1911] SOHLOMILCH'S THEOREM IN BESSEL'S FUNCTIONS 23
We now superpose an infinite number of components, analogous to (2) with the same origins of space and time, and differing from one another only in the direction of f, these directions being limited to the plane xy, and in this plane distributed uniformly. The resultant is a function of t and r only, where r = V(&'2 + y"**), independent of the third coordinate z, and therefore (as is known) takes the form
s = a0 4- ai<70 (r) cos t -f ci2J0 (2r) cos 2t + asJ0 (3r) cos St+ ..., .. .(5)
reducing to (1) when t = 0*. The expansion of a function in the series (1) is thus definitely suggested as probable in all cases and certainly possible in an immense variety. And it will be observed that no value of £ greater than TT contributes anything to the resultant, so long as r < TT.
The relation here implied between F and / is of course identical with that used in the purely analytical investigation. If <£ be the angle between f and any radius vector r to a point where the value of / is required, £==rcos<£, and the mean of all the components F(g) is expressed by
2 [*"•
T JO
The solution of the problem of expressing F by means of/ is obtained analytically with the aid of Abel's theorem. And here again a physical, or rather geometrical, interpretation throws light upon the process.
Equation (6) is the result of averaging F(%) over all directions indifferently in the xy plane. Let us abandon this restriction and take the average when £ is indifferently distributed in all directions whatever. The result now becomes a function only of R, the radius vector in space. If 6 be the angle between R and one direction of f, £ = R cos <9, and we obtain, as the mean
............(7)
. o
where jP,' = F.
This result is obtained by a direct integration of F (£) over all directions in space. It may also be arrived at indirectly from (6). In the latter/(r) represents the averaging of F(^) for all directions in a certain plane, the result being independent of the coordinate perpendicular to the plane. If we take the average again for all possible positions of this plane, we must recover (7). Now if 6 be the angle between the normal to this plane and the radius vector R, r = JR, sin B, and the mean is
f "f(Esio.0)sm0d8............................(8)
Jo
* It will appear later that the a's and b's are equal.rom the expression for the