DISTRIBUTION OF INTENSITY IN BROADENED SPECTRUM LINES. 
477 
following a different law. A wedge whose length lies along y would produce a curve 
of critical intensity I c which, outside both axes, would have an equation 
or 
I c = 2l 0 cosh q a e- qx - pyiana 
py tan a + qx — log e (2l 0 cosh qaj I c ) 
and be straight. But between the axes the equation is 
or 
I c = 2l u cosh qx . g-^-pytana 
py tan « = log e (2l,,/[ c ) —qa+ log e cosh qx, 
which is curved. We can verify at once from these equations that dyfdx is 
discontinuous and changes sign when x = <r, and that the form of the curve of 
intensity I c is as shown in the figure (fig. 4). The summits of these curves, however, 
are not the summits which the separate components would individually show. The 
upper parts of the photographs of indicate this appearance very precisely, so that 
the strongest components of form a symmetrical pair. 
Before proceeding to the effect of superposition of such pairs, together with a 
possible central component, we must prove a very general theorem. The main 
characteristic of all the curves which have been photographed, not only of hydrogen 
lines, but of helium and lithium, is that they contain no point at which the curvature 
is towards the axis. In other words, if y is measured as in the last figure, d 2 yldx 2 is 
always positive whatever the sign of dy/dx. Apparently the only exponential 
arrangement which is physically possible and possesses this property in general is the 
class of curves dealt with in the theorem. A proof of this statement would occupy 
some space, and we therefore merely prove that this class has the necessary 
property. 
Consider a set of lines, n in number, with any rates of attenuation q u q 2 , ..., q n , 
whose axes are coincident. Their central intensities are I l5 I 2 , ..., I„. If k = p tan a, 
the bounding curve of the photograph they produce is given by the equation 
r = n 
r = I 
when they are all simply exponential. Accordingly, by differentiating twice, 
3 T 2 
