Hyper geometrical Series and Frequency Distributions. 389 



For the present purpose we need only to introduce S/3, and 

 with sufficient accuracy we may take 



8(U 2 -'2gy) = -28/3 cos 2x {26) 



We suppose 5/3= — •000,000,2. so that the new value of /3 

 is —'000,052,6. Introducing corrections according to (2tf) 

 and writing only the last two figures, we obtain column 5 of 

 Table L, in which the greatest discrepancy is reduced from 

 10 to 4 — almost as far as the arithmetic allows — and becomes 

 but one-millionth of the statical difference between crest and 

 trough. This is the degree of accuracy attained when we 

 take simply 



TJr=?/ — ae~ ■■' < "OS .v — /3 e ~ 2y cos 2x — <ye~* y cos 3 a, . (2 7 J 



with a.— ^ 6 , g and 7 determined by Stokes' method, and 

 ft determined so as to give the best agreement. 



XXXIX. The Amplication of Solid Hypergeometrical Series 

 to Frequency Distributions in Space. By S. D. Wicksell, 

 Dr. Phil, Lund. Sweden*. 



IN the number of this Journal issued in September 1914, 

 Dr. L. Isserlis, under the above title, published a paper 

 on the fitting of hypergeometrical series to correlation 

 surfaces. The problem to describe curves of variation by 

 aid of hypergeometrical series was treated as long ago as 

 1895 by Prof. Pearson in his classical Memoir : " Skew 

 Variation in Homogeneous Material/' Phil/Trans, vol. clxxxvi. 

 Later, in 1899, Prof. Pearson gave a fuller discussion of the 

 hypergeometrical series (Phil. Mag. vol. xlvii.). It is this 

 paper that is the starting-point and chief place of reference 

 of Dr. Isserlis. On the whole, the hypergeometrical series 

 and its special case for ?z=co, the binomial series, play a 

 dominating part in Prof. Pearson's celebrated theory of 

 variation of one variate. As a consequence hereof, it was 

 natural that the attempt should be made to employ solid 

 hypergeometrical series as a means to describe also surfaces 

 of correlation. Hereby, however, a fact has evidently been 

 overlooked that greatly limits the range of applicability of 

 any hypergeometrical or multinomial types of correlation 

 functions. Of course, there must be some identical relations 

 between the moments that should be more or less fulfilled 

 in all cases of application. Dr. Isserlis also produces several 

 such relations. But it is evident that he has not ascribed 

 too much importance to these limitations. In the theory of 

 variation of one variate there are similar conditions, but they 

 * Communicated bv the Author. 



