The Rev. C. Graves oti a system of Triple Algebra. 119 



delicate colours may be attacked, but this must be a rare 

 case. 



I am more desirous of seeing ships built of an incombustible 

 material, the means of escape at sea being few, and confined 

 to few ; and whilst there is any hope of doing it easily, I 

 scarcely think it proper for any one to neglect what informa- 

 tion may exist on the subject. 



XV. On a sgsfem of Triple Algebra, and its apjdicatio?! to 

 the Geometry of three Dimensions, By the Rev. Charles 

 Graves, A.M., Professor of Mathematics in Trinity College, 

 Dublin^. 



WHETHER we can directly evolve out of the ordinary 

 double algebra such a more general algebra as will 

 answer all the requisitions of the geometry of three dimensions, 

 is a question into which I am not about to enter at present. 

 But Mr. Cockle's discussion of it in the last Number of the 

 Philosophical Magazine, and his use of an imaginary whose 

 square is equal to positive unity, have reminded me of a system 

 of triple algebra which I devised some time ago, and which, 

 I am inclined to believe, will find morefavour with others than 

 it has (lone with myself. The outline of it was stated in a 

 letter addressed by me to the late Professor MacCullagh in 

 April 1845, since which time I have suffered it to lie without 

 further development or application. For my own part, I com- 

 mend the superior power, symmetry, and flexibility of Sir 

 William Hamilton's quaternion theory, of the excellence of 

 which use has only more and more convinced me. I find, 

 however, that there are mathematicians who continue to object 

 to it on the ground that they cannot conceive the nature of 

 the operations by which his symbols i,j, k transform extra- 

 spatial, scalar quantity into directed linear magnitude; and 

 who protest against the sacrifice of the commutative character 

 of multiplication. To those who cannot bring themselves to 

 waive these objections, I submit the following system. It has 

 the advantage of being easily and completely interpreted ; and 

 I may add that, in the treatment of some geometrical questions, 

 it has proved itself not inferior even to the quaternions. 



1. Imagine an operation denoted by the symbol (5), and 

 of such a nature that — 



1. 5(1) is a quantity not homogeneous with the real unit, 

 so that no equation of the form 5(a) + 6 = 0, where a and b are 

 real, can subsist, unless a and b be separately equal to zero. 



* Communicated by the Author. . 



