MATHEMATICS: M. B. PORTER 
335 
tion of the galvanometer ensues. The heating current is divided be- 
tween the two strips, and by suitable resistance coils the circuit is ad- 
justed once for all so that whatever the strength of the heating current 
it produces equal dissipation of energy in the two strips. If now after 
closing the shutter the heating current is graduated until the deflection 
formerly produced by radiation is reproduced by electrical heating, the 
energy dissipated in either strip is the measure of the absorbed radia- 
tion. In the two strip pyranometer the secondary deflection by indirect 
heating is unimportant, because of the symmetry of the arrangement. 
However, to avoid this source of error altogether the exposure is limited 
to 30 seconds, and a full minute is allowed to lapse before introducing 
electric heating. 
Numerous measurements of the sky-radiation have been made from the 
North Tower of the Smithsonian Institution. On fine days the sky-radi- 
ation alone received on a horizontal surface ranges from 0.07 to 0.13 calo- 
ries per square centimeter per minute. On cloudy days, not thick enough 
for rain, the values run from 0.20 to 0.30 calories according to the kind 
of cloudiness prevailing. Measurements were made on the reflection 
from new fallen snow, and for total solar and sky radiation this proved 
to be 70%. 
In the simpler form the instrument is so sensitive that it could be 
used in the deep shade of a forest, or with screens of selective transmis- 
sion, so that it would be suited to botanical as well as meteorological in- 
vestigations. As in the case of the silver disk pyrheliometer, the Smith- 
sonian Institution may undertake to prepare pyranometers at cost 
(approximately $150) where valuable investigations may be promoted 
thereby. 
NOTE ON LUCAS' THEOREM 
By M. B. Porter 
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TEXAS 
Received by the Academy, May 5, 1916 
In a recent note^ I ventured to give a proof, which I thought might be 
new, of Lucas' Theorem and one of its more immediate generalizations 
to rational functions. Professor Bocher has kindly called my attention 
to the fact that the same proof had previously been given by him^ and 
that the extension to rational functions was a special case (in the method 
of proof as well as in the results obtained) of other of his results.^ I 
called attention to the interesting fact that this proof of Bocher 's applies 
without modification at once to integral functions of class zero. It is 
the purpose of this second note to show that it also applies without modi- 
