202 Proceedings of the Royal Irish Academy. 



vectors ; they may then be denoted by /? and y. I^ow e^ is the 

 expression for the circular versor, having the angle T^, and the axis 

 up ; similarly ev is the expression for the circular versor, having the 

 angle Ty, and the axis Uy. The product e^ey does not in general 

 allow the order of the factors e^ and ev to be changed ; consequently, 

 if e^+y is an equivalent expression, it cannot allow the order of the 

 logarithms /3 and y to be changed in yS + y. The sum of tLese 

 logarithms is non-commutative, just as much as the factors of wliich 

 they are the logarithms ; and from this fact I conclude that the 

 square or any power of ^ + y must be so formed that the order of 

 P prior to y is preserved. 



There is another line of argument which proves very conclusively 

 that in the expansion of e^ + y the powers must be expanded so as to 

 preserve the order of the logarithms. It is known that e-^eye^ ex- 

 presses an angle, the magnitude of which is Ty, and the axis of which 

 is Uy, turned by an angle of 2T(i round Z7/3. Now, according to the 

 principle which I am advocating, 



Were the trinomial - ^ + y + j3 treated as a sum of vectors having no 

 real order, it would reduce to y, and ey is known not to be equiva- 

 lent to e~Peye^. But it is a remarkable fact that when the powers 

 of - y8 + y + y8 are formed so as to preserve in the several partial 

 products the natural order of the vectors, the terms of the series for 

 the cosiae are independent of y, while those for the directed sine 

 involve y. This was shown at length in my paper on The Fundamental 

 Theorems of Analysis generalized for Space. 



Consider now the light which the generalized exponential theorem 

 throws on the subject of differentiation. Let B denote a Hamilionian 

 vector, that is to say, the product of a tensor and a quadrantal versor. 

 The tensor may be denoted by h. As a quadrantal versor is equiva- 

 lent to an imaginary axis,^ it may be denoted by J- 1;8. Hence 

 B = h J- 1/3. First of all, what is the differential coefficient of e^, 

 supposing B to vary both in magnitude and axis. Suppose B to 

 change into B + dB ; then e^ becomes e^ + ^^. By the generalized 

 Exponential Theorem 



{dBy {dBf 



= eB U+dB + 



2! 3! 



1 " On Hyperbolic Quaternions," Proc. E.S.E., 16th July, 1900. 



