412 Journal of Agricultural Research vu. iii.no. 5 



it does only the assumption that if we equate the area and moments of 

 a theoretical cur\-e to the area and moments of a series of observations 

 we shall get a reasonable fit of the curve to the obser\^ations. Experience 

 with the method in the hands of different workers in England and Amer- 

 ica has abundantly demonstrated that this assumption is entirely justi- 

 fied in the fact. 



In the papers cited, and in others also, Pearson has given the equations 

 for the calculation of the constants from the moments in the case of 

 (a) skew frequency cur\-es in general, (b) sine curves, (c) parabolas of all 

 orders, (d) the point binomial, (c) hypergeometrical series, etc. There 

 has been lacking, however, the determination of the equations connect- 

 ing moments and constants for the general family of logarithmic curves 

 of the type 



}' = a + bx + ex- + d\og(x+ a) 



and its modifications. I suggested some time ago to Mr. Miner that he 

 attack the problem, which, while theoretically simple and straightfor- 

 ward, proved rather laborious in the actual carrying out. This he has 

 done, with the results set forth in this paper. 



Raymond Peari, 



MOMENTS OF A LOGARITHMIC CURVE 

 GENERAL CASE 



Let y,, v,, J3, . . . , y^, . . . , yjn he a. series of ordinates with cor- 

 responding abscissae x,, x^, x^, . . . , Xf;, . . . x,„ to which is to be fitted 

 the curve y = a + bx + c logmX. 



Let the unit of calculation = x^ — a;, and the origin be placed at 

 2Xi — %2. The abscissae expressed in units of calculation and taken from 

 the new origin will then be i, 2, x\, . . . , x'^., . . . .v'„. In calcu- 

 lating the moments each ordinate must first be multiplied by the base, 

 li,r.^, of the rectangle of which it is the mid line. Otherwise the mo- 

 ments will represent not the whole area, but strips of unit base of which 

 the ordinates are the mid lines. For the first three ordinates Ik is 



I, I, 2X3—5, respectively; for higher ordinates it is found from the 

 equation fe,,^= 2%',; -4j;\._,-f4A:',,_- ... - (- i)* 43;',-^ (- i)t 5. The 



upper limit of the area is given by 2x'^-2x'm-j+ . . . -(-i)* 2x'3 + 



(_i)* -~ = i-{-(j^ where / is the integral and q the fractional portion of 



the number. 



Let M„ represent the Hth moment about the origin as above chosen. 

 Then 



Mn = j^^\a + bx + e log.„x) :,Mx=-^[(/ + ?)''«-0y^'] 



+ c !og,„p 





