B40 



ANALYTIC GEOMi:n;y 



[Cm. i. 



Then if the OCX n I i nates 

 of any point P in the 

 systems are 



and 



2 = 



x 1 = 



V Fio.148 Z' = Q'P, 



then projections of OP and the broken line OQQ'P upon <>X. 



, in turn, will be equal; hence, 

 f/'eosa._,-f z' 



OF, 



* = x' cos YI + V ' cos -Y2 + * f cos Y 3 . 



These formulas are for transformation from the first sys- 

 tem to the second. But, also, by projecting OP and 

 OMM'P upon OX*, OY', OZ', respectively, 

 d = x cos a x 4- y cos^j + z cosy^ 

 y 1 = xcosaa-hycos^-fzcosyj, . . . [ !">.] 



Z f = X COS 8 -f ^ COS f$i + 2 COS y g , 



and these formulas are for the reverse transformation, from 

 the second system to the first. 



NOTE. It is to be remembered that in the transformations of [11] and 

 [15], twelve conditions exist, by eq. [4] and eq. [12], three of ea> 

 tiie following types, 



COS a O, + C08 a <4 + 008*0, = 2, 



cos 2 a, -f cos 3 /3,+ cos 2 y, = 1, 



COS Oj cos a, + cos/?! cosft -I- cos y t cosy, = 0, 



008 Oj COS /?, + COS O, COS fa + COS 03 COS f} 3 - 0. 



These equations are not independent, however, but reduce to six 

 independent equations. 



