434 Mr. J. Cockle on Quadruple Algehra 



direct means, as to those which are opake, as Mr. Joule has 

 done for magnetic substances. For the proofs drawn from 

 diamagnetism and sonorous vibrations are only indirect, 

 although the last appear to be tolerably conclusive. 



I shall soon return to this subject, in reference to some 

 researches on the relation between diamagnetism and the in- 

 duced currents, on which I am at present engaged. 



LII. On Systems of Algehi^a involving more than one Ima- 

 ginary ; and on Equations of the Fifth Degree. By James 

 CockLe, Esq., M.A., Barrister-at-La'w'^. 



CONCEIVE two imaginaries, such that their respective 

 squares are equal either to positive or to negative unity. 

 Then the product of two linear functions of these imaginaries 

 is not of the same form as its factors. The product of the 

 imaginaries prevents this similarity, and obstructs the forma- 

 tion of a System of Triple Algebra on the basis just men- 

 tioned. But, if we invest the last-named product with the 

 character of a third imaginary, and assume that its square is 

 equal either to positive or to negative unity, four systems of 

 Quadruple Algebra will present themselves, in each of which 

 the product of two linear functions of the three imaginaries 

 will, in general, have the same form as its factors. 



Let « and /3 respectively represent the first and second, and 

 7 the third imaginary. Then y=«/3 or /3«. But as, in qua- 

 druple algebra, aj3 is not always equal to /3«, I shall select the 

 former as the expression for 7. 



Let A denote a linear function of a, j6, and y; in other 

 language, let 



A = w + «jr + /By + y2;, 



then, in one of the four systems of quadruple algebra above 

 alluded to, the expression A bears the name of a quaternion. 

 In the remaining three systems the respective terms tessarine, 

 coquaternioji, and cotessarine may be applied to it. At least 

 I have suggested such a nomenclature in No. 1360 of the 

 Mechanics' Magazinef, where I have shown the existence of 



* Communicated by the Author, In connexion with his paper published 

 at pp. 406-410 of the preceding (34th) volume of this Journal, Mr. Cockle is 

 desirous of referring the reader to two articles subsequently communicated 

 by him to the Mechanics' Magazine, and which will be found at pp. 634 

 and 558, 559 of vol. 50 of that work. 



t See pp. 197, 198 of the current (51st) volume of that work. I had, 

 however, previously employed the term " Tessarine " both in that and in 

 the present Journal. 



