492 



THE AMERICAN NATURALIST [Vol. LV 



2. Long-winged females. The corresponding table to 

 1, but for females. 



3. SJiort-ivinged males. This table includes all our data 

 up to June, 1921, on the normal duration of life at 25° C. 

 of males belonging to the Quintuple stock. These flies 

 carry the wing mutation vestigial ; hence the designation 



short-winged." 



4. Short-winged females. The corresponding table to 

 3, but for females. 



We have tried a number of ditferent plans for the 

 graduation of these tables, and wish to acknowledge 

 gratefully the helpful suggestions of our colleague in the 

 department. Dr. Lowell J. Eeed, in connection with this 

 phase of the work. It was first found that a rather 

 satisfactory result could be obtained by fitting a logarith- 

 mic parabola of the type 



to the q.r data. Working from this as a basis we finally 

 decided that, as a practical matter, results on the whole 

 most satisfactory could be got by the following type of 

 graduation. 



\ogl^ = ea^{a-\- bx -\- cx- -{^ daf). (i) 



This amounts to asserting that the instantaneous death 

 rate increases with age as a modified logarithmic func- 

 tion of X. 



The actual equations for the four calculated L lines of 

 Tables II to V inclusive, together with the absolute num- 

 ber of individuals on which the curves are based, are as 

 follows: 



Long winged (4,586 flies) : 



log i^ = e «>«»*^'^»^ (3.0041905 — .02937911a- -h .000140245j-= 



— .0000015897r''). (ii) 



Long winged 9s (5,426 flies) : 



