612 ON THE PEOBLEM OF RANDOM VIBRATIONS, AND [441 By parity of reasoning when E and F are interchanged, the same integral is proportional to J0(f), and may therefore be equated to A0'J0(e)J0(f), where A0' is now an absolute constant, whose value is at once determined to be unity by making e, or f, vanish. 'The lemma f Jo Jo v/(e2 +/2 - 20/cos (?) dO = 2-rrJ, (e) J0 (/) (25) is thus established*. We are now prepared to investigate the probability •* n v > ^D ^2> • • • <"n) that after n stretches 11} lz, ... ln taken in directions at random the distance from the starting-point 0 (Fig. 2) shall be less than an assigned magnitude r. The direction of the first stretch ^ is plainly a matter of indifference. On the other hand the probability that the . angles 6 lie within the limits 0X and d: + d6l , #„ and 02 + d62, ... 6n^ and Sn_^ + dOn_^ is Ai Fig. 2. which is now to be integrated under the condition that the nth radius vector sn shall be less than r. . Let us commence with the case of two stretches ^ and lz. Then P*(r; l^l^^ Air the integration being taken within such limits that s2< r, where sa* = l*+ lz2 - 2lil2 cos 0lt The required condition as to the limits can be secured by the introduction of the discontinuous function afforded by Weber's integral. For r I Ji (rx) J0 (szx) dx Jo vanishes when s.2 > r, and is equal to unity when s2 < r. After the introduction of this factor, the integration with respect to 6^ may be taken over the complete range from 0 to 27r. Thus /*5*7r Pffi dxJ^ (rx) J0(s2ai). * Similar reasoning shows that if D0 (g) represent a symmetrical purely divergent wave, provided that/> e.s<?)) is a potential satisfying everywhere the equation V2 + l=0, and accordingly may be expanded Q round G in the Fourier series '