Electromagnetic Wave-surface. 399 
Nearly all our equations are symmetrical with respect to 
capacity and permeability; so that for every equation con- 
taining some electric variables there is a corresponding one, 
to be got by exchanging electric force and magnetic force, &e. 
And when the forces, inductions, &c. are eliminated, leaving 
only capacities and permeabilities, these may be exchanged in 
any formula without altering its meaning, although its imme- 
diate Cartesian expansion after the exchange may be entirely 
different, and only convertible to the former expression by 
long processes. 
If either » or ¢ be constant, we have the Fresnel wave- 
surface. Perhaps the most important case besides these is 
that in which the principal axes of permeability are parallel 
to those of capacity. ‘There are then six principal velocities 
instead of only three, for the velocity of a wave depends upon 
the capacity in the direction of displacement as well as upon 
the permeability in the direction of induction. For instance, 
if 41, fo, 3 and cj, C2, cz; are the principal permeabilities and 
capacities, and the wave-normal be parallel to the common 
axis of u, and ¢, the other principal axes are the directions of 
induction and displacement, and the two normal velocities are 
(cooz)—2 and (¢34g)7?. 
The principal sections of the wave-surface in this case are 
all ellipses (instead of ellipses and circles, as in the one-sided 
Fresnel-wave) ; and two of these ellipses always cross, giving 
two axes of single-ray velocity. But should the ratio of the 
capacity to the permeability be the same for all the axes 
(41 / = 2 | c=p3/¢3), the wave-surface reduces to a single 
ellipsoid, and any line is an optic axis. There is but one 
velocity, and no particular polarization, If the ratio is the 
same for two of the axes, the third is an optic axis. 
Owing to the extraordinary complexity of the investigation 
when written out in Cartesian form (which I began doing, 
but gave up aghast), some abbreviated method of expression 
becomes desirable. 1 may also add, nearly indispensable, 
owing to the great difficulty in making out the meaning and 
mutual connections of very complex formule. In fact the 
transition from the velocity-equation to the wave-surface by 
proper elimination would, I think, baffle any ordinary alge- 
braist, unassisted by some higher method, or at any rate by 
some kind of shorthand algebra. I therefore adopt, with some 
simplification, the method of vectors, which seems indeed the 
only proper method. But some of the principal results will 
be fully expanded in Cartesian form, which is easily done. 
And since all our equations will be either wholly scalar or 
wholly vector, the investigation is made independent of qua- 
2H 2 
