Hatai, Weight of the Brain. 171 



In the formula {?>), when the variable x is less than a constant 

 C, or in this case 8.7, we can take the logarithm of the real positive 

 nnmber (C) and pnt a negative sign before it, i. e., .569 log( — C) = 



— .509 log C where — C = (x — 8.7). 



With the foregoing nnderstanding, the formnla ean thns be ap- 

 plied even in the case of a rat, the body weight of which is less 

 than 8.7 grams. 



There remains, however, a second defect in this formula (3) which 

 cannot be overcome. 



When the value of x lies between 7.7 and 9.7 grams, the formula 

 fails to represent the observed values on account of sudden change 

 in the course of the resulting curve. Although this interval is very 

 small when we consider the whole extent of the curve, yet it prevents 

 the general ai^plication of the formula. 



In Chart I, Plate II, in the paper by Donaldson, '08, the curve 

 representing the change in the brain weight between the body weights 

 of 5 and 10 grams was completed by simply joining the two points, 

 both of which had been carefully calculated by the formula (3), and 

 it was not until we came to consider the formula in another con- 

 nection that we appreciated the impossibility of applying it to this 

 interval. 



I have now obtained a revised formula which is free from the 

 foregoing objections. At the same time it should be stated that 

 the values obtained by this new formula do not differ from the 

 values so far as computed by the previous formula (3), or as 

 given by the ideal line by which the curve was previously com- 

 pleted. 



I shall jDresent first the theoretical considerations touching the 

 revised formula. Let us consider the series 



2 o ^^ ^-^Ll+z" l + z-'-'J 



(6) 



where i|;(z) and (^(z) are (some) functions of z. The sum of the 

 first n terms of S becomes obviously 



" ^^ '^ l+z° 



