then substituted in the second equation gives rise to an ordinate u', which when substituted in the first, gives rise to a third abscissa e", and so on to infinity. That is to say, in order to represent the continued alternate employment of the two preFig. 14.

Fig. 13.

ceding equations, we must draw through the point u a horizontal line up to the curve, and through e the point of intersection draw a vertical as far as the straight line, through the point of intersection u' draw a horizontal up to the curve, through the point of intersection e' draw a vertical as far as the straight line, and so on to infinity, descending more and more towards the point sought.

287. The foregoing figure (13) represents the case in which the ordinate arbitrarily chosen for u is greater than that which corresponds to the point of intersection. If, on the other hand, we chose for the initial value of u a smaller quantity, and employed in the same manner the two equations e arc tan u, u = = we

should again arrive at values successively closer to the unknown value. Figure 14 shews that in this case we rise continually towards the point of intersection by passing through the points u eu e' u" é", &c. which terminate the horizontal and vertical lines. Starting from a value of u which is too small, we obtain quantities e é é" e"", &c. which converge towards the unknown value, and are smaller than it; and starting from a value of u which is too great, we obtain quantities which also converge to the unknown value, and each of which is greater than it. We therefore ascertain

successively closer limits between the which magnitude sought is always included. Either approximation is represented by the

formula

€ = arc tan

...

[are tan are tan are tan )}].

When several of the operations indicated have been effected, the successive results differ less and less, and we have arrived at an approximate value of e.

in a different order, giving them the form u=tane and e= λu. We should then take an arbitrary value of e, and, substituting it in the first equation, we should find a value of u, which being substituted in the second equation would give a second value of e; this new value of e could then be employed in the same manner as the first. But it is easy to see, by the constructions of the figures, that in following this course of operations we depart more and more from the point of intersection instead of approaching it, as in the former case. The successive values of e which we should obtain would diminish continually to zero, or would increase without limit. We should pass successively from e" to u", from u" to e, from e' to u, from u' to e, and so on to infinity.

The rule which we have just explained being applicable to the calculation of each of the roots of the equation

which moreover have given limits, we must regard all these roots as known numbers. Otherwise, it was only necessary to be assured that the equation has an infinite number of real roots. We have explained this process of approximation because it is founded on a remarkable construction, which may be usefully employed in several cases, and which exhibits immediately the nature and limits of the roots; but the actual application of the process to the equation in question would be tedious; it would be easy to resort in practice to some other mode of approximation.

F. H.

18

289. We now know a particular form which may be given to the function v so as to satisfy the two conditions of the problem. This solution is represented by the equation

Nx

The coefficient a is any number whatever, and the number n is nX such that = 1-hX. It follows from this that if the tan nX initial temperatures of the different layers were proportional to sin nx the quotient they would all diminish together, retaining between themselves throughout the whole duration of the cooling the ratios which had been set up; and the temperature at each point would decrease as the ordinate of a logarithmic curve whose abscissa would denote the time passed. Suppose, then, the arc e being divided into equal parts and taken as abscissa, we raise at each point of division an ordinate equal to the ratio of the sine to the arc. The system of ordinates will indicate the initial temperatures, which must be assigned to the different layers, from the centre to the surface, the whole radius X being divided into equal parts. The arce which, on this construction, represents the radius X, cannot be taken arbitrarily; it is necessary that the arc and its tangent should be in a given ratio. As there are an infinite number of arcs which satisfy this condition, we might thus form an infinite number of systems of initial temperatures, which could exist of themselves in the sphere, without the ratios of the temperatures changing during the cooling.

290. It remains only to form any initial state by means of a certain number, or of an infinite number of partial states, each of which represents one of the systems of temperatures which we have recently considered, in which the ordinate varies with the distance x, and is proportional to the quotient of the sine by the arc. The general movement of heat in the interior of a sphere will then be decomposed into so many particular movements, each of which is accomplished freely, as if it alone existed.

Denoting by n,, ng, ng, &c., the quantities which satisfy the nX

equation

1-hX, and supposing them to be arranged in

order, beginning with the least, we form the general equation

Vx= a1e-knit sin

n ̧x + αe¬kn2 sin n ̧x +a ̧e-kn, sin n ̧æ + &c.

If t be made equal to 0, we have as the expression of the initial state of temperatures

vx = a ̧ sin n ̧x + α, sin n ̧x+a, sin n ̧x + &c.

The problem consists in determining the coefficients a1, a, a, &c., whatever be the initial state. Suppose then that we know the values of v from 0 to x = X, and represent this system of values by F(x); we have

x =

F' (x) = 1 (a ̧ sin n ̧æ + a ̧ sin n ̧æ + a ̧ sin n ̧æ+ a ̧ sin n ̧æ+ &c.)'... (e).

X.

291. To determine the coefficient a,, multiply both members of the equation by x sin nx dx, and integrate from x = 0 to x = mx sin nx dx taken between these limits is

The integral (sin

If m and n are numbers chosen from the roots n1, ng, nz,

We see by this that the whole value of the integral is nothing; but a single case exists in which the integral does not vanish,

0

0

namely, when m = n. It then becomes and, by application of known rules, is reduced to

1 Of the possibility of representing an arbitrary function by a series of this form a demonstration has been given by Sir W. Thomson, Camb. Math. Journal, Vol. II. pp. 25–27. [A. F.]

It follows from this that in order to obtain the value of the coefficient a,, in equation (e), we must write

the integral being taken from x = 0 to x = X. Similarly we have

=

1

2 fx sinn, F(x) dx-a,(X-sin 20,x).

2

In the same manner all the following coefficients may be deter

mined. It is easy to see that the definite integral 2 få sin nx F (x) dæ

always has a determinate value, whatever the arbitrary function F(x) may be. If the function F(x) be represented by the variable ordinate of a line traced in any manner, the function x F(x) sin nx corresponds to the ordinate of a second line which can easily be constructed by means of the first. The area bounded by the latter line between the abscissæ x O and x = X determines

=

the coefficient a,, i being the index of the order of the root n.

The arbitrary function F(x) enters each coefficient under the sign of integration, and gives to the value of v all the generality which the problem requires; thus we arrive at the following equation

This is the form which must be given to the general integral

in order that it may represent the movement of heat in a solid sphere. In fact, all the conditions of the problem are obeyed.