XXXV— XLVI] Introduction lxi 



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



1 di/ x — a , . . . 



ydx f{x) 



we may suppose f(x) expanded in a series of powers of x, and so 



1 dy _ x — a ... 



y dx c + c x x + c 2 x 2 + ... + c n x n + 



then a, c , c lt c 2 , ... c H ... can be uniquely determined from the 'moment co- 

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

 other constants /3 1; /3 2 — 3, /3 3 , /3 4 — 15, ... which vanish for the Gaussian curve, and 

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

 c , Ci, c 2 ...c n ... converge, if, as usual, these constants are less than unity, the 

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

 involves the (n + 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 c 2 then our differential equation is of the form 



ldy = x-a y . 



ydx Co + dx+CvX* 



and we need only & = (ifff** and /3 2 = /J. 4 /fi«-, where fu, /li 3 , /j. 4 are the second, third 

 and fourth moment coefficients about the mean. 



1 dij oc ■■ • ct 



If we take the form — r- — . we reach the Gaussian, in which each con- 



ydx c 



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 — ^ = — , 



y dx c + 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 - -M- ■ ■ : , then contributory cause-groups are 



' ydx c + CiX + dx* J -. 



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 



1 di] sc — a 



form - -~= was adopted to allow of this wide generalisation of the 



y dx c + CiX + c 2 ar r ° 



Gaussian hypothesis. 



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



A,(even) = ( M +l){i / S n _ 1 + (l+ia) y 8„_ 2 j/(l-H«-l)«) 0"). 



£„ (odd) =(n+ 1){£& A l -. + (l+i«)/S«- 2 }/(l-H»-l) a ) 0»i). 



where a = (2ft-3ft-6)/08,'+3) (liv), 



in terms of lower /S-constants. 



