174 
Proceedings of the Boyal Soeiety 
A, we sliall have a corresponding increase in the number of undula- 
tions comprised within the limits of the sheet ; the tendency again 
being toward blackness. On making the approach from either side, 
that is, on taking the functions of a -f 8a and of a - Sa, and attri- 
buting to the variations 8a an infinitesimally small value, we find, 
on both sides, the functions to be black ; yet, on making 8a abso- 
lutely zero, we find whiteness instead of blackness. 
Such being the general features of the curve, we may obtain its 
details by an examination of one of the half-waves, such as AOB. 
Eor this analysis it will be most convenient to place the origin of 
co-ordinates at the point O. We shall therefore write — 
2/ = 0H 
x = BP 
I — arc BP 
r = radius of curvature at P 
a = inclination of curve at P 
s = surface BOHP . 
We shall also write — 
Y = OA 
X = OB 
L = length of BA 
B = radius of curvature at B 
A = inclination at A 
S = area BAO , 
for the limits of these quantities. 
Since, in such an arrangement, the angular tension at the point 
P is proportional to the ordinate HP, the radius of curvature there 
must be inversely proportional to the same ordinate ; and therefore 
we must have 
rx-c^ . . . . ( 1 ) 
where c is a constant determined by the dimensions of the spring. 
This equation contains the analytical definition of the curve. 
Since, from the nature of curvature, 
dl = r . da 
the generic equation (1) may be written 
X . dl — r^. da , 
( 2 ) 
