358 



in the region of its greatest abundance, is the ditiference in the hourly 

 velocity of development (based on the time to complete the stage) at two 

 mean temperatures differing by one degree Centigrade,* these two being 

 averages of temperatures varying at an average rate of approximately one 

 degree per hour in the medial range, e. g., between 20° and 30° C. Prac- 

 tically, the medial range of the conditions in the region where the inves- 

 tigator finds the species thriving is used as standard, and the developmental 

 unit is approximately established by the use of data from experiments 

 which simulate these conditions as nearly as possible. Furthermore, there 

 is a great amount of individual variation, even in the most carefully 

 selected stocks, which necessitates the use of large lots in order to arrive 

 at dependable averages. The variation of the alpha value renders the 

 calculations very laborious. The problem is still further complicated by 

 the fact that the developmental total is not a constant, but varies for 

 different individuals of the same generation and for different generations 

 of the same year. (See definitions of terms, pp. 330-333.) 



Von Oettingen, in his attempts to find the threshold of development, 

 assumed a series of alpha values, calculated time-temperature products 

 for each one, and selected that one which gave the most nearly constant 

 products for dififerent mean temperatures. He also calculated the prob- 

 able error in his method. Reibisch ('02) calculated the alpha value by 

 the formula (x — a )y ^ k, where x is the temperature and y is the time. 

 Krogh ('14), in his work with Johansen on fish eggs, discovered that the 

 threshold so calculated is not the real one, and he undertook in 1914, by 

 studying the time required for embryonic stages of frog development, to 

 determine the relation of the actual threshold to the alpha value obtained 

 by Reibisch. He found that the graph representing the velocity of 

 development is flattened out at the lower end and falls of? at the upper 

 end, whereas it had always been assumed to be straight. He worked over 

 the published data of Loeb and Wastenys, performed experiments on sev- 

 eral additional animals, and thus compiled a table showing the straight- 

 line limits of two species of echinoderms, six species of fishes, one frog, 

 one aquatic insect, and one land insect. This discovery was the culmina- 

 tion of a long series of papers on fish eggs bv Apstein ('11), Dannevig 

 ('94), Earll ('78), Reibisch ('02), Williamson ('10). and Johansen and 

 Krogh ('14). Up to the present time all of this work on fishes appears 

 to have been ignored by entomologists, as also the work of phenologists, 

 by investigators of both insects and fishes. 



Velocity curves. As was pointed out in PART TWO, pp. 334-338, 

 temperatures above alpha may be summed for that part of the velocity 

 curve which is a straight line, but not outside the straight-line limits. Tem- 

 peratures would probably never have been summed except for the coinci- 

 dence that, for a part of the temperature range, "efifective temperatures" 

 and amounts of development are numerically equivalent. Whenever the 

 re.sults were satisfactory, it was, in fact, amounts of development and not 



• The Fahrenheit scale is used in this paper. 



