in our Proposition) is true, at any rate for rational functions of p, V1 – u cos w, and N1 - u* sin w, and is also most important. This objection, however, deserves to be examined with care, which we now propose to do in the course of our demonstration.

. 1

L. S" (+3cP,+...+(2i+1) cPet ...} dpi' dc.

-1 o

29. Multiply both sides of the last equation by the double element du'dw', and integrate, p24 (1—c*)du'dw'

(1+c+- 2cp)* The property of Laplace's Functions proved in Art. 26, shows that every term of the series on the right, except the first, vanishes of itself, independently of the other terms; and therefore (as was before intimated) the terms cannot accumulate. The first term is 47: and therefore the integral of the fraction on the left, that is

* 3+ (1 c) du'da'

= 47.

(1 + 02 – 2cp)" It is remarkable that this result is altogether independent

[ocr errors]
[ocr errors]
[ocr errors]

of c.

30. The truth of this may be shown also by integrating the fraction on the left. This cannot readily be done with the co-ordinates as at present chosen. But it may be done by a simple transformation, and a change in the way of taking the elements.

Suppose a sphere of radius unity described about the origin of co-ordinates. Let O' and w' be the angular co-ordinates to a point P, e' (or cosu') measured from a fixed point A along a great circle of the sphere, and w' the angle which this great circle makes with another and fixed great circle through A. Then do'. da' sin 0.

- du'dw', is an infinitesimal element of the surface of the sphere at P. Take D a point within the sphere, and let CD=c, and sup

[ocr errors]


pose CD meets the sphere in Q when produced forwards, and in q when produced backwards. Let u and w be the co-ordinates of Q. Then p (see its value, Art. 27) is the cosine of the angle which CP and CQ make with each other: and the distance of P from D=V1 +0 – 2cp. Let

2cp. Let it be the angle which the plane CPQ makes with CAQ, that is, the angle AQP. By changing the origin of the angles from A to Q, and dividing the surface of the sphere into new elements, beginning from Q as the origin, the element at P, with these new co-ordinates cos p and y, will be - dpdyr.

By reverting to the meaning of integration we see that the integral under consideration = (1 - 0) < limit of sum of all the elements of the surface of the sphere divided respectively by the cubes of their distances from D. But this, by the change of co-ordinates, also (1 – c) dpdy which

= 27
(1 +c+- 2cp)''

(1+0 – 2cp)* 1-c? 1

1-C 1 1 =21

+ const. = 27 W1+- 2cp = 47, whatever value chas. This coincides with the former result.




(1c) |

[ocr errors]
[ocr errors]

31. This integration helps us to see by what process c disappears from the result; and it will assist us in the latter part of the present demonstration.

The quantity 1 - p is the versed-sine of the arc QP, and is measured along the line QCq. Let this line be divided into n parts each equal to h, so that n.h=the diameter =2, n being very large and h very small. Draw perpendiculars to the diameter through these divisions cutting the circle QPq in a series of points; and call the distances of these points from D, beginning from Q,

1-c, s', s", s'"...... sln=1, 1+c. Suppose P is at the acth division; then

1-p=x.h, and d(1-P) or - dp= (+1) h - xh = h.

Then by mere expansion, omitting the squares and higher powers of h as they vanish in the limit with reference to the first power, we see the truth of the following;

d(1-P) (1+0* — 2cp) {(1-c)' +2c (1 – p)}'

- dp

[subsumed][ocr errors]

By giving x its successive values from 0 to n-1, and adding together all the resulting values of this expression and taking the limit, we have the integral with respect to p. It matters not in which order we effect the integration. Hence the whole integral

(1 - cm) dy dp
(1 +c* — 2cp)

.. +...+


n being made infinitely great,

-L* -{I-. 17o+G-4) +...+(2)

" {G – 7) +...+(---)}]


[ocr errors]

df [2+



Here it will be seen that the terms within the last brackets mutually destroy each other whatever be the value of c. It may also be observed that were this not the case, that whole part of the expression would vanish for the particular value c=1 (which is the only case we shall have to use), whatever the value of the sum of the terms following the multiplier 1-c', so long as that sum is not infinite.


32. Suppose now F(u', w') is any function of r' and w', then

F(u', w') du'do' (1 - 0)

(1 + c* — 2cp)* {1+

2i ; } w

The reasoning above enables us now to prove that the integral on the left-hand side = 47F (u, w), which directly leads to the theorem we are wishing to demonstrate.

The function F(u', w') at the point Q is F (u, w), call it F: let F", F"... F(") be its values at the points of junction of the successive elements along the great circle QPq. Then by

1 1 1 multiplying the successive values of F"... and adding them together, we have

F(u', w') du'do'

(1 + 0* — 2cp)

F(u', w') dy dp (1 –c)

(1+02 - 2cp)

[ocr errors]


glæ+1)) by F, F',

[ocr errors]

21 (1


[ocr errors]


[ocr errors]
[ocr errors]

n being made infinitely great.


1-C The fractions

diminish successively in value, being the ratios of QD to the successive values of DP. When c=1 each of them vanishes; and in the limit none of the factors F" F, F" F', ... become infinite. Hence the



integral = Lod4.2 F, when c=1, = 47 F (4, as) because F(x, @)

is a function of u and w only, and is altogether independent of t.

:: ATF(, 6) = L S"{1+3P,+...+(2i+1) Pet...} F(",0%)dji' dea';



غوا F(1



[ocr errors]

26 (F (u', w')

3F(u', w'). +

P, +......



47 The general term of this, viz.

27 2i+1

F(u', w') Pidu'dw',

47 which we will call Fi, is a function of u and w; and evidently satisfies Laplace's Equation in u and w, because P, does so. Hence, this is a Laplace's Function, of the eth order: and the result is, what we were to demonstrate, that any function of u and w may be expanded in a series of Laplace's Functions ; or,

Flu, w) = F,+F,+F,+ ...... +F+ ...... 33. Those who are at all acquainted with the controversy which followed the first discovery of these remarkable functions by Laplace, will understand why we have entered so fully upon

the subject. Laplace's demonstration in the Mécanique Céleste was by no means conclusive. This Mr Ivory pointed out in the Philosophical Transactions for 1812; and in the Volume for 1822 he threw considerable doubt upon the applicability of the theorem to functions that are not rational and entire functions of M, VI-cosw, V1-? sin w. Poisson has written much upon the subject. In the first edition of the author's Mechanical Philosophy the last method of Poisson was followed, as given in his Théorie Mathématique de la Chaleur; in which he effects the integration of the fraction on the left-hand side by the artifice of substituting for it an integrable, but entirely different fraction in its general form, but which coincides with it in the particular case for which he requires it in the result, viz. when c=1. In the Second

« ForrigeFortsett »