.*(art. “Probability,” § 48). - ↑
*Loc. cit.*pt. I. § 4; and app. 6. - ↑
*Loc. cit.*p. 122 seq. - ↑
*Loc. cit.*pt. ii. § 7. - ↑ The second by Burbury, in
*Phil. Mag.*(1894), xxxvii. 145; the third by its author in the*Analyst*for 1881; and the remainder by the present writer in*Phil. Mag.*(1896), xii. 247; and*Camb. Phil.**Trans.*(1905),*loc. cit.* - ↑ Compare the formula for the simple case above, § 4.
- ↑ On the irregularity of the dice with which Weldon experimented,
see Pearson,
*Phil. Mag.*(1900), p. 167.