crystalline Reflexion and Refraction. 21 
Lemma il. Let & 7, ¢ denote, as before, the displacements of a particle 
whose initial coordinates are x, y, 2; and after putting 
= danas OG ae dé dy 
“= dz dy’ Y= da dz’ 7 = dy da’ (c) 
suppose the axes of coordinates, still remaining rectangular, to have their direc- 
tions changed in space, whereby the quantities x, y, z will be changed into 
x’, y’, 2’, answering to the new coordinates .’, y/’, 2’, and to the new displacements 
&, 7, ¢; then will the quantities x’, y’, z’ be connected with x, y, z by the 
very same relations which connect the coordinates 2’, 7’, 2’ with x, y, z, or the 
displacements &, x’, ¢’ with & », ¢. 
That is to say, if the axes of x, y, z make with the axis of 2’ the angles 
a, B, y, with the axis of y/ the angles a’, f’, 7’, and with the axis of 2 the angles 
a’, B’, " respectively, we shall have 
x= x’ cosa-+yY’ cosa’ +7’ cos a’, 
Y¥ = x’ cosB + Y’ cos p’ +2’ cos Bp”, (D) 
Z= x’ cosy + Y’ cosy + z’ cos 7’, 
and 
x’ =xcosa+ycosB+z cosy, 
y’=xcosa’ + ycos pf’ +z cos7, (D’) 
z’ = xcosa’ + ycosp’ + z cosy’; 
just as we have, for example, 
E=# cosa+7 cosa + cosa”, 
9 = & cos 6 + 7/' cos 6’ + & cos B%, (d) 
€= & cosy+ 1 cosy’ + ¢ cosy”, 
and 
x = «cosa +ycosB+ z cos ¥, 
y¥ =2008 a + ycosp’ + z cosy, (d’) 
2 = «cosa + ycosp’ + z cosy’. 
For, the change of the independent variables x, y, z into 2’, y/, 2 gives us 
the equations 
