Prof. A. Macfarlane on ' Outlines of Quaternions.' 135 



refractive indices for visible rays, and render the results of 

 Colin and Arons and the independent results of Tereschin for 

 water not improbable a priori. Indeed, I think much more 

 evidence will be needed than Prof. Fessenden has given before 

 they are doubted. 



Prof. Fessenden also states that pure water insulates as well 

 as india rubber 1 The highest recorded resistance for water is 

 far below that of indiarubber. 



I would not have troubled you with this note were it not 

 that anything appearing in the Philosophical Magazine carries 

 authority, and if inaccurate is calculated to mislead. 



X. On Colonel Hime's ' Outlines of Quaternions.' 



To the Editors of the Philosophical Magazine. 

 Gentlemen, 



I CANNOT agree with your reviewer in holding that 

 paragraph 11 of the l Outlines ' is not quaternions. The 

 equation 



as your reviewer says, may be nonsense ; nevertheless it 

 follows from the fundamental principles of quaternions as 

 laid down by Hamilton and Tait. It is said that i 2 = — 1 and 

 f 2 = —1; therefore it follows that i 2 =j 2 , unless in quaternions 

 the axiom does not hold which says that things which are 

 equal to the same thing are equal to one another. But from 

 i 2 =j 2 it follows that i= +j or —j, for we are told that it is 

 only the commutative law of all the laws of algebra which 

 breaks down, and here that peculiarity does not enter. By 

 extending the same reasoning to the other two pairs, we obtain 



How does your reviewer demonstrate that it is nonsense? 

 He says that i, j, k have already been defined as co-perpen- 

 dicular unit-vectors, and to say that they are equal is to rob 

 them arbitrarily of their most characteristic feature, so that 

 they are no longer what they were defined to be. Yet a few 

 sentences before this same writer defends Hamilton's no less 

 arbitrary violation of the definition of a vector, which does 

 not indeed rob it of any of its meaning, but piles on it what 



