crystalline Reflexion and Refraction. 39 
Again, let the coordinates 2, ¥,, 2, have reference to one set of refracted waves, 
and x, y;, 2, to the other, the axes of z, and 2, being perpendicular to the 
respective waves, and their positive directions being those in which the waves 
are propagated. Suppose the displacements to be parallel to y,, y;, and to be 
denoted by 7, 7; respectively. Then if the axes of x, y,z make with the 
axis of x, the angles a, 8,35 Yo) and with the axis of x, the angles a’,,, 6/., 7 
we have, by the formule (F), 
dn, ON 1 
i COS Go) 
dz, oT ay ee 
dp dip 
eS 2 (2) 
aa 08 Bi») + 47 dz, 08 Bix», 
dn, aS } 
Z = — C08 Yo) + > COS Ya) 
dz, ot den a a 
and thence, by the relations (19), 
dh 
a Tig = (a°l cos a.) + b’m cos B,., + €°N COS Y,.)) 
*Lcos a’,, + bm cos B'.) + ¢7n cos Ys); 
dy: 
i! 
dn, , « ; é 
es a ae COS a, + bm’ cos By, + €°*n' COs ¥,.)) 
= = 
>’ cos ais) + b?m’ cos Bi, + €°n’ COS ¥/2))- 
Suppose the nese which generates the wave-surface of the second medium 
to have its centre at O, and to be touched in the points Q and Q’ by two planes 
which cut the axes of w, and x; perpendicularly in the pomts P and P’; the 
lengths OP and OP” being expressed by s, s’, and the lengths OQ and OQ’ by 
r,r’. Let the axes of 2, y,, 2, make with the direction of OQ the angles 
4) Boy Yo and with the direction of OQ’ the angles a}, 6;, y,. Then, from the 
equation (H) in Lemma IV., it is manifest that 
@1 cos a... + bm cos B,,, + €7n COS Y., = 7S COS a, 
@1c0s a/y) + b’m Cos Bio, + €'n COS 2) = 1's’ COS a, 
al cos a... + b’m’ cos B,, + €’n’ cos Yi.) = 7s COs B,, 
at cos a, + bm’ cos Bio) + c'n' Cos Y/2) = 7's’ cos 8}. 
