Cambridge Philosophical Society. 227 



having a symmetrical function of a, b, c, and, throwing away this 

 stipulation, points out a variety of triple systems, partially or wholly 

 interpreted. 



Sir William Hamilton's quaternion algebra is not entirely the 

 same in its symbolical rules as the ordinary algebra : differing in that 

 the equation AB=BA is discarded and AB= — BA supplies its place. 

 Those of Mr. De Morgan's system, which are imperfect, all give 

 AB=BA, but none of them give A(BC) = (AB)C, except in particu- 

 lar cases. 



Mr. De Morgan gives systems of triple algebra, which he distin- 

 guishes into quadratic, cubic, and biquadratic, according as the in- 

 vented imaginary units represent square roots, cube roots, or square 

 and fourth roots, of the negative real unit. It would not be easy in 

 an abstract to give any account of these, but among them are 

 found, — 



1 . An imperfect quadratic system, strongly resembling the common 

 double algebra, and which would, but for its imperfect character, be 

 at once recognised as the proper and natural extension of the inter- 

 pretation of imaginary quantities to three dimensions of space : the 

 ultimate symbol for a line is I (cos 9 + sin 9 ^a)~^)- 



2. An imperfect quadratic system, very like the former one, except 

 in having a peculiar inversion in the operation of multiplication, and 

 a somewhat remarkable mode of representing what would by analogy 

 be called arithmetical multipliers. 



3. A perfect quadratic system, the interpretation of which has con- 

 siderable resemblance to that of the first-mentioned system, and is 

 completely attainable, though not of great interest. 



4. Three perfect cubic systems, each irreconcileable with the 

 others, though closely connected with thei];i. Each system presents 

 a triple trigonometry, the cosine and two sines of which are each a 

 function of two angles ; but these can be easily expressed as func- 

 tions of common circular and hyperbolic sines and cosines. The in- 

 terpretations of these systems are very imperfect, and appear to pre- 

 sent great difficulty, but their symbolical character is unimpeachable. 



5. A perfect biquadratic system, which is of a redundant charac- 

 ter, that is, its fundamental form represents a line drawn in space 

 from a given origin, with a symbol to spare, which may represent 

 the time of drawing it, its density, its tendency to a given position, 

 &c. at pleasure. Many interpretations are attainable, but Mr. De 

 Morgan does not pretend to say that he knows the one which ought 

 to be adopted. It is singular that every attempt to reduce this 

 algebra, by assigning a condition among the subsidiary symbols of its 

 fundamental form, leads to an imperfect algebra. The system first 

 mentioned in this abstract is one such result, and fails in its rules of 

 multiplication, as before mentioned, Another is obtained, which is 

 perfect as to its rules of multiplication, &c., but fails in its rules of 

 addition. 



Mr. De Morgan concludes by giving some formulae which may be 

 useful to those who would try to interpret algebra of three dimen- 

 sions by the use of solid angles in the place of plane ones. 



Q2 



