118 On the Geometrical Interpretation of Quaternions. 

 If Q, Q' ? Q" be any quaternions, the condition 



*Q+/3Q'+7Q"=o 

 is equivalent to the system 



WW + fiw' + y>w" = 



aoo + j3#' + y^" = 

 from the last three of which may be deduced 



if, 



*i 



y 



# 



y, 



3/" 



^ 5 



#* 



/ 



which is the condition that the three lines, whose direction- 

 cosines are proportional to x, ?/, js, ..., lie in the same plane ; 

 in other words, the planes of rotation of the three quaternions 

 are all parallel to one straight line. 



If Q, Q!, Q" represent the rotations BC, CA, AB of the 

 spherical triangle ABC, the quaternions 



*Q"-yQ=Q 2 

 /3Q-*Q'=Q 3 



will represent arcs drawn from the angular points A, B, C, 

 and cutting the opposite sides in points whose segments are in 

 the ratios /3 : y, y : u 9 a: (3 respectively, and the resulting con- 

 dition 



"Qi + /3Q 2 + yQ 3 =0 



shows that the three planes of rotation intersect in a common 

 line, for they all pass through the same point, viz. the centre 

 of the sphere ; consequently the three arcs all meet in a point. 

 If 



a = /3=y, 



the points where the arcs Q 15 Q 2 , Q 3 meet the sides of the 

 triangle will be the middle points of those sides, and the con- 

 dition 



Qi + Q 2 +Q 3 =o 



will express that the three arcs meet in a point. This theorem 

 includes all the corresponding theorems with respect to plane 

 triangles. 



