XIII. DIFFERENTIATION IN THE QUATERNION ANALYSIS. BY ALEXANDER MACFARLANE, D.Sc., LL.D. [COMMUNICATED BY MR. JOLY, ROYAL astronomer of Ireland.] [Read JUNE 25, 1900.] Ir is a prevalent belief in the learned world that the quaternion analysis has not taken that place among the methods of research which was predicted for it by its celebrated founder. Few mathematicians, excepting Professor Tait, have been able to bend the bow of Ulysses. Writers on the subject appear to share the opinion. explicitly enunciated by Professor Hardy1 that the writings of Hamilton contain the suggestion of all that will be done in the way of quaternion research and applications. The very greatness of the Elements appears to have had a deterring effect; for it has been considered a great storehouse of all the results that can be harvested in the quaternion field. It is commonly thought that the field is but a small corner of the mathematical domain, and that Hamilton has gone over it so thoroughly that there is little left to glean after him. However, there are good reasons for believing that the field is not narrow, but is, in all probability, nearly coextensive with the mathematical domain. My opinion is that Hamilton's estimate of the importance of the discovery which he communicated to this Academy at the memorable meeting of November, 1843, is under, instead of over the mark, and that this will be demonstrated by the course of development of mathematical analysis in the coming century. But, first of all, the bow must be examined to find out why it is so difficult to bend. I do not suppose that Hamilton considered that he had arrived at a finality of wisdom on the principles of the analysis. He did not intend to give to the world a dead analysis, however classical. Anyhow, the analysis can receive only good from a free and independent discussion of its principles; if they are perfect, their perfection will thereby become the more apparent and convincing; if they are imperfect, it is certainly desirable that the imperfection 1 In the preface to his Elements of Quaternions. 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 multiplicity of value." The other place is in the treatment of differentiation. He says at page 411: "The functions of quaternions, which have been lately differentiated, 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 expressions may not become too complex." Space differentiation, as taught by Hamilton, certainly presents novel difficulties; 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 differentiation; 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, therefore and (q + Aq)2 = q2 + qAq + Aqq + (Ag)2; hence, the limiting value when Aq = 0 is q+0g/0 which is indefinite, because the second term is not independent of Aq; consequently, there appears to be no differential coefficient. The new definition makes the differential of q' to be qAg + Aqq; and this expression cannot be reduced to 2qAq, because the products q▲g and Aqq 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 coordinate 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, €9+q' = ees' only when q and q' are coplanar; the general formula is = 2 €99 + term of the second derived from e2e?' is + terms of the third and higher orders. The order derived from e+ is (q2 + q22 + 2qg'). Now, if (q + q')2 = q2 + q'2 + qq' + q'q, (q + q')2, while that we get the above difference of the second order; but if (q + q′)2 = q2 + q'2 + 2qq', there is no difference of the second order. Similarly, if (q + q')3 = q3 + 3q2q′ + 3qq'2 + q'3, there is no difference of the third order. And, generally, if the nth power of 9 +q is formed after the formula for a binomial of scalar quantities, but subject to the condition that in each partial product g is always preserved anterior to q', there will be no difference of the nt order; and the exponential theorem generalised for space will retain the simple form which it has for the plane, namely, 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 Hamiltonian vectors; they may then be denoted by ẞ and y. Now eß is the expression for the circular versor, having the angle Tẞ, and the axis UB; similarly ey is the expression for the circular versor, having the angle Ty, and the axis Uy. The product eßer does not in general allow the order of the factors es and ev to be changed; consequently, if eß+ is an equivalent expression, it cannot allow the order of the logarithms ẞ and y to be changed in ẞ+y. The sum of these logarithms is non-commutative, just as much as the factors of which they are the logarithms; and from this fact I conclude that the square or any power of ẞ+ y must be so formed that the order of B prior to y is preserved. There is another line of argument which proves very conclusively that in the expansion of es+y the powers must be expanded so as to preserve the order of the logarithms. It is known that e-Beveß expresses an angle, the magnitude of which is Ty, and the axis of which is Uy, turned by an angle of 2TB round UB. Now, according to the principle which I am advocating, Were the trinomial - ẞ + y + ẞ treated as a sum of vectors having no real order, it would reduce to y, and ev is known not to be equivalent to e-Beveß. But it is a remarkable fact that when the powers of - B+y+ẞ are formed so as to preserve in the several partial products the natural order of the vectors, the terms of the series for the cosine are independent of y, while those for the directed sine involve 7. This was shown at length in my paper on The Fundamental Theorems of Analysis generalized for Space. 1 Consider now the light which the generalized exponential theorem throws on the subject of differentiation. Let B denote a Hamiltonian vector, that is to say, the product of a tensor and a quadrantal versor. The tensor may be denoted by b. As a quadrantal versor is equivalent to an imaginary axis, it may be denoted by 18. Hence B=b18. First of all, what is the differential coefficient of B, supposing B to vary both in magnitude and axis. Suppose B to change into B + dB; then eB becomes eB+dB. By the generalized Exponential Theorem 1 "On Hyperbolic Quaternions," Proc. R.S.E., 16th July, 1900. deb In form the differential coefficient does not differ from that for db it follows that dB" = n B-1dB. Hence, if B denote any vector logarithm, real or imaginary, and n a positive integer, The above symbol B denotes a vector in the Hamiltonian sense; it is really an imaginary vector, and can be analysed into b-1ẞ, where b denotes the magnitude, and ẞ the axis. Now dt' but, according to the results of the above investigation, |