XXXV XLVI] Introduction Ixi 



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



1 du x a 



-^"-TTT 



y dx f(x) 



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



_ _ 



ydx c +c 1 x + c*a; i + ... + c n x n + ..." 



then a, c , c lt c t , ... c n ... can be uniquely determined from the 'moment co- 

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

 other constants /9,, /3 2 3, $,, /9 4 15, ... which vanish for the Gaussian curve, and 

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

 c , c,, 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 n 

 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 a . If we stop 

 at c s then our differential equation is of the form 



1 dy _ x a 



.(li), 



y dx c + 



and we need only /9, = fi^jn? an ^ & = /* 4 //i, 2 , where ft, 2 , p,, /j. t are the second, third 

 and fourth moment coefficients about the mean. 



If we take the form - ^ = '.we reach the Gaussian, in which each con- 

 y dx 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 + c^x 



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 -fim , then contributory cause-groups are 



- 



ydx 



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 fm -- - was adopted to allow of this wide generalisation of the 

 y dx c + c,x + c t x* 



Gaussian hypothesis. 



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



/ 8 n (even) = (n + l)[i/3 n _ 1 + (l+ia) ; 9 n _ J )/(l-Hn-l)a) ............ (Hi), 



/S. (odd) =(n+l){iA/8-, + (l+i)/8^}/(l-i (-!)) ...... 0"i), 



where a = (2& - 3/3, - 6)/(& + 3) ........................... (liv), 



in terms of lower /3-constants. 



