200 Proceedings of the Royal Irish Academy. 



thould be removed. There are two places in the Elements of Quaternions 

 where further investigation seems desirable. The quaternion analysis 

 is intended to be applicable to space of three dimensions, but at these 

 two places Hamilton restricts the analysis to the plane. 



The first place is in the treatment of logarithms. He says at page 

 386:— 



" In the present theory of diplanar quaternions we cannot 

 expect to find that the sum of the logarithms of any two proposed 

 factors shall be generally equal to the logarithm of the product ; but 

 for the simpler and earlier case of complanar quaternions that algebraic 

 property may be considered to exist with due modification for multi- 

 plicity of value." 



The other place is in the treatment of differentiation. He says at 

 page 411 : — 



"The functions of quaternions, which have been lately diffe- 

 rentiated, may be said to be of algebraic form ; the following are 

 a few examples of differentials of what may be called, by contrast, 

 transcendental functions of quaternions ; the condition of complanarity 

 being, however, here supposed to be satisfied, in order that the expres- 

 sions may not become too complex." 



Space differentiation, as taught by Hamilton, certainly presents 

 novel difiS.culties ; there is, in general, no differential coefficient ; 

 recourse is made to a new definition of a differential, and under 

 certain conditions only is there an analogue to Taylor's theorem. 

 "What is the source of these difficulties ? It is, according to Hamilton, 

 the non-commutative character of quaternion multiplication. He says, 

 at page 391 of the 'Elements : — 



' ' The usual definitions of differential coefficients and of derived 

 functions are found to be inapplicable generally to the present 

 calculus, on account of the non-commutative character of quaternion 

 multiplication. It becomes, therefore, necessary to have recourse to 

 a new definition of differentiation, which yet ought to be so framed 

 as to be consistent with, and to include, the usual rules of diffe- 

 rentiation ; because scalars as well as vectors have been seen to be 

 included under the general conception of Quaternions." 



The essence of the difficulty will be seen by taking the simple 

 instance of the square. According to Hamilton, 



{q + Aqy = q'^ + qAq + Aqq + (Aqf ; 



therefore {q + AqY - q^ = qAq + Aqq + {Aq)-, 



and {{q + Aqy - q'^} / Aq = q + Aqq / Aq + Aq; 



