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LIX. On the Sy7nhols of Algebra, and on the Theory of Tes- 

 sarines. By James Cockle, Esq., M.A., of Trinity Col- 

 lege, Cambridge, and Barrisier-at-Law of the Middle 

 Temple*. 



AT page 436 of the last (33rd) volume of this Journal, I 

 proposed to include under the genus imaginary two 

 independent species of quantity which I distinguished by the 

 respective terms unreal and impossible. But, if we extend the 

 meaning of the word "imaginary" so as to make it compre- 

 hend all quantity that is not real, a third species of quantity, 

 for which I would suggest the name of ideal, must be added 

 to the two already included in the common genus of imagi- 

 naries. And perhaps it will be conducive to distinctness of 

 conception and to convenience if we admit a fourth species of 

 imaginaries, which I propose to call typal, from their squares, 

 &c. running into certain types. 



The peculiar symbols of the quaternion theory of Sir W. 

 R. Hamilton, of the octave theory of Mr. John T. Graves 

 and Mr. Cayley, of the triple algebra of Professor De Morgan, 

 and of the pluquaternion theory of the Rev. T. P. Kirkman, 

 all belong to the species ideal. It may be stated as the cha- 

 racteristic of ideal quantities, that, in their combinations, they 

 do not follow the laws of that ordinary algebra to which Mr. 

 De Morgan applies the term Double Algebra. The imagi- 

 naries in some of the systems of Mr. Cayley's theory of couples 

 (the systems marked C, D, C, and D', Phil. Mag., S. 3, 

 vol. xxvii. pp. 39-40) are ideal. And so, in general, are those 

 of the last two systems (E and E', lb. p. 40) in Mr. Cayley's 

 paper. Mr. J. T. Graves would call the couples involved in 

 the above systems anomalous. 



But in other of the systems of Mr. Cayley (those marked 

 A, B, A', B', lb. p. 38 et seq.) the imaginaries axe typal', and 

 this is also the case in Mr. J. T. Graves's theory of couples 

 {lb. vol. xxvi.). Borrowing a term from the learned writer 

 last mentioned, we may call all the couples mentioned in this 

 paragraph normal. The characteristic of typal quantities is, 

 that, although in their laws of combination they follow the 

 rules of ordinary algebra, yet the types or conditions by which 

 they are defined are not consistent with that algebra. Ktypal 

 differs from an impossible quantity in this : that the ordinary 

 algebra /orcf-s impossible quantity upon our notice, and defines 

 it by means as purely algebraic as those by which unreal 

 quantity is defined ; while on the other hand, typal (as well as 



* Communicated by Dr. John Cockle, FJl.C.S. 



