Makch 27, 1908] 



SCIENCE 



511 



This law appears to hold over a wide range 

 of ordinary working intensities, breaking 

 down only for very low and for excessively in- 

 tense stimulation. 



Fechner proceeded further by assuming that 

 the above constant was proportional to the 

 corresponding increment 8B to the sensation. 

 Hence 8L/L = cSB and by integration 

 B^o (log L — log Lo). 



In this form or in similar forms differing only 

 in the choice of integration constant, Fech- 

 ner's law has been accepted by psychologists 

 for half a century. 



There are two very serious if not fatal de- 

 fects in this deduction. In the first place, the 

 increments 8L and SB are finite quantities and 

 by no means infinitesimal increments ap- 

 proaching zero as a limit, such as would be re- 

 quired for such an integration. The least 

 perceptible increment to the stimulus (8L) 

 is determined by the sensibility of the sensory 

 organ concerned. At the threshold value it is 

 as large as L itself, while at moderate intensi- 

 ties it bears a fixed ratio to L. The value of 

 85 is entirely arbitrary, dependent upon the 

 unit chosen in which to measure it. It may 

 be greater than unity in special cases. In the 

 second place, c is not a constant but a func- 

 tion of L. At low intensities approaching the 

 threshold value it varies rapidly with L. 



There appears to be no direct method for 

 overcoming these defects. A method of avoid- 

 ing them altogether has however occurred to 

 the writer and been applied to the visual case 

 in a way that may be perfectly satisfactory to 

 I>sychologist and mathematician alike. 



Consider any physical instrument — a gal- 

 vanometer for instance, capable of indicating 

 on a scale the amount of an external stimulus 

 affecting it. The derivative of scale reading 

 with respect to the stimulus will be a measure 

 of the sensibility of the instrument at all parts 

 of the scale. Conversely, the general integral 

 of sensibility will give the scale reading as a 

 function of the stimulus. 



In the visual case we have sensibility to 

 find scale reading. The best data on sensi- 

 bility are those of Konig and Brodhun' cover- 



^ Berlin Sit«., 1888, 917-931. 



ing about twenty different intensities for each 

 of six different wave-lengths. The writer has 

 elsewhere" shown that these data may be repre- 

 sented by the function 



P = dL/L = P^+ {l — P^){LJL)n 



where P„i is the minimum value of P, i„ is 

 the threshold value in light units and n a 

 niimber varying from one third to two thirds 

 with wave-length. The reciprocal of the least 

 perceptible increment SL or 1/LP is a meas- 

 ure of the desired sensibility of the eye to 

 differences of intensity. Hence we have for 

 the scale reading or, in this case, the visual 

 sensation of brightness, 



B = /^ 1^ = £ log [1 + P»(i«io-" - 1)]'^" 



where Z^ is a constant dependent upon the 

 unit of sensibility chosen. 



This general form includes Weber's law and 

 Fechner's law as special cases for moderate 

 intensities, but holds for low intensities down 

 to the threshold of vision. Weber's law 

 8L/L = constant may be extended to cover low 

 intensities by writing 



8L/L = P^+ ( 1 — PJL-nLo". 



P. G. Ndtting 



Bureau of Standards, 



Washington, D. C, 



December, 1907 



ASTRONOMICAL NOTES 



FLUCTUATIONS IN THE SUN'S THERMAL RADIATION^ 



Many scientists have attempted in the past 

 to show that periodical fluctuations occur in 

 meteorological phenomena, presumably de- 

 pendent on changes , in the solar radiations. 

 The two most plausible periods of solar change 

 are the sun-spot period, whose mean value is 

 about eleven years, and the time of the synodic 

 rotation. Professor Newcomb develops an- 

 alytical methods for the investigation of fluc- 

 tuations in a fixed period, and also when the 



= Bull. Bureau of Standards 3, 62. 



" Simon Neweomb, " A Search for Fluctuations 

 in the Sun's Thermal Radiation through their 

 Influence on Terrestrial Temperature," Transac- 

 tions of the American Philosophical Society, N. S., 

 Vol. 21, V. 



