ESENTATION. 353 



The curve takes on very closely such forms as are given 

 by the following statistical equations:— 



U = A (1 4- */o)~ (l-x/b)™\ or its equivalent 



y = A(t/a) m [(c-t)(c-a)\»; 



y = A(l-x 2 Ja 2 )™ 



y = A (1+xfaF* e-»* 



U = A(l+.r 2 /a 2 )- m e- t -- 1 - 



»»4.(*-«fr- 

 and of course exactly expresses 



y = A xr™ e-** 



y = A *-W 



5. Logarithmic homologue of curve.— The synthesis of 

 the curve is brought into clearer light by examining its 

 logarithmic homologue, viz. the curve the co-ordinates of 

 which are the logarithms of the co-ordinates of the given 



Thus the nature of the products of the ordinates of the 

 two generating curves can readily be seen by taking the 

 logarithm of botli sides of (1), thus :— 



log y = log A + m Xogx + nx* (11) 



thus with positive or negative values of both m and », the 

 numerical sum of the variable quantities is indicated ; but 

 the antilogarithm where the signs are negative is the 

 reciprocal of the antilogarithm where the signs are 



Similarly if m be + and n -, or vice versa, the numerical 

 difference is indicated, the antilogarithm in the latter case 

 being the reciprocal of the antilogarithm in tlie former. 



The types of curves fully written out, included by the 

 expression considered, are as shewn in the schedule above 

 given. As m, n or p approach the values there indicated, 

 the curve approaches that defined in the schedule, and the 



