(D.) 



(F.) 



12 Sir W. R. Hamilton on Quaternions. 



3. With the assumed relations (A.)) (B.), (C.)) we have the 

 four following expressions for the four constituents of the pro- 

 duct of two quaternions, as functions of the constituents of the 

 multiplier and multiplicand : 



w/' = to to' — X x' —y y^ — z z',- 

 x" ='w.v' + xiscf + 1/ z'— zy', 

 y = >wi/' + i/'vi:/-{-s!x' — xz', 

 z" ='w z' + ziii:^ + o:y' ~y xKJ 

 These equations give 



W/'2 4- ^"2 +y'2 + Z"^ =z (tt;2 + ^2 +y + ^2) (.^2 + ^2 ^.y2 ^ ^/2j . 



and therefore 



1^" = ^^', ; (E.) 



if we introduce a system of expressions for the constituents, 

 of the forms 



TO = ju. cos 9, 



a: = jtA sin 9 cos <p, 



?/ = jM, sin 9 sin <p cos rj/, 



s = jo. sin Q sin <p sin vp,- 

 and suppose each [x, to be positive. Calling, therefore, ju. the 

 modulus of the quaternion Q, we have this theorem : that the 

 modulus of the product Q" of any two quaternions Qand Q', is 

 equal to the product of their moduli. 



4. The equations (D.) give also 



to'to" + a;' a;" -^T^y'^ + s;';s" = TO (to'^ + x^'^ + ^^ + ;j;'2)^ 

 TOTO" + XX» +yyll + z ^r" = to'(to2 + ^'^ + ?/2 + ;s2j . 



combining, therefore, these results with the first of those equa- 

 tions (D.), and with the trigonometrical expressions (F.), and 

 the relation (E.) between the moduli, we obtain the three fol- 

 lowing relations between the angular co-ordinates 5 4) \I/, 6' (p' \I/', 

 fl" (p" vP" of the two factors and the product : 

 cos 9" = cos 9 cos &' — sin 9 sin 9' (cos (p cos <p' + sin <$> sin 4>' cos {^ — \I/')), ^ 

 cos 9 =:cos9'cos9" + sin9'sin9"(cos<|)'cosi$" + sin<^'sin^"cos(\j/'— 4/")), ^ 

 cos 9' =cos9"cos9 + sin9"sin9(cos4>"cos<$ + sin<^"sin45Cos(\|/" — 4/)). J 

 These equations (G.) admit of a simple geometrical con- 

 struction. Let xyz be considered as the three rectangular 

 co-ordinates of a point in space, of which the radius vector is 

 = ja.sin9, the longitude =\I/, and the co-latitude =4^; and let 

 these three latter quantities be called also the radius vector, 

 the longitude and the co- latitude of the quaternion Q; while 9 

 shall be called the amplitude of that quaternion. Let R be the 

 point where the radius vector, prolonged if necessary, inter- 

 sects the spheric surface described about the origin of co-or- 

 dinates with a radius = unity, so that ^ is the co-latitude and 



