PHYSICS: NICHOLS AND HOWES 
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to a certain sub-class [X], of [H]. If the forms of [H] are not all generic 
from this new point of view, we may adopt a new canonical form for the 
exceptional ones, and continue in this way. 
We have observed that the coefficients of a unique canonical form are abso- 
lute invariants, and moreover one-valued invariants, in the sense, that their 
values are uniquely determined as soon as the coefficients of the form F are 
given. In the ordinary theory of algebraic invariants, it is at once apparent 
that these invariants are algebraic functions of the coefficients of F. Con- 
sequently it follows from their one-valuedness that, in this case, these invari- 
ants are rational functions of the coefficients. In the theory of invariants of 
linear differential equations, the uniquenes of a canonical form gives rise to 
invariants which are rational functions of the coefficients of the differential 
equation and of their derivatives. 
There are many cases in which a ^-valued canonical form is obtained rather 
than a unique one. That is, if we resume our terminology, the sub-class [<i>] 
contains not merely one, but exactly k forms 3>i, $2, . . . , $h eacn of which is 
equivalent to F by a transformation of the group G. The coefficients of these 
canonical forms will still be absolute invariants of F, but they will be k-valued 
functions of its coefficients. It is obviously possible to find an equation of de- 
gree k with one- valued invariants as coefficients, of which these ^-valued 
invariants are the roots. In the theory of algebraic invariants we obtain in 
this way irrational invariants, as roots of an equation whose coefficients are 
rational invariants. 
TYPES OF PHOSPHORESCENCE 
By Edward L. Nichols and H. L. Howes 
Department of Physics, Cornell University 
Communicated, August 1, 1918 
The existence of phosphorescence of exceedingly short duration was long 
ago revealed by the phosphoroscope of Becquerel but, until very recently the 
afterglow of luminescent bodies has been studied quantitatively, only where 
it is of comparatively long duration. Curves of decay were supposed to be 
all of the same character. It was assumed that the law of diminution of 
brightness, as expressed by»the equation 
Oi + M 2 (a 2 + b 2 t) 2 
was of general application and that the phosphorescence of various substances 
differed only in color, brightness and duration. 
The measurements of Waggoner 1 and of Zeller 2 on phosphorescence of short 
duration tended to confirm this view. On the other hand the observations of 
