425 



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

 separate equations, 



w =. w, X = x', y =: y', ■? = z'. 



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

 ternion; and the four real quantities w, x, y, s he calls the con- 

 stituents thereof. Quaternions are added or subtracted by 

 adding or subtracting their constituents, so that 



q-{-q'=w + w' + i(x + x) +i (y + y') + ^(s + «')• 

 Their multiplication is, in virtue ot the definitions (a) (b) (c), 

 effected by the formulas 



qq' = q" = w" + ix" -^-jy'' + k%", 



tv" = ww' — xx' — yy' — zz', 

 x" — wx' + xw' + y% — zy', 

 y" =r wy' + yw' -{• zx' — xz', 

 z" — wz' + zw ^xy' — yx', 

 which give 



and therefore 



if we call the positive quantity 



(D) 



M = /mj^ 4- ^"^ + 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 



\ (F) 



w:= fi cos V, 

 X z= fj. sinO cos f, 

 y = fi sin 6 sin cost//, 

 « = j« sin sin ^ sin i// ; j 



then, because the equations (d) give 



w'w" + ^'y + tj'y" + s's" = w {w'^ + ^' + 2/'"'+ ■^")» 

 m)m;" + xx" + yy" + zz" = w'{w'^ + a;2 + 2/^ + s^), 

 2 p 2 



