IN NON-EUCLIDEAN SPACE. BY A. MC AULAY. d 



to a quaternion according to the following description ; 

 if u is the position unitat of P and v is the position 

 nnitat of Q, vii~'^ is a nnitat signifying a length PQ in 

 the sense PQ measured along the straight line PQ ; and 

 similarly if x is any scalar, xvu~^ is a quaternion signify- 

 ing the same line-segment. 



The following five fundamental statements are almost 

 obvious : — 



(1) A (i'M~i) = length PQ, or what is the same 

 thing S( vir'^) = cos PQ. For in elliptic space 

 cosPQ = cosOP.cosOQ + sin OP. sinOQ.cosPOQ 

 iind we have S(z'K«) = Sm. Sw — S. YuYv. 



[Important note on the establishment of our methods. 

 Virtually all our proofs are based on the two cosine 

 formulae for the two spaces. Certainly many tacit 

 geometric assumptions are made below, (such as those 

 referi'ing to common perpendiculars between lines and 

 planes), but these, did space permit, could easily be 

 stated explicitly and proved by present methods.] 



(2) If P is taken as origin in place of O, the new 

 position unitat of Q is understood to be vu~^ If lo is 

 the position unitat (origin O) of a third point E. ; then 

 whether we take O for origin or P the line-segment QR 

 is signified by the same unitat. For wn~'^ (y?<"^)~i 

 = wv~^. 



(3) Two interpretations of qrq-^. If q and r are 

 taken to represent quaternions (or line-segments) with 

 axes through O there is first 

 the usual interpretation of 

 qrq~'^ as a third quaternion 

 with axis through O. It is 

 what r becomes by conical 

 rotation round the axis of q 

 through an angle 2Kq. But p. 

 the diagram (in Avhich the 



lines mean straight lines and OAC and DAB are both 

 bisected at A so that the lines are all in one plane) 

 shows that qrq~'^ may also be interpreted as the line- 

 segment (DC in the diagram) obtained by translating 

 the line-segment r through O (OB in the diagram) along 

 the axis of q (in the diagram OA is the line-segment 

 ■q through O) through a distance equal to '2Aq. The 



