212 
BIOLOGY: PEARL AND PARKER Proc. N. A. 
. .m) are constants. Then as in the above case we show that the Hnear 
element can be put in the form 
ds' - {dx^Y + . . + {dx^^y + gij dx' dx' {i, j = m+1, . . n), 
where gij are independent of x^, , x"". When m = w — 1, equation. 
(6.7) is reducible to the euclidean form. 
7. If are the contra variant components of a vector in a Rieman- 
nian geometry, its co variant components Ai, are given by 
A' = g'« A^. (7.1) 
When this expression is substituted in (3.1), we obtain 
^-rZA^-A,'t^ = 0. (7.2) 
Ox^ dx^ 
When we are dealing with a non-Riemannian geometry we say that a field 
of parallel covariant vectors is one which satisfies (7.2). 
The conditions of integrability of (7.2) are 
B^jk = 0. ' (7.3) 
In order that the A's given by (7.3) shall satisfy (7.2) it is necessary that 
Bfjki = 0. (7.4) 
As in the case of contravariant vectors, it can be shown that when there 
are m {< n) independent sets of solutions of (7.3) which satisfy (7.4) there 
exist m fields of parallel, covariant vectors. 
The methods of §§ 4, 5 cannot be applied to the case of covariant fields. 
ON THE INFLUENCE OF DENSITY OF POPULATION UPON THE 
RATE OF REPRODUCTION IN DROSOPHILA^ 
By Raymond Pkari. and Sywia L. Parker 
ScHooi. OF Hygiene and Pubi^ic Hejai^th, Johns Hopkins University 
Communicated, May 5, 1922 
It has long been known that degree of crowding of organisms in a given 
space, or the density of the population, has an influence upon various 
vital processes of the individuals composing the population. In the 
matter of growth Semper^ and before him Jabez Hogg^ showed that volume 
of water apart from food and other conditions has an influence upon the 
rate. This subject has again been studied recently by Bilski^. Farr^ 
showed that there is in man a definite relation between density of popu- 
lation and the death rate. This old work of Farr's has recently been gone 
over carefully and confirmed by Brownlee.® Drzwina and Bohn^ show 
that a particular concentration of a toxic substance, just lethal for a single 
