( 61) 
This is a quadratic equation in s", which, with given /, m’and n’, 
furnishes 2 values for s* and so also for s’. 
It is easy to deduce the equation of the wave-surface in tangential 
coordinates from this equation. We first seek the transformed wave- 
surface. Let 0 = A'a'+ B'y'+ C'z'+ 1 be a tangent plane to the 
transformed wave-surface; we consider A’, B' and C!' as the coordi- 
nates of that plane. The direction-cosines U, m' and n’ and the 
distance s° to O are for that plane: 
ee ery C' 1 
ers —- r¥ = — = — 
q q =) q 
where 7? = A® + B? + C®. If this is substituted in equation (53), 
taking into account that s'= s" ur uy uz, We find: 
A? B? 
Ar lly My eg —1 ay 0 ur Uy Mz g? — 1 
en 
C12 
4 = 
Az 0” fly fly tz g'® — 1 
If we now direct our attention to the wave-surface itself, we 
find easily, the coordinates of a plane being the reciprocal value of 
the portions cut off from the axes with inverse sign, that if we choose 
the principal directions as axes of the coordinates: 
Meee B CoB, 5 Cr mas 
= 1 cae G on 1 es se 
Further ag —= —=- and also a= =, a,= - . Since 
bi 8e Gre ne 
Mart? + wy y? Huze=l is the equation of the magnetic ellipsoid 
on the axes, Wurgy uz is the inverse value of the product of the 
axes or of the volume of a parallelopiped on three conjugate dia- 
meters; if we take for these diameters the principal directions, we 
find ue My {2 = , Where w is the volume of a parallelopiped 
WEm Cm Um 
on the principal directions, the sides of which are all 1. The 
tangential equation of the wave-surface changes therefore, after 
removing the fractions, into : 
