FOE DETEEMINING THE EADIATING. POWEES OF SURFACES IN AIE. 391 
assigned portion of the bottom of the calorimeter (or the corresponding portion of the 
hemispherical surface of which the radiating surface was a diametral plane) is not easily 
determinable, on account of the complicated integration to which the investigation 
leads. It is easy, however, to compare very approximately the amount which thus falls 
on a very narrow annulus of the hemispherical surface contiguous to the diametral plane 
formed by the radiating surface, with the whole quantity which radiates from that 
surface in a given time. Thus let 
S^=the quantity of heat which radiates in a unit of time from the small element 5S 
of the radiating surface, and falls on the indefinitely small area lo of the above-mentioned 
hemisphere ; 
I = the intensity of normal radiation ; 
E= angle of emanation, and therefore I.sinE= the intensity of radiation in a 
(Erection making an angle E with the radiating surface ; 
f = angle of incidence on u ; 
r — distance of u from SS ; 
^ = projection of r on the radiating diametral surface ; 
6 = angle which ^ makes with the diameter of the diametral plane passing through 
the projection of u on that plane ; 
g'= the quantity of heat which radiates on the element a; 
then 
and if denote the distance of w from the diametral plane, 
• T? • 
sinE= - > 
r 
Also 
The angle <p is that which r makes with the normal to the surface at u. Instead of 
expressing its general value, suppose co to be situated within the diametral annulus 
above mentioned, and therefore very near to the diametral plane ; it may then be con- 
sidered as lying on the axis of assuming the origin of coordinates to be at the 
extremity of the diameter from which & is measured. The normal to the point u may 
then also be considered as parallel to that diameter, the axis of x, and we shall have 
cos 9 = 
gcosfl 
r 
hq'z^luZi ^4 cos 
=I<y2j 
- ("2^g2)2 Cos^^%; 
Hence 
