426 

 we have 



cos 6" = cos cos 0' — sin9s!n6'(cos0cos0' + sin^sin^'cos(;|/ — >//')), "j 

 cas9 =cos9'cose"-l-sin0'sine"(cos0'cos0"+sin0'sin^"cos(i|'' — ^')), r (g) 

 cose= cos 0" cos 6 + sin 0" sin 9 (cos ^"cos0 + sin ^"sin 6 cos (;//" — <//)). J 



Consider x, y, z 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 uniiy ; call 

 R the representative point of the quaternion q, 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 B 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 Q - cos 0' cos Q" 4- 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, K' = B', Yi"-TT-B"; (h) 



these angles are therefore 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 sr (or its in- 

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

 the semiaxis of -+- t/, with respect to the semiaxis of -f ^ : 

 that is, according as the positive direction of rotation in 

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



