XXXV— XLVI] Introduction Ixi 



If the differential equation to the uui-modal frequency distribution be 



y dx fix) 



we may suppose /(*•) expanded in a series of powers of x, and so 



1 dy X — a ,,s 



- ~f-= (1). 



y ax Co + CiX + c^x- + . . . + c„a;" + . . . 



then a, Cq, Cj, c^, ... Cn--- can be uniquely determined from the 'moment co- 

 efficients ' of the frequency distribution. These constants are functions of certain 

 other constants /3i, /So — 3, /S3, /34— 15, ... which vanish for the Gaussian curve, and 

 are small for any distribution not widely divergent from the Gaussian. Further 

 Co, Ci, C2...c„... converge, if, as usual, these constants are less than unity, the 

 factors of convergence being of the order V/3-coastant. As a matter of fact c„ 

 involves the (»i + 2)th moment coefficient, and thus we obtain values of the 

 c-constants subject to very large errors, if we retain terms beyond c.^. If we stop 

 at Ci then our differential equation is of the form 



Idy ^ x-a y, 



y dx c^ + CiX+CiX"^ 



and we need only ^1 = fi^jfj,^ and ^.y = /xJfj.J', where /i,, M3> /^4 ^-fe the second, third 

 and fourth moment coefficients about the mean. 



1 dv 30 ^ ct 



If we take the form — f^ = , we reach the Gaussian, in which each cou- 



y dx Co 



tributory cause-group is independent, and if the number of groups be not very 



large, each cause-group is of equal valency and contributes with equal frequency 



results in excess and defect of its mean contribution. If we take — r^ = , 



y dx Co + CiX 



then each contributory cause-group is still of equal valency and independent, but 



does not give contributions in excess and defect of equal frequency. 



Finally if we take - ~ = '- , then contributory cause-groups are 



•' y dx Co+CiX+CiX^ •' or 



not of equal valency, they are not independent, but their results correlated, and 

 further contributions in excess and defect are not equally probable. The use of this 



form — =^ = was adopted to allow of this wide generalisation of the 



y ax Co + CiX + c^x- '■ ° 



Gaussian hypothesis. 



If we adopt it, every /3-constant is expressible by means of the formulae : 



/3„ (even) = («+!) [i/3,._i -f (1 + ^a) /3„_.,1/(1 - K« " 1) «) (lii). 



je„(odd) =(n + l)liA/8n-. + (l+i«)/8„-.)/(l-H»-l)«) (liii). 



where ci = {2^^-3/3,-6)/{^.,+ S) (liv), 



in terms of lower /3-constants. 



