122 The Rev. C. Graves oti a system of Triple Algebra, 

 we shall have 



i/':=xi/-\-i/a^-{-zz' V (3.) 



:::"={a;—i/)z' + :2{x'—y')J 



From these relations it would not be difficult to determine 

 directly the mode of constructing the product line ; but the 

 result will be more readily arrived at by means of the follow- 

 ing method. 



Let ^, tji ^ be the projections of the line (-rj/z) upon OA, OB, 

 and OZ the axis of Z. Then I, m, n, the unit lines upon these 

 axes, are respectively expressed in terms of.? and s* by means 

 of the equations 



7_ 1+^(1) „,_l-^(t) ^^_ 5i(l-6(l) ). 



and we may wiite 

 instead of 



l^ + mri-\- n^ 



'}. (*.) 



And for the laws of combination of the imaginaries /, ?n, n, we 

 have the equations 



Im — O, /« = 0, ?ww=v'^2.« 



Hence the product of two lines (^jj^) and (^V^'} is 



Now the part of this which is on the axis OA, viz. \^2.l^^', 

 is in length a fourth proportional to the projections of the real 

 a?-unit and the two factor lines upon that axis. 

 So again the part 



which lies in the plane perpendicular to OA, is, in Mr. War- 

 ren's sense of the word, a fourth proportional to the projec- 

 tions upon that pla?ie of the same three lines. 



7. From the geometrical interpretation which has been as- 

 signed to the symbol s, we can derive what seems to me to be 

 a satisfactory explanation of the vanishing of the product of 

 the factors 1—5(1) and l+s(l), although these factors are 

 each different from zero. 



In consequence of s being distributive, the expression 

 (1_5(1))(1 -|-.s(l)) means the difference between 1+5(1) and 

 5(1+5(1)). Now, as the line l+s(l) coincides with the axis 

 OA, round which rotation takes place, it is unaffected by any 



