Application of Solid Hyper geometrical Series. 401 



Bv (9U „ _ m(l-q)(nq-m) 



B } (21) P*«- q (n-l) 



and by (24) 



_ m(l — q)(L — 2q)(nq-m)(nq — 2m) 



These equations lead to a quadratic for q, 



L m p 2 o -I L p 2 o m m l " p 2 o J 





2^20 



m1 = 0. . . . (75) 



A similar quadratic can be obtained from the other 

 marginal values. If q. q' are the values of q obtained from 

 the quadratics we may adopt \Zqq ( for the value in the 

 fitting ; denoting this by Q, we have 



_m , _m 



r ~Q" ' = Q" 



and n is given by (23), since 



_ rr>q(\-q) 

 P "~~ ,<-\ • 



A simplified solution of this character in which a priori 

 values are assigned to certain constants has to be employed 

 with great care. In the case of the 25000 deals at whist 

 (ordinary shuffling) dealt with above, the two quadratics 

 are 



</ 2 ~--62330? + '11028 = and </--G 1221? + -11217 = 0. 



Both have imaginary roots. It is thus impossible to fit this 

 table with a double hypergeometric series whose start is 

 at (0, 0). 



It is a matter of interest to observe that the roots are 

 imaginary because p 2 o and p 02 have values 1*66130 and 

 1*68318, differing by about 10 per cent, from the corre- 

 sponding theoretical value 1'8640 deduced from the theory 

 of chance on the hypothesis of perfect shuffling. The effect 

 of the faulty shuffling is to diminish the " standard 

 deviations " in the table, i. e. to crowd the table closer to 

 the mean values. There is a deficiency of high frequencies 

 and of very low frequencies, so that the actual table cannot 

 be fitted with a theoretical double series which allows for the 

 occurrence of " no trumps " in the hands of both partners. 

 It will be observed that the general solution of the preceding 

 paragraph gave a start at about (*5, *5). 



Phil. Mag. S. 6. Vol. 28. No. 165. Sept. 1914, 2 D 



