744 
DR. PLUCKER ON A NEW GEOMETRY OF SPACE. 
If especially 3=-, the last four equations become 
r=7 J cos a — s' sin a, ' 
g=g' cos a— o’ sin a, 
s=r f sin a+s' cos a, 
• • ( 34 ) 
and the expression 
<r=g' sin cos a, 
sg — r<r 
will not be altered by the transformation of coordinates [29], 
38. Again, let OX and OZ turn round OY ; let a! and a be the angles formed by 
these axes in their new position, OX' and OZ', with OZ, and a'— «=$■. In the new 
system of coordinates the primitive equations of the ray become 
(z 1 sin a -\-cd sin a!)=(z' cos cos a')r-{-g, 
y'=(d cos cos «')s-f-c\ 
From the first of these equations we derive 
a/(sin a ' —r cos a') = — 2 '(sin a— r cos a'j+f, 
whence 
^ sin a — r cos a 
sin a! — r cos a! 
§ = 
sin a — r cos oc 
(35) 
(36) 
After replacing in the second equation of this number of by (r'z'-{-g'), we obtain 
y=(cos a -l-r' cos a' )sz' sg 1 cos a'), 
whence 
s' = (cos a + r' cosa')s, 
< 7 '=(r-]-sg' cos oi ; 
and by eliminating r' and g' by means of (35) and (36), 
j ’s sin ■& 
sin «' — r cos a 
(erg — nr) cos «' + c sin a! 
sin a! — r cos a! 
From (35)-(38) we derive 
from (36) and (37), 
s'p'—r’o'— ( sp — r<r ) cosct + <r siu “ • 
? sin u' — r cos «' ’ 
sin S-. 
(37) 
(38) 
(39) 
(40) 
