306 Proceedings of Royal Society of Edinburgh, [june 20, 
we have 
2 
9 
lz x+ Zz.^x + z 2 . 3z x 
x — A 
{ 
Z Z 2 Z 3 (J> 
1 .2 .3 ~ r I.2.3.4.5. 6 ~ 1 9 + t C 
J 
which is related to recurring functions of the third order. 
There is thus opened up a wide field for research, rendered, 
however, unprofitable by the absence of application to physical 
phenomena. 
3. Note on the Biological Tests employed in determining 
the Purity of Water. By A. W. Hare, M.B. Edin. 
(Plates X., XI.) 
(From the Public Health Laboratories, Edinburgh University.) 
Part I. 
There are two experimental paths by which the facts relating to 
organic impurity in water may be approached, the chemical and the 
biological ; and there are two aspects in which these facts may be 
regarded, the catalytic and the fermentative. The series of observa- 
tions made in following the one path is the necessary complement 
of that met with in the other. The path of chemical investigation 
has been well cleared, and is easily traversed ; that of biological 
inquiry is still beset by many impediments, and is as yet by no 
means a safe one. It is the object of many recent researches, and 
of this paper, to lessen these difficulties. The pollution of water 
with organic substances is inevitably associated with the presence 
of those organisms whose function it is in the economy of nature to 
disintegrate such materials. These organisms in their turn are 
dependent upon organic matter for their continued existence ; the 
Bacteriacem to which they belong being distinguished from allied 
forms by an entire absence of chlorophyll, which obliges them to 
feed only on organic substances. The relation of these two factors 
of pollution, each to each, is therefore a necessary one ; and due 
regard must in all cases be paid to each in determining the degree 
of pollution that their coexistence implies. It is further of 
