Macfarlane — Differentiation in the Quaternion Analysis. 201 



hence, the limiting value when A^- = is 5' + 03-/0 which is indefinite, 

 because the second term is not independent of ^q ; consequently, there 

 appears to he no differential coefficient. The new definition makes 

 the differential of q^ to he ql^q + ^qq ; and this expression cannot be 

 reduced to 2q^q, because the products q^q and ^qq are in general 

 non-commutative. 



It is evident that Hamilton's reasoning all depends on the truth of 

 the rule according to which the square of a binomial is formed. He 

 writes the square as two successive factors {q + dq) {q + dq), applies 

 the distributive rule, and preserves the order of the factors in the 

 partial products. This is the reason why dq is posterior to q in one 

 term and anterior in the other. But when dq is by nature posterior 

 to q, as is the case when q denotes a logarithm, that cannot be the 

 true rule for forming the square. No doubt a sum of arbitrary coor- 

 dinate vectors is independent of order, but that is no good reason for 

 assuming that an expression such as q + dq is independent of order 

 when it denotes an index. 



The investigation of this question leads directly to a consideration 

 of the other peculiarity mentioned above. According to Hamilton, 

 g9+«' = 0<ig<i' only when q and q' are coplanar ; the general formula is 



e«+«' = e'^e'i' + ^^ ^^ + terms of the third and higher orders. The 



term of the second order derived from e'^^' is ^{q + q'Y, while that 

 derived from eV' is ^{q^ + q'"^ + 2qq'). ]N"ow, if 



{q + q'y = q^ + q''^ + qq' + q'q, 

 we get the above difference of the second order ; but if 



{q + q'f = f \ q'^ + 2qq', 

 there is no difference of the second order. Similarly, if 



{q + q'y = q^ + 3qY + oqq'' + q'^, 



there is no difference of the third order. And, generally, if the w"* 

 power of q -\- q' is formed after the formula for a binomial of scalar 

 quantities, but subject to the condition that in each partial product q 

 is always preserved anterior to q', there will be no difference of the 

 n*^ order ; and the exponential theorem generalised for space will 

 retain the simple form which it has for the plane, namely, 



^9+3' = e<lg'l\ 



If we look further into the matter, we shall find good reasons for 

 believing that it must be so. Suppose that q and q' are Hamiltoniau 



