of Qiiater7iions. 495 



we have q(i — qn\ and since C'A' is equal to AC, we see also 

 that g'^', (^q represent equal rotations, but generally in different 

 planes ; these planes, however, being equally inclined to the 

 planes of «7, qK 



Consider now in particular the case in which AB, BC are 

 both quadrants. Then it is obvious that C'A' is in the same 

 great circle with AC, so that if 5" represent the rotation AC, 



—f, will represent the rotation C'A', and we have therefore, in 



this case^ qq'^iq'q)'^- Now let q, represent (not AC but) 

 CA, so that g, q', q^ represent the rotations by which the sides 

 of the triangle would be described in a cyclic order ; then we 

 shall have, for ani/ triangle, 



,1 1,1 



g'!Z=-. QQ,= Y' ^'^ q' 

 and if we further suppose all the sides of the triangle to be 

 quadrantSf we shall also have, as we have just seen, 



qq'=Qp q,Q=Q'i q'q,=^Q' 



Before we proceed further, there is one important remark 

 to be made with respect to the symbol ( + J*, or — ^; namely, 

 that the effect of the rotation denoted by it, is, regard being 

 had to the conventions above established, independent of the 

 direction of the axis r \ since a semi-revolution, in a;2j/ plane 

 in which it is possible, merely reverses the direction of the ope- 

 rand line. We may therefore in all cases substitute the general 

 symbol — for — ^, and write (+^)i=— . To illustrate this 

 further, let q represent any rotation AB, and let us examine 

 the meaning of the product — r'S'. The arc AB must be so 

 placed in its own plane that its end B shall be at the intersec- 

 tion of that plane with the plane of the rotation — ^/; then the 

 operand line must first describe AB, and then half a circum- 

 ference in the latter plane, after which it will obviously be 

 again in the same great circle with AB, cutting the sphere in 

 the point opposite to B. Now the same effect would have 

 been produced by a single rotation in the plane of the rotation 

 g, through an angle S + tt described positively, or tt — 9 de- 

 scribed negatively, 6 being the angle of the rotation q. If, 

 then, g'=( + ^)% we have 



in which we may henceforth drop the subscript r*, and write 

 simply —q on the first side. 



It is still more obvious that similar remarks apply to the 

 case of the symbol -1-^; or that we may always drop the sub- 



