425 



and to, X, y, z, to', x', y', z' are real, is equivalent to the four 

 separate equations, 



w = w', X — x', y =■ y', ^ = ^'• 



Sir W. Hamilton calls an expression of the form q a qua- 

 ternion; and the four real quantities to, x, y, z he calls the con- 

 stituents thereof. Quaternions are added or subtracted by 

 adding or subtracting their constituents, so that 



Q + q' = m; + tc' + i(a; + x') +i [y + y') + k{z + z'). 

 Their multiplication is, in virtue of ihe definitions (a) (b) (c), 

 effected by the formulae 



qq' = q" = w" + ix" +jy" + kz". 



w" = ww' — xx' — yy' — zz', 

 x" = ivx' +XU/ + y%' — ssj/, 

 y" = toy' + yw' -^zx' — xs', 

 z" - wz' + zw'-{-xy' — yx', 



(D) 



7/2 _ („fi 



(,«2 ^ ^.2 + y2 + ^2) (,^/2 + ^n + y/2 + ^r2)^ 



(E) 



which give 



and therefore 



fl" = fifx', 



if we call the positive quantity 



M= Vw''+x-'+y' + z\ 

 the modulus of the quaternion q. The modulus of the pro- 

 duct of any two quaternions is therefore equal to the product 

 of the moduli. Let 



«; = /u cos 0, 



x = n sin 6 cos </», 

 y = ju sin sin <f> cosi//, 

 z ■=: fi sin 6 sin (/> sin ;p ; 

 then, because the equations (d) give 



w'w" + x'x" + y'y" + «'«" = "^ («^" + ^' + f + ^")' 



ww" + xx" + yy" -\-zz" = w'{v? + x" + / + »'), 

 2p5 



(F) 



