ilANJJOM FLIGHTS IN ONE, TWO, OB THREE DIMENSIONS 625 It is .some cheek upon the formula* to compare the exact results for n= 6 in ((54) with those derived for the case of n great in (70), although with such a modi-rale value of n no precise agreement could be expected. The following table gives the numerical results for ldPK/dr in the two cases: TABLE II. n = 6. rll From ((51) From (70) i) ..... "•2~>()()r"/l~ ^483^;^ •r) ... •Oft'MK) •O.r)88(i 1 ..... •200.r) •ii(K>7 ~2 ..... •4 1 (!7 •4 If!!) tt ..... •2! M) •25)22 4 ..... •os;i3 •.[().r).r) r> ..... •oo(ir)2 •007 Hi (J ..... •ooooo ...... So far us the principal term in (TO) is concerned, the maximum value ore-urn whrii /•//— 2. If. will la1 .seen (,hut tho agreement of the two Formuk) in in fact very good, *«i lon^ UH /•;/ ditcH not, much exceed \/->i. AH the, maximum value of r/l for which tSn» Irm* rrsull. differs from xevo, is a] )proaehei I, the agreement neces-Miirily falls i»tV, Hi-ymid r/l — n, whe.u the true value, iw xero, (70) yields finite, thouh small, I',S. March 'Jrd. ••-•-In (45) we have the cxpreHHion for [ihe probability of a iv.miltniil i>"1 wlii-n a large number (•//.) of iHiqx-riodic vibrationH are combined, nlutw rcprcwnlnlive poinlH are distributed at random along the circumference ufa cirrh» uf rmlitiH /, HO that the, comporumt amplitudoN are all equal. It IB of iiitt.Tt'Kt tiM*xtt«nd t,ho inveHtigation to cover the cane of a number of groups in which lh<« umpliludeH are different, nay a group of pi components of amplitude /i, n group containing ?% of amplitude 19, and HO on to any number of groups, hut ill way* umW llw roatriction that every p is very large. The total number (2J/;) niny Htill br denoted by n, The roBult will be; applied to a caae where flu- number of groupH in infinite, the reprewintative points of the components being diHlribuled at random over the area of a circle of radius L. Wo start from («U }, now taking- the form The fh«rivation of the limiting form proceeds as before, where only one I WUH connulcrutl. Writing 8l « |pAa, «8 » %pA\ etc., we have K. VI. 40 f(:4 appears to have been devised to meet the case when s = 0 and the integral,, though finite, does not converge to a definite value when x = <x>. If, however, ,s < -1, or <0, respectively, according as the cosine factors in (38) do or do not produce a constant term, the integral (38) has been shown to be finite; it is -also convergent; and the integrals obtained by omitting before each successive differentiation the factor to be differentiated, viz.': cos (%ir-rx-lqir) where g<p, are also finite (cf. Todhifnter's Integral Calculus, 1889, Arts. 214, 284). In these circumstances it would appear that-_(38) is itself valid, and that it is unnecessary to consider the integral obtained by "the next following operation" (s =p + f - %ri). It would seem then that the above considerations are sufficient to justify the differentiation by which 0,t is obtained (jp = 0, s- -f), and a fortiori that for <fe (# = 0, s=-2), etc. W..IY.S.]elative index is then much restricted, the