220 Sir W. Rowan Hamilton on Quaternions. 



whether the cause of the motion is gravity, or the fluidity of 

 the lowest part, or the fluidity in general. Without attaching 

 much value to such distinctions, we may reply, that the cause 

 of the motion is gravity ; the cause of the steadiness of the 

 motion is the melting of the lower surface; the cause of the 

 glacier character of the motion is the plasticity. 



I am, dear Sir, 

 Trinity College, Cambridge, Your faithful Servant, 



February 18, 1845. ^^ WhEWELL. 



XXXI. On Qjiaternions', or on a 7iew System of Imaginaries 

 in Algebra. By Professor Sir William Rowan Hamilton, 

 LL.D.y P. R.I. A., Correspo7iding Member of the Institnte of 

 France, and Royal Astronomer of Ireland. 



[Continued from vol. xxv. p. 246.] 

 12. A QUATERNION, Q, divided by its modulus, ja, 

 -^^ may in general (by what has been shown) be put 

 under the form, 



jK,~ ^ Q = cos 9 + /r sin 9 ; 



in which 6 is a real quantity, namely the amplitude of the qua- 

 ternion ; and i^ is an imaginary unit, or square root of a nega- 

 tive one, namely that particular root, or unit, which is distin- 

 guished from all others by its two directional coordinates, and 

 is constructed by a straight line drawn from the origin of co- 

 ordinates to the representative point R ; this point R being on 

 the spheric surface which is described about the origin as 

 centre, with a radius equal to unity. Comparing this expres- 

 sion for ju.""^ Q with the formula (M.) for the product of any 

 two imaginary units, we see that if with the point R as a posi- 

 tive pole, we describe on the same spheric surface an arc P' P" 

 of a great circle, and take this arc =7r — 6= the supplement 

 of the amplitude of Q ; and then consider the points P' and P" 

 as the representative points of two new imaginary units /p/ and 

 2p//, we shall have the following general transformation for any 

 given quaternion^ 



Q = jo, iyi p// ; 



the arc P' P" being given in length and in direction, except 

 that it may turn round in its own plane (or on the great circle 

 to which it belongs), and may be increased or diminished by 

 any whole number of circumferences, without altering the value 

 ofQ. 



13. Consider now the 'product of several successive quater- 

 nion factors Qi, Qgj . . under the condition that their ampli- 



