746 
DE. PLUCKEE ON A NEW GEOMETEY OF SPACE. 
round OZ in its primitive position through an angle a such that ZX in its new position 
contains OM. Accordingly we obtain 
whence 
cos a = 
cosg 
sin £ ’ 
tan 2 a= 
1 — cos 2 £ — cos 2 £ cos 2 >] E 2 
cos 2 0 cos 2 0 D 2 
By making use of the formulae (34), the equation of the complex (7) becomes 
(A cos 05 -|- B sin a)r '—( A sin 05 — B cos a )s' 
+ (E cos 05 + D sin u)g ' — (E sin 05 — D cos oo^o^ — C F — rV)=0, 
and may be written thus, 
AV+B's+C'+D'<7+F(s£-r<r)=0, (47) 
in omitting the accents of the new coordinates and in putting 
E cos 05+ D sin 05=0, ] 
A'=(AD-BE) C -^, B'=(AD+BE) C -^,| (48) 
D'=(D 2 + E 2 )^, C'=C, F'=F. | 
E ; 
41. In order to give within ZX to OZ the required direction along OM, the formulae 
(44) are to be used after having replaced a by £. Accordingly the equation (47) is 
transposed into the following one, 
A'(sin £+F cos £)— BV+C'(cos Z > — F sin C Q 
+ D'((slf'— /</) sin ^+o-' cos £)+F((s'g>'— rV) cos a’ sin £) = 0, 
and may be written thus, 
AV+ B"s + C" + F"(sg—rc) = 0, (48*) 
on omitting the accents of the coordinates and putting 
D' cos £=F' sin 
A"=(A'F-C'D')^, 
B"= — B', 
C"=(A'F'+A'D')^, 
(49) 
F"=(D' 2 +F 2 ) C -^. 
42. Finally, the origin may be moved within XY to a point the coordinates of which 
are #° and Accordingly the equation of the complex, on replacing f and c r by g-\-x° 
and <r+^°, becomes 
(A" — F'y )r + (B" + F V)s + C" + F"(sf — nr) = 0, 
