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Ex. 2. The mean distances of all points within a sphere from

3 a its centre = ī:

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Ex.3. Find the mean inclination to a given plane of a system of planes, whose number is infinite, regularly distributed in space.

Let the positions of these planes be determined by means of their normals, and let us suppose all these to be drawn through the centre of a given sphere : then, as these planes are regularly distributed in space, the number of normals to them contained within a given portion of the surface of the sphere varies as that portion, and consequently we may consider the surface-element of the sphere to express the point of section of the normal of a particular plane with it. Now if a = the radius of the sphere, and (a, 0, 0) is the place of this element in reference to the system of polar coordinates explained in Art. 165, a’ sin 6 do do is by (69), Art. 243, the area-element of the surface; so that if the given plane is the plane of (x, y), o is the angle at which the other plane is inclined to it. Consequently in this problem The numerator of (36) = a? ["" /*o sin 0 de do

= 27 a; The denominator of (36) = a* /** / "sin 6 do do

= 27a". as the mean angle = 1 = 57.29578°; that is, is equal to the angle, the subtending arc of which is equal to the radius.

In the eight large planets, the mean value of the inclination of the planes of their orbits to the plane of the ecliptic is 2° 19' 64"; and in case of seventy-nine small planets the mean inclination of the planes of their orbits to that of the earth's orbit is 7° 41'3". Consequently as these observed results are so far below the mean a priori determined as above, we are necessarily led to infer a physical connection between these planets and the plane of the ecliptic; that is, the doctrine of chances indicates thus far a physical law which binds these several bodies to the solar system.

On the other hand in the case of the comets, so far as a general inference can be drawn from the calculated elements of 190, which bave been observed between the years 1556 and 1861* ; the mean inclination to the plane of the ecliptic is about 50°, which is so near to the a priori mean as calculated above that no connection can be hence inferred as to a physical law of relation. Many more comets have been seen, but the above are all which have been observed between these years, and whose elements have been calculated. Also of the 18 which have elliptic orbits, and whose periodic times are known, at least approximately, the inclinations are very small, so that hereby is shewn a very strong probability of a law of relation between them and the solar system. And if these comets are excepted, the mean inclination of the others rises considerably above 50°.

Ex. 4. Find the mean of all squares inscribed in a given square.

Let a = the side of the given square, and let x = the distance from one of its angles of the angle of the inscribed square ; so that the side of the inscribed square = (2x2 – 2 ax + a2). Hence

The numerator of (36) = / "(2x2 – 2ax +a) dx

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Ex. 5. Find the mean of all the focal radii vectores of an ellipse, which are drawn at equal angular intervals. Let the equation to the ellipse be r =


1-e cos

; then
The numerator of (00) =J_, 1-ecos 0
The numerator of (36) = /* a (l – e) do

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• Chambers' Hand-book of Astronomy. Appendix III. Murray, London, 1861.

.:. The required mean value = a (1-e?)* = 6, that is, is equal to half the minor axis of the ellipse.

Ex. 6. The mean length of all parallels of latitude on a sphere whose radius = a, drawn at equal angular intervals from the centre, = 4a.

Ex. 7. Another interesting problem of mean value is that of the distribution of double stars in the celestial vault, and it is also important on account of the inference drawn from it by W. Struve as to the physical connection of such binary systems.

Let n = the number of stars up to a given order, say, to the eighth order inclusive; in which case n = 100,000, more or less. So that the greatest number of pairs of these = "" > ; let r = the number (very small) of angular seconds of separation of the two members of a double star ; so that a pol expresses in seconds the area of the circle of which r is the radius. Let x = the number of double stars which the space ape occupies, when these pairs are regularly distributed; then as the surface of the whole sphere expressed in seconds = 47 (206265)?,

n(n-1) pe

H 8(206265)2' which assigns the number of double stars which, within a radius of r", the celestial vault ought to exhibit.

Now Struve has found 311 double stars between the north pole and 15° of south declination, the angular distance between the members of which does not exceed 4"; whereas if such stars were regularly distributed, there ought, as he finds, to be one at most. We are therefore obliged to infer that the distribution in pairs is not fortuitous, but that there exists a true physical law connecting the two members of such a binary combination.



SECTION 1.Reduction of Multiple Integrals by simple

application of the Gamma-function. 276.] An examination of the processes required for the complete solution of the problems of the two preceding chapters and of other similar questions shews that they depend on the determination of the value of multiple integrals; and that the problems are only solved when these integrals are evaluated. Now in many cases the element-functions are of certain special forms, the general forms of which can be determined by means of certain other integrals which have already been evaluated; such as the Gamma-function and the integral-logarithm. In other cases the order of the multiple integral can be so far reduced, that the solution depends on a single integration; when it is said to be reduced to a quadrature. There are other processes of simplification which are frequently of great importance. All these we propose to investigate in the present chapter.

277.] The limits of the definite integrals which ordinarily occur are either constant, or are determined by an equation which fixes the range of all the integration processes; and according as the limits are assigned in one or other of these two modes, so will the method for the reduction of the integral vary. I will first take the case in which all the limits are constant; and explain the process, devised by M. Cauchy, by which the value of a given multiple integral may be made to depend on that of one or more single integrals.

Let the integral, which I will denote by 1, be a multiple integral of the nth order, and of the form 1=11....... dz dy dx ;

(1) in which n variables x,y,z, ... are involved; where m is a positive quantity ; P and Q are functions of the variables of the form P = P,. P,.P, ...;

(2) Q = Ro + Qx+Q,+Q: +...,

(3) where Ro is a constant; P, and Qx are functions of x only; P, and Q, are functions of y only; and so on; where q is always a positive quantity, and may have an impossible part, if the real part is always positive; and where all the limits of integration are constant.

Now according to an artifice due to M. Cauchy, which has already been applied in Art. 131, I propose to replace q- in (1) by a definite integral, with constant limits, in terms of a variable t which is independent of the former variables. For this purpose we have by (250), Art. 122, .

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Let P and q be replaced by their values given in (2) and (3); and separating the variables as the limits are constant, we have

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To simplify this, let us make the following substitutions for the definite integrals, each integral being taken with its proper limits;

Pre=&idx = u, P8-eedy = v, p.e-vedz = w, (7) so that u, v, w, ... are functions of t; then

1 pos
I m-le-201...wvu dt;


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and thus the multiple integral given in (1) is reduced to a single integral.

278.] The following is an example of this general theorem; in which however I take only three variables, as the process is the same in all cases.

P = 28–179-1zr-1e-(ax+by+cz), I

Q = k tax+By+yz, where p, q, r, a, b, c, k, a, ß, y are positive constants; and let us suppose oo and 0 to be the limits of integration for each of the variables; then


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