Prof. Powell on the Theory of the Dispersion of Light. 305 
-angles to the axis, and of such lengths that I L shall be of 
equal length to the semicircle A, and J M equal to B, and KN 
equal to C; from the point O draw the straight line O L, and 
‘also from P to M: it will be seen that O L is the approxi- 
mate line of the developed soffit, and P M that of the inter- 
mediate development. Add Q, R, and S, which are the centre 
lines of the three developments. 
It will be seen that when these developments are placed as 
in an arch, these three lines Q, R, S_ being parallel with the 
axis, will be in a plane perpendicular to the axis, and, there- 
fore, that all the points in each spiral will be vertical with the 
axis, and also with one another. 
Through any point in P M draw a straight line V at right 
angles with P M, which straight line shall extend to the axis 
of the cylinder. 
At the point where it intersects R, a line T perpendicular 
to the axis intersects Ralso: this last perpendicular line cuts 
the three lines Q, R, S at the points where the lines U, V, W, 
which meet in X, intersect Q, R, S. 
The joints are then drawn upon the three developments 
eee with the lines U, V, W, and at such distances that the 
ines Q, R, S shall be cut into equal parts. Of course, care 
must be taken to divide the approximate line of the soffit into 
a given number of stones. ‘The angle X will be that which 
the intrados form with the axis of the cylinder, and the angle 
U W will give the wind of the bed. On this principle and 
by the rules here given, it is nearly as easy to work the stones 
of a skew bridge as those of any other. 
Park Village East, London, March 17, 1836. 
LXIII. Further Observations on M. Cauchy’s Theory of the 
Dispersion of Light. By the Rev. Bapen PowELt, M.A., 
E.R.S., Savilian Professor of Geometry, Oxford. 
(Continued from p. 28.) 
I PROCEED to illustrate the further researches to which 
I alluded in my last paper; relative to the development of 
the theory of dispersion, and simplifying the process of M. 
Cauchy. 
In order to consider the subject in its simplest form, let us 
confine our attention to a plane wave perpendicular to the 
axis of z, with vibrations parallel to the axis of y. Then the 
displacements § and ¢ will vanish, and the differential equation 
