76 Professor Hamitton’s Therd Supplement 
tan. ¢ = 375, ; (ES) 
they enable us therefore to determine, for any given value of 4, that is, for any pro- 
posed direction of the small final line &/, or BB;, the corresponding value of 9’, that 
is, the direction of the initial plane of ray-lmes, having for equation 
y =a tan. ¢. (U") 
Thus the final line / and initial plane ¢’ revolve together, but not in general with 
equal rapidity ; and arbitrary rectangular directions of the one do not in general give 
rectangular directions of the other, because the conditions 
“ Ui PY ui Ui 
oe tan. ¢, ere tan. >, 
tang fee pois ita eee poe nla 
OD, oa oF ‘ 
ox ty ang ox by meter WV) 
T ; f T 
hh ty» H=ei+z, | 
(in which x is the semicircumference to the radius unity,) give the following formula 
for the angle ¢,, 
Q (2 # + =) cotan. 26, = 
Gee enna 
which is not in general satisfied by arbitrary values of that angle. There are how- 
ever in general two rectangular final directions determined by this formula, which 
correspond to two rectangular initial planes; and if we take these rectangular direc- 
tions and planes respectively for the directions of 2, y, and for the planes of w' 2’, 
yz, we shall have 
oa Situs ie 
oy ah Scie (x ) 
We may alse in general satisfy, at the same time, by a proper choice of the semiaxes 
of co-ordinates, the following other conditions, 
ay 7 Z Se 7 By” Or) 
By this choice of co-ordinates, the relations (S$) are simplified, and become 
