280 Mr. G. Boole's Notes on Quaternions. 



may nevertheless be employed, with the understanding that 

 certain factors shall be rejected or other reductions performed, 

 and thus they may lead to correct results. But the employ- 

 ment of such a form of the process seems to involve a depart- 

 ure from the principle, that the laws of the sign shall consti- 

 tute in every respect an exact counterpart to the laws of the 

 thing signified. 



Sir William Hamilton's theory of the application of qua- 

 ternions appears to be based upon the relations which their 

 elements bear to the angles and angular points of spherical 

 polygons ; and similar to this is the basis which Prof. Graves 

 has adopted for the not less interesting theory of triplets and 

 multiplets. In all these systems we meet with such theorems 

 as the following, viz. that the product of two given points is a 

 certain third point, &c, by which it is meant that a certain 

 expression, having a determinate reference to the latter point, 

 is the product of two expressions having a similar determinate 

 reference to the two original points. I believe that upon exami- 

 nation it will be found that these systems of interpretation are 

 founded upon a principle of Naming, as the one which I have 

 proposed is founded upon a principle of Operation. And I think 

 it not foreign to the subject to remark, that the symbolical 

 forms of common language as exhibited in the calculus of logic 

 may indifferently be referred to the one or the other of these 

 modes of conception. 



Laws of Quaternions. 



The laws of quaternion multiplication are founded, as is 

 well known, upon the following relation : 



e 2 = — 1 /= — 1 £*= — 1 ij—k jk—i ki-j 



Ji= —k kj= — i ik = —j ; 



but it may be shown that the three last of these laws are con- 

 sequences of the former ones considered as of universal appli- 

 cation. 



For \{ij—h universally, let the subject bejy, then 



or 



ify-kjy\ 



but^sa — 1, therefore 



and similarly for the others. 

 Lincoln, September 3, 1848. 



