OF ATTRIBUTES IN STATISTIC*. 269 



(ABCD . . .) = <, (a,,**^, ... a.) 



(aBCD . .) = ^j(a,,0j,a 8 , ... a.) 



<fec. Ac. 



or 2" equations altogether. But if m < 2", we can express the constants in terms of 

 the frequencies by means of the first m equations, and then insert their values in the 

 remaining equations, thus obtaining 2" m necessary relations between the fre- 

 quencies. If the surface we are dealing with is symmetrical, there will be only 2""' 

 independent ultimate groups, and m must be less than 2""' if special relations are to 

 subsist between the groups. 



Now this is the case in the normal surface itself. The standard deviations will not 

 appear in any of the multiple integrals, which must be functions solely of the correla- 

 tion coefficients, 7-, 3 , r 18 , r w , &c., and the total frequency. That is to say n variables 



give 1 + ' ' ., constants that appear in the expressions for the total frequencies of 

 the ultimate groups. This gives the following figures : 



n 1 + i^inl 2"-' 



2 2 2 



344 



478 



5 11 16 



6 16 32 



There must therefore be one relation subsisting between the ultimate fourth order 

 groups in normal correlation besides the mere equality of contrary frequencies five 

 relations between the fifth order groups, sixteen between those of the sixth order, 

 and so on. If we could find these relations the expression of fourth order frequencies 

 in terms of third, sixth in terms of fifth, and so on, would cease to be indeterminate 

 as in the general case of equality of contraries. Mr. SHKPPARD'S theorem could then 

 be extended to the case of groups of any order in normal correlation, which would 

 give results of great interest "for calculating certain chances, e.y., the chance of a man 

 being above average when his father, father and grandfather, father, grandfather, 

 and great-grandfather were above average. 



Finding myself quite unable to solve the above problem, I have handed it to 

 Professor KAKL PEARSON ; he informs me that the relations sought depend on 

 equations between the area, sides, and angles of the generalised spherical triangle in 

 hyper-space, but the problem has not yet been solved. It is curious that investiga- 

 tions into the theory of logic should lead to properties of hyper-spherical triangles 

 or tetrahedra. 





