and on Equatiofis of the Fifth Degree. 435 



these four systems, discussed them, and pointed out their cha- 

 racteristics. I here propose to advert for a moment to the 

 same subject, to consider it under a slightly different aspect, 

 and also to exhibit, for convenience of comparison, the mo- 

 dular expressions of all the systems. We have, then, 



1. The Quaternion System of Sir W. R. Hamilton^ in 

 which 



a2=_i, |3«=-1, and «/3 = y; 



but in vk'hich also 



contrary to what we should have inferred from the equations 



hence, the quaternion system is abnormal, or does not obey 

 the laws of ordinary algebra. The modulus of a quaternion is 

 the positive square root of the expression 



IIO^ + X^ + J/^ + Z^. 



2. The Tessarine System — a normal system in which 



a2=-l, /32=1, and y^=-l=u^fi^ 



The true modulus of the tessarine A is the positive square 

 root of 



(M?+^)2 + (a? + 5r)2. 



3. The Coquaternion System, in which 



a2= — 1, /32=:1, and y^=l; 

 but the last relation is inconsistent with the conditions 



and the coquaternion system is abnormal. The modulus of 

 A, considered as a coquaternion, is the positive square root of 



4. The Cotessarine System, in which 



«2=l^ /32=,1^ ^2^ 1^5,2^2^ 



and which is a normal system, having for its modular form the 

 positive square root of 



It is to be borne in mind, that in all the above systems 

 y=aj3; that, whenever the double sign ( + ) occurs, the sign 

 of the term is indifferent and quite independent of that of the 

 preceding or following term ; that w, a^, y, and z are real 

 quantities, positive, negative, or zero ; and that, in multiplying 



2 F2 



