DE. PLUCKEE ON A NEW GEOMETEY OE SPACE. 
743 
If the primitive system of coordinates be only displaced parallel to itself, the coordi- 
nates of the new origin being (x°, y°, z°), we obtain 
x’=x—x\ y'=y—y\ z'=z—z°; 
and by substituting in the last equations, 
x=r , z + (f' + x°— r'z ), 
y=s'z+(<T f +y 0 -s‘ f z) ; 
whence, by comparison with the primitive equations, 
We have further 
s=s, 
g=4+a?—rz 0 , 1 
<r=o'+y°- S z°.) 
sg—ra— (s'g' — rV) + x°s —y°r. 
( 30 ) 
( 31 ) 
If x°=0, y°=0, and accordingly the origin move along OZ, the expression (sg—ra) 
remains unaltered [29]. 
37. If OY and OX turn round OZ, forming in the new position OY', OX' the angles 
a! and a with OX, we have 
x=x' cos a-\-y' cos vi=rz -j-f, 
y—x ] sin a+y sin ot—sz-\-c ; 
whence, on putting («'— a)=^, 
, rsina' — s cos a! , . o sin — tr sin a f 
X— — ^ Z- f 5 r— , 
sin £ 1 sin 3 
y=— 
r sin cc — s cos « 
sin $ 
>sma— <r sin « 
sind 
We immediately derive from these equations of the ray in the new system (a/, y', z'), 
whence 
/ sin §=r sin cx! — s cos a', 
g' sin $=g sin ex!— a cos a!, 
— s sin S=r sin a — s cos a, I 
— a sin §=g sin a — <r cos a, ] 
r^r 1 cos a-\-s' cos a', j 
g=g' cos a-fV cos I 
s=r' sin a-j-s' sin ex', j 
c=g' sin sin a', j 
( 31 *) 
( 32 ) 
(sg— re) = (s'g' — rV) sin 
5 k 2 
and 
( 33 ) 
