ordinate (the mode, etc) is asymmetrical, it is obvious that 

 the fundamental assumption which furnishes the ordinary 

 "probability curve " must be modified, since if the y axis 

 be (arbitrarily) taken at the maximum or minimum ordinate 

 the rate of variation of frequency is not the same for 

 negative as for positive values of x. 



We may suppose that any sufficiently small length of the 

 curve may be represented, with any required degree of 

 precision, by some probability curve 



y = Ce -^ = e -^ + o (3) 



in which 



e c = C,orc = logeC (3a) 



Since under the hypothesis x has determinate values, k, 

 the modulus of x, may be regarded as exact only for the 

 part of the curve in question, and to have a different value 

 for other parts of the curve. Obviously we may suppose 

 that both Core and /c 2 are assignable functions of x t that 

 is to say (3) becomes 



tf = F(*)r^«ore^W^) (4) 



and the functions F or F and f may of course be as com- 

 plex as the case may require and include any necessary 

 constants. When it is legitimate to suppose that 



F(x) = <• + /ilog( ± .r) (5) 1 



f(*)= V** (5a) 



these expressions substituted in the above give 



» = •-*/**♦•♦**»<*■» (6) 



hence putting 

 p = 2-«; n= - 1/y ; A'^e a or log A = a ; m = /I. .(6a) 

 the above expression is seen to be equal to 



It is evident however that as x approaches a zero value 



/^ (or in) would also require to approach a zero value, in 



1 If x be negative log - (•) may generally be regarded as log + x. 



