106 Prof:. F. Y. Edge worth on a Variant Proof of 



This may also be written 



. sin (2k\ / x) /sin (2ic*/a) 

 v = A i — f—± I — £ — f—± . cos (at + e) 



(2k\/oc) /sin (2tcy/ 

 \tcs/x I 2/c^/a 



which shows usjthat rj vanishes for 2/c v /x = stt, where s is a 

 positive integer. 



(6) m=l, nsl. The depth and the breadth both vary 

 directly as x. We then have 



v= A Jity «) / J.byo , cos(<rf+e) , 



(7) m = £, « = 2. The depth now varies as ^/x and the 

 breadth as x 2 . Hence we have 



, = A J -^»/ 



^-^cos(^ + e). 



<r' 



(8) m = l, n = 2. The breadth again varies as <# 2 but the 

 depth now varies as x. Therefore we have 



. J 2 (2tc*/x) I J 2 (2/c x /a) / • N 



x I a 



We may note that if m==l, that is the depth varies as x, 

 then equation (7) becomes 



J n (2fc^x) / j ?l (2/c v / q) 



x n/2 / a n/ 



X. JL Variant Proof of the Distribution of Velocities in a 

 Molecular Chaos. By Professor F. Y. Edgewokth, F.B.A., 



All /Souls College, Oxford*. 



THE received proofs of Maxwell's law involve the 

 fundamental axiom of Probability, the stability of an 

 average comparatively with its constituents. But the higher 

 theory of Probability which deals with deviations from 

 averages seems not to have been fully utilized. It follows 

 from this theory that if there are a number of quantities 

 from time to time assuming different values independently 

 at random according to (almost) any law of frequency, then 

 the average or sum or generally (almost) any linear function 

 of these elementary quantities tends to fluctuate according 

 to the normal (or " Gaussian ") law of frequency. The 

 randomness attributed to the elements is not incompatible 



* Communicated bv the Author. 



