426 

 we have 



cos0"izicos0cos6' — sill 9 sine' (cos cos^' + sin0siii0'cos(i|/ — j//')), "| 

 cos9 =:cos9'cose" + sine'siii6"(cos0'cos0"+sin0'sin0"cos(»//' — i^")), r- (g) 

 cos0'^ cos0"cose + ^i"^"^'"^('^°^^ "'^°^0"t'^'"0 "^'"* cos()^" — i//)). J 



Consider x, y, s as the rectangular coordinates of a point 

 of space, and let r be the point where the radius vector of 

 x,y, z (prolonged if necessary) intersects the spheric surface 

 described about the origin with a radius equal to unity ; call 

 R the representative point of the quaternion a, and let the 

 polar coordinates and i//, which determine R upon the 

 sphere, be called the co-latitude and the longitude of the re- 

 presentative point R, or of the quaternion q itself; let also ' 

 the other angle Q be called the amplitude of the quaternion; 

 so that a quaternion is completely determined by its modulus, 

 amplitude, co-latitude, and longitude. Construct the repre- 

 sentative points r' and r", of the other factor q', and of the 

 product q" ; and complete the spherical triangle rr' r", by 

 drawing the arcs rr', r'r", r" r. Then, the equations (g) 

 become 



COS0" = COS0COS0' — sin sin 0' cosrr', 

 cos - cos Q' cos Q" -f sin Q' sin Q" cos r'r", 

 COS0' = cos0"cos0 +siu0"sin0 cosr'^r, 

 and consequently shew that the angles of the triangle rr' r" 

 are 



R = 0, R' = Q', K"-ir- d" ; (h) 



these angles are therefoi'e respectively equal to the ampli- 

 tudes of the factors, and the supplement (to two right 

 angles) of the amplitude of the product. The equations (d) 

 show, further, that the product-point r" is to the right or 

 left of the multiplicand-point r', with respect to the mul- 

 tiplier-point R, according as the semiaxis of -f- s (or its in- 

 tersection with the spheric surface) is to the right or left of 

 the semiaxis of 4- «/> with respect to the semiaxis of -\- x: 

 that is, according as the positive direction of rotation in 

 longitude is towards the right or left. A change in the 



