(op. cit. p. 182); "that the individual errors, though not following the same law, are practicably (sensiblement) of the same order of magnitude, and each contributes little to the total error" (op. cit. p. 183). Recapitulating these conditions he concludes that "in this case the resultant error suivra sensiblement la loi de Gauss”*.

The reasoning whereby the (proper or normal †) law of error has been found as an approximation to the actual locus (which results from the composition of several independent. elements) may be extended to obtain a closer approximation by taking into account the first of the terms which have been neglected.

Put S2k2, S denoting summation with respect to all the elements and k2 the mean square of deviation (not now identical for all the elements). And for S(k4-3k22) put K2. Then R may be written.

The first and main term of the integral is the normal error-function

1

which integral may be replaced by 1, K,

d'y
dx

may be extended to terms of smaller orders so as to form a descending series, the "generalized law of error" for even functions.

The transition to odd components is effected by obtaining likewise an odd descending series; the whole of the odd series like the part of the even series which remains after the first main term-tending to vanish as the number of the

Op. cit. § 4, Leçon xvi. Cp. end of § 11, Leçon xiv.

†The term "law of error" in this Paper may be understood according to the context either as the "generalized law," or only the first and main term of that approximation, the so-called "Gaussian," or "normal," law.

Camb. Phil. Trans. loc. cit. (1905). Statistical Society, 1906, "Law of Great writer.

Cp. Journal of the Royal
Numbers," by the present

components is increased. Thus the first term of the odd series may be written

=

where K1, Sk3*, is the mean third power of deviation for any element from its mean value; the mean value being taken as the origin for each of the elements, (and for their sum) †. When the differentiation is performed it is seen that the term is affected with a coefficient which diminishes as the number of components, say s, is increased. For example, if each element is a binomial assuming the value 0 or a with respective probabilities q and p,

k3= {(q—p)sa3 /(} spq)3⁄4a3,

that is of the order 1/spq. Thus not only Laplace's proof of the law of error for even frequency-functions, but also Poisson's proof thereof for odd functions, together with his determination of a second approximation ‡, are found to hold good upon certain conditions. One who considers that those conditions are very generally realized will regard as misleading the Astronomer Royal's statement: "The conclusion is therefore unwarranted and there is no proof at all that peculiarities of the functions f efface themselves in the final result" (loc. cit. pp. 167-8).

III. It remains to notice two other proofs § of Laplace's law which have been likewise approached and missed by Dr. Sampson.

There is first Morgan Crofton's proof-either by way of a partial differential equation T, or more directly or more directly that the continued superimposition of frequency-curves of any form will result in the normal law of error. Dr. Sampson applies Morgan Crofton's method of composition to certain

It will be noticed that the subscripts of the k's correspond to powers (of the component errors). The subscripts of the K's correspond to corrections of the normal function.

Mutatis mutandis, if the elements are weighted.

Given by Todhunter, op. cit. pp. 567-8.

Two among several variant proofs which are referred to in the article on "Probability" (Ency. Brit.) §§ 104-111.

|| Laplace rather than Gauss deserves to be the eponym of the law of error when, as throughout in this paper, it is considered as resulting from the random combination of numerous independently fluctuating

constituents..

Cited in the article on "Probability" (Ency. Brit.), § 109.

** See Phil. Trans. 1870.

frequency-curves obtained by grafting an oscillating element on the normal law, as thus

He shows that when curves of this type are compounded the divergence from the normal law tends to disappear. This curiosum may have some bearing on astronomical observations. But it may be doubted whether observations extending to infinity--other than the law of error-have much concrete significance.

The result of continued composition may often be more advantageously contemplated by means of Laplace's method above indicated. Let the frequency-function for one element be 1(a), for another (a), ranging between given limits, which may be infinite. If the functions are even †, put p11(ax) cos ar. And let Spida (between extreme limits), (a). Let 02(a) be formed likewise from 2(x). Then for the compound of the two frequencies we have

S® 0,1(2) × 0:(a) cos arda.

This method may be employed to obtain an answer to a question which Dr. Sampson has raised, but has not rightly answered: namely, what frequency-curves enjoy the property that when two of a family are compounded the result belongs to the same family. Dr. Sampson appears to think that the normal law of error is the only curve which possesses this property in perfection, "with any generality" (loc. cit. p. 170). He is not aware that the property appertains to a wide class of which the normal law is a species: namely, the class for which (a) (as above defined) is of the form exp-at. There is here disclosed a variant proof of the law of error. If there is a final form resulting from continued composition, it must be reproductive; and reproduction is the proprium of curves for which (a) is of the form exp-at. A further condition (borrowed from Morgan Crofton) which must be fulfilled by the sought form limits t to the value 2. Using the value of (a) thus obtained, viz. exp-a2, obtain the normal law .

we

A simple case of the more general type proposed by Dr. Sampson (loc. cit. p. 170). Note that a is less than unity, and A is taken so that the integral of the function between limits +∞ and is unity.

† As in the case above instanced. For the case of odd functions see Camb. Phil. Trans. loc. cit. p. 53 (1905).

See Camb. Phil. Trans. 1905, "Law of Error," Part 1. § 4, and Appendix, § 6.

1

Dr. Sampson is thrown off the track by a little slip in a mathematical operation such as the best may incur when not put on their guard by prior knowledge of the subject. Dr. Sampson experiments on frequency-functions of the form (or the more general form which is presented π(1+x2) when we take a for the parameter of x*). Here it may be observed (x) is of the form e-a (or e-aa), and accordingly we know a priori that the continued superposition of a components of the type will have for result

1

a

π (1+x2) Dr. Sampson, employing Morgan Crofton's method of composition by way of integration, finds this result true for two components; but not true in general, for any number of components. That is, in our notation, if the frequency

function

result is

1
π(1+x2)

1

π (a + 1)2 + x2›

but not for larger values of a.

when a=1;

In Dr. Sampson's own symbols, the abscissa, and the form of the resultant frequency-curve,

This result is evidently untenable; since it imports that if we put together two frequency-curves, each symmetrical about a central point and thence descending continuously, the central ordinate for the compound will be zero! But if we resolve the expression above equated to (O) into two rational fractions according to the general rule for integration, we shall find that while the denominator of the result is rightly given by Dr. Sampson, the numerator ought to have been (a + 1) [✪2 + (a2 − 1)]; the expression in square brackets being a factor of the denominator. Dividing out we obtain, as we ought, for the result

Substituting in the last-written expression for x, x/a and dividing the expression by a (substituting for dr, dx/a).

This is not the only passage in the paper under consideration where an inappropriate conception has led to inexact work. In connexion with his peculiar notion as to the nature of an error of observation, the writer appears to hold that if the extent of an error varies with the time according to the law y sint, the frequency of error between the extreme values which it can assume is constant. Not so, surely; the number of errors per unit of time being constant, the number of errors per unit of space is propor-tioned to The frequency-curve is

dt

αξ

that is to

not a straight line but a symmetrical curve of the fourth degree with an infinite ordinate at the centre and contact with the abscissa at the points distant t from the centre in either direction.

In this connexion it is remarked by Dr. Sampson, "it seems probable that Laplace and Poisson were on the wrong lines." To those who have followed the preceding comments it will seem more probable that Dr. Sampson is on the wrong lines.

L. On Transpiration from Leaf-Stomata.
By HAROLD JEFFREYS, M.A., D.Sc.§

WITH most of Sir Joseph Larmor's note in the April Philosophical Magazine I am in agreement. It is clear that the amount of water evaporated from the stomata of a leaf in a given time must be less than that from the whole surface of a wet leaf of the same size; it must also be less than the amount that could be evaporated from all the stomata if their total number were the same and all were so much separated that they could be treated as isolated.. Hitherto only the latter criterion has been used to indicate the upper limit, and the important question is to decide whether the former imposes a further restriction or not.. When the air is at rest the question is simple, for both limits

*The following criticism is due to Professor Bowley.

"For y sint .... the distribution is represented by n=1 for −1<<1 and n=0 beyond those limits (fig. 3).” Fig. 3 and the context bear out the interpretation here given.

The number which multiplied upon A a small fraction of (the unit of) the abscissa, gives the (proportionate) number of observations occurring between x and x+AE (or x+4x and x-14%).

§ Communicated by the Author.