59 



terpretation of equation (1): the latter of equations (2) and 

 (3). 



The following eases deserve special attention : 



If either of the factor lines coincides with the symmetric 

 axis, the product line must also coincide with it. 



If either of the factor lines is contained in the symmetric 

 plane, the product line must also be contained in it. 



But if one coincides with the symmetric axis, and the 

 other with the symmetric plane, the product line will vanish. 



Startling as these consequences may appear, they are to 

 be explained by reference to the geometric meaning of the 

 symbols t and ^ ; both of which, according to the interpreta- 

 tion assigned to them, are inoperative to move a right line out 

 of either the symmetric plane or the symmetric axis. 



The analytical difficulty raised by the last case seems to 

 force us to admit that the vanishing of a product does not ne- 

 cessarily imply the vanishing of a factor. In the present in- 

 stance it is caused by the vanishing of one of the moduli of 

 multiplication belonging to each of the factors. 



The length of the product line is equal to the product of 

 the lengths of the factor lines only in the case where the vector 

 angle of one of the factors is equal to the vector angle of the 

 real linear unit (where p or pi r: 6'). 



Having thus interpreted the results of multiplication by 

 means of the existing trigonometry, Mr. Graves proceeded to 

 show how the use of a new kind of trigonometry gives in- 

 creased symmetry and flexibility to the present theory of 

 algebraic triplets. 



The foundations of this new calculus are thus laid. Using 

 the exponential development we shall find 



C* z= A -I- <^ -I- i^v 

 where 



^ = ' + its + riafeo + «"• 



f 2 



