be ascertainable from the morphological constants unless the degrees of resistance presented in different directions are known; except, however, the cases of the more symmetrical systems in which the positions of these axes are fixed by symmetrical considerations.

A similar observation applies to the absorption-figure for monochromatic light, which is also an ellipsoid.

The fact that the elasticity-figure of crystals is a surface of a higher order than an ellipsoid is due to its being the outcome of a compounding and averaging whose scope is more limited and not so uniform as that above referred to.

11. On a Species of Tetrahedron the Volume of any member of which can be determined without employing the proof of the proposition that Tetrahedra on equal bases and having equal altitudes are equal, which depends on the Method of Limits. By M. J. M. HILL, M.A., D.Sc., F.R.S., Professor of Mathematics at University College, London. The object of this communication is to prove the existence of the species of the tetrahedron mentioned in the title.

Art. 1. A proof is first given of the known proposition, that if the edges B A, CA, DA of the tetrahedron ABCD be produced through A to E, F, G respectively, so that BA-AE, CA=AF, DA = A G, then the tetrahedra ABCD, AEFG are of equal volume.

Art. 2. From the above proposition it is deduced that if the edge DA of the tetrahedron ABCD be perpendicular to the plane A B C, and if D A be produced to E, so that DA = A Ê, then the tetrahedra ABCD, ABCE are of equal volume.

Art. 3. Now let A B C D be a tetrahedron, and let DH, CK be drawn equal and parallel to B A.

Join H A, AK, K H, H C.

Then if B H be perpendicular to the plane A CD, it follows, by applying Art. 2 twice over, that the tetrahedra ABCD, ADC H are of equal volume.

In like manner if DK be perpendicular to the plane A Ĉ H, it follows that the tetrahedra A DCH, A HCK are of equal volume.

A

K

B

H

Hence the tetrahedron ABCD is one third of the prism, having the same base and altitude.

The two conditions

(1) That BH is perpendicular to the plane A C D, and (2) that D K is perpen

dicular to the plane A CH-result in the expression of the lengths of the six edges of the tetrahedron in terms of two positive quantities a, k as follows:

AC a√9-3k2;

AD BC=2a;

AB=BD=DC=a√1+k2.

Hence o< k < √3.

The faces B DA, BDC are equal isosceles triangles.

The faces AC B, ACD are equal scalene triangles.

The planes B C A, B C D are perpendicular to each other, and so are the planes ADB, ADC.

The planes AC B, A C D are inclined at an angle of 60°.

The planes ABC, ABD are inclined at the acute angle whose cosine is √3-k2, and so are the planes CDA, CD B.

The planes BDA, BDC are inclined at the angle whose cosine is † (k2 −1), which is obtuse if o<k<1, but acute if 1<k<√/3.

The volume of the tetrahedron is a3k2√3−k2.

12. On Absolute and Relative Motion. By Prof. J. D. EVERETT, F.R.S.

Though there is no test by which we can distinguish between absolute rest and uniform velocity of translation, D'Alembert's principle furnishes a test by which deviation from such uniformity can be detected. Every deviation produces the same effects which would be produced by bodily forces opposite to the actual changes of velocity. The intensity of the apparent bodily force is equal in each case to the absolute acceleration.

What is called centrifugal force is an apparent bodily force directed outwards from the centre of curvature of the body's path, and having an intensity equal to the distance from this centre, multiplied by the square of the absolute angular velocity. Angular velocity, unlike velocity of translation, involves acceleration; and by comparing the accelerations of different points of a rigid body we can measure the absolute angular velocity of the body. The slope of a conical pendulum and the concavity of the surface of the liquid in a revolving vessel are phenomena which depend on absolute velocity of horizontal rotation; and another measure of horizontal angular velocity is furnished by differences of pressure at different points in a horizontal tube full of liquid.

13. On the Magnetic Field due to a Current in a Solenoid.
By W. H. Everett, B.A.

The case of a solenoid of circular section is the only one hitherto investigated, and this has been done by considerations derived from magnetic shells. In this paper the problem is approached by a more direct method, and general solutions are obtained in a form which can be readily worked out to numerical values. Special application is made to the case of a rectangular (or polygonal) solenoid, the component forces being expressed in finite terms. For a very long solenoid of any form of section the longitudinal force in either of the end sections is shown to be exactly the same at all points, and in any solenoid the longitudinal force is shown to be more uniform in the end sections than in the medial section. As a particular case the method gives the component forces due to a plane circuit at any point in its field; and a simple expression is found for the force, at any interior point, due to a circular current.

14. On the Law of Error in the Case of Correlated Variations.
By S. H. BURBURY, F.R.S.

If we have a great number, N, of independent magnitudes, each liable to variation according to any law of its own, but remaining always finite, the chance that their sum, each divided by N, shall lie between x and a+da is proportional to e-hda, where h is a constant. The proof of that proposition is originally due to Poisson. The first application of it was to errors of observation, each of the 'magnitudes' aforesaid being such an error, and the N observations being supposed independent.

Modern writers, among others Mr. F. Galton and Mr. F. Y. Edgeworth, have substituted for the single magnitude given by each independent observation a group of mutually dependent or 'correlated' magnitudes, and for the single square forming the index in e- a quadratic function, i.e., sums of square and products of the correlated magnitudes. If, for instance, they be denoted by x and y, the expression corresponding to e-hdr will be (ax2 + bxy + +ey3)dxdy. The coefficient b expresses the fact of correlation' between x and y.

The object of the following paper is (1) to extend the purely mathematical investigation hitherto applied to the case of single magnitudes to the case of groups of magnitudes, the members of each group, although 'correlated' with one another, being still supposed independent of any other group-in this I only follow the lines of the known proof; and (2) to show how in certain cases the method may be extended to groups which do not possess this property of mutual independence.

1. Let fa(α1.

...

PART I.-ARTICLES 1-15.

an.

bn,

an), or fa, be a continuous function of the variables a, Let fo(b1. bn), or f, be the same or a different function of the variables b1. and so on to fa(Ir In), there being N functions. Denote by Pa the integral

...

...

2. Assume the b's to be independent of the a's, so that the variables a are not contained explicitly or implicitly in fo... or fq, the variables b are not contained in fa, or in f... . fq, and so on.

Then

·-SS·

In form now n linear functions whose type is

...

where the coefficients are numerical and of the order of magnitude

be required to find the value of

1

Let it

N

subject to the condition that the linear functions lie respectively between limits 81 + . . . 8 + ds1, &c.

19

4. To do that, substitute 8, En for G1

an by the linear equations, and then perform the integration according to bc, &c. That reduces the integral to $(81 don, where p denotes some function.

...

sn)ds

Now let

Then if 01 = 02

=0=0, p=1. If any 00, p<1.

From this it follows that unless every = 0, each of the quantities X,Xь .

less than unity.

...

Xq is

an)

7. Now let a1 an be any set of correlated variables, and let fa(α1 · da,... dan be the chance that they shall simultaneously lie between the limits .a,+ da,, &c. Let fo(b,... b) have the corresponding meaning for another set of correlated variables independent of the a's, and so on for fe, &c. Then, if we form the linear functions

the chance that they shall lie between the limits r1 and r1+ dr, &c. is, by Proposition I.,

multiplied by a constant independent of r1

rn.

8. In this expansion substitute for Xa &c. their values, and expand the exponential factor.

If any differs from zero, each of the factors X is, by Prop. II., less than unity, and therefore (since we are dealing not with the product of integrals, but with the integral of the product Xa... Xa) we may neglect in that expansion all powers of 0 above the second. The expansion is then reduced to the form

in which

R=A,0,2+B1220,02+ &c. — (1201 + . . . +1°'„On) √ − 1,

in which A, B12, &c., are known.

And performing the integrations according to 0, expression

en, we obtain as result the

in which the coefficients are known quantities when the functions fa, fb, &c., and the X's, u's, &c., are known.

It thus appears that the chance of the linear functions

is a quadratic function of (r-r1), (r1-r2), &c. The exponential form always occurs in the result, whatever be the forms of the functions fa, fo, &c. Only the coefficients of the quadratic function are determined by the forms of these functions.

PART II.

9. Up to this point we have assumed that any one of our sets of correlated variables, e.g. a1 ̧.· an, or, as we may call it, any one' association,' is independent

of the variables in any other. It is now proposed to treat the several associations a bn, &c., as representing the state of the same material system, defined by n variables, at successive points of time. That is, if x ..an be the variables defining the system, fada dan is the chance that when t = 0 they shall lie within the limits

x1 = a1 and x1 = a1 + da1, &c.
x2 = a, and x2 = a12 + da2, &c.

Similarly fodb, dbn is the chance that when tr they shall lie within the limit xb, and a1 = b1 + db1, &c., and so on. Then generally the variables b1 . . . b1 are not independent of a because given that when t=0, a, a,, &c., that affects the chance that when t=T, x, shall b1, &c.

...

an,

We cannot therefore at once apply our investigation of Part I. to the series of functions fa, fb, &c.

10. It is now shown that in cases where the condition of complete independence is not satisfied, another condition, called the modified condition of independence, may be satisfied. That is, namely, that although the b's are not independent of the a's for such values of the time r, yet it may be the case that when r= or >T, df where T is a definite time, the b's are independent of the a's, or vanishes. It is da

shown that if that modified condition be satisfied for every pair of associations, or states of the system, separated by an interval of time not less than T, we may legitimately apply the method of Part I. to the whole series of associations, N2 in number, at intervals of time at or exactly as if they were mutually independent. If, that is, we form n linear functions of the type

T
N'

...

r1+dr1,

and find the chance that they shall respectively lie within the limits r1 &c., by the method of Part I. we obtain correct results so far as the exponential form is concerned. The result is of the form e-dr1 drn, with R a quadratic function of (r11), (12 −1°2), &c.