CHAP. III.] DIFFERENTIALS OF VECTORS. 101 (6.) We may differentiate a vector-function twice (or oftener), and so obtain its successive differentials. For example, if we differentiate the derived vector p', we obtain a result of the form, dp'=p'dt, where p′′=Dip′ = Di2p, by an obvious extension of notation; and if we suppose that the second differential, ddt or det, of the scalar t is zero, then the second differential of the vector p is, dp = ddp = d. p'dt = dp′. dt = p′′. dt2 ; where dr2, as usual, denotes (dt)2; and where it is important to observe that, with the definitions adopted, d2p is as finite a vector as do, or as p itself. In applications to motion, if t denote the time, p" may be said to be the Vector of Acceleration. (7.) We may also say that, in mechanics, the finite differential dp, of the Vector of Position p, represents, in length and in direction, the right line (suppose PT in Fig. 32) which would have been described, by a freely moving point P, in the finite interval of time dt, immediately following the time t, if at the end of this time t all foreign forces had ceased to act.* (8.) In geometry, if p = (t) be the equation of a curve of double curvature, regarded as the edge of regression (comp. 98, (12.)) of a developable surface, then the equation of that surface itself, considered as the locus of the tangents to the curve, may be thus written (comp. 99, II.): p=$(t) + up' (t); or simply, p = p(t) + dp (t), if it be remembered that u, or dt, may be any arbitrary scalar. (9.) If any other curved surface (comp. again 99, II.) be represented by an equation of the form, p= (x, y), where now denotes a vector-function of two independent and scalar variables, x and y, we may then differentiate this equation, or this expression for p, with respect to either variable separately, and so obtain what may be called two partial (but finite) differentials, dép, dyp, and two partial derivatives, D, Dyp, whereof the former are connected with the latter, and with the two arbitrary (but finite) scalars, dx, dy, by the relations, And these two differentials (or derivatives) of the vector p of the surface denote two tangential vectors, or at least two vectors parallel to two tangents to that surface at the point P: so that their plane is (or is parallel to) the tangent plane at that point. (10.) The mechanism of all such differentiations of vector-functions is, at the present stage, precisely the same as in the usual processes of the Differential Calculus; because the most general form of such a vector-function, which has been considered in the present Book, is that of a sum of products (comp. 99) of the form xa, where a is a constant vector, and x is a variable scalar: so that we have only to operate on these scular coefficients x.., by the usual rules of the calculus, the rectors a.. being treated as constant factors (comp. sub-art. 2). But when we shall come to consider quotients or products of vectors, or generally those new functions of vectors which can only be expressed (in our system) by Quaternions, then some few new rules of differentiation become necessary, although deduced from the same (or nearly the same) definitions, as those which have been established in the present Section. As is well illustrated by Atwood's machine. (11.) As an example of partial differentiation (comp. sub-art. 9), of a vector function (the word "vector" being here used as an adjective) of two scalar variables, let us take the equation, in which p (comp. 99, (3.)) is the vector of a certain cone of the second order; or more precisely, the vector of one sheet of such a cone, if x and y be supposed to be real scalars. Here, the two partial derivatives of p are the following: so that the three vectors, p, Dip, Dyp, if drawn (18) from one common origin, are contained (22) in one common plane; which implies that the tangent plane to the surface, at any point P, passes through the origin o: and thereby verifies the conical character of the locus of that point P, in which the variable vector p, or OP, terminates. (12.) If, in the same example, we make x = 1, y=-1, we have the values, whence it follows that the middle point, say c', of the right line AB, is one of the points of the conical locus; and that (comp. again the sub-art. 3 to Art. 99, and the recent sub-art. 9) the right lines OA and OB are parallel to two of the tangents to the surface at that point; so that the cone in question is touched by the plane AOB, along the side (or ray) oc'. And in like manner it may be proved, that the same cone is touched by the two other planes, BOC and COA, at the middle points a' and B' of the two other lines BC and CA; and therefore along the two other sides (or rays), oa' and OB': which again agrees with former results. (13.) It will be found that a vector function of the sum of two scalar variables, t and dt, may generally be developed, by an extension of Taylor's Series, under the form, it being supposed that dt = 0, d3t = 0, &c. (comp. sub-art. 6). Thus, if pt at2, (as in sub-art. 1), where a is a constant vector, we have dot = atdt, d2pt = adt2, d3pt = 0, &c.; and $(t + dt) = a(t + dt)2 = {at2 + atåt + adt2, rigorously, without any supposition that dt is small. (14.) When we thus suppose At=dt, and develope the finite difference, Ap(t) = $(t + dt) — ¢(t), the first term of the development so obtained, or the term of first dimension relatively to dt, is hence (by a theorem, which holds good for vector-functions, as well as for scalar functions) the first differential dpt of the function; but we do not choose to define that this Differential is (or means) that first term: because the Formula (100), which we prefer, does not postulate the possibility, nor even suppose the conception, of any such development. Many recent remarks will perhaps appear more clear, when we shall come to connect them, at a later stage, with that theory of Quaternions, to which we next proceed. BOOK II. ON QUATERNIONS, CONSIDERED AS QUOTIENTS OF VECTORS, AND AS INVOLVING ANGULAR RELATIONS. CHAPTER I. FUNDAMENTAL PRINCIPLES RESPECTING QUOTIents of vectors. SECTION 1.-Introductory Remarks; First Principles adopted from Algebra. ART. 101. The only angular relations, considered in the foregoing Book, have been those of parallelism between vectors (Art. 2, &c.); and the only quotients, hitherto employed, have been of the three following kinds: I. Scalar quotients of scalars, such as the arithmetical frac II. Vector quotients, of vectors divided by scalars, as in Art. 16; III. Scalar quotients of vectors, with directions either simi β lar or opposite, as = x in the last cited Article. But we now a propose to treat of other geometric QUOTIENTS (or geometric Fractions, as we shall also call them), such as for each of which the Divisor (or denominator), a or os, and the Dividend (or numerator), ẞ or оB, shall not only both be Vectors, but shall also be inclined to each other at an ANGLE, distinct (in general) from zero, and from two* right angles. 102. In introducing this new conception, of a General Quotient of Vectors, with Angular Relations in a given plane, or in space, it will obviously be necessary to employ some properties of circles and spheres, which were not wanted for the purpose of the former Book. But, on the other hand, it will be possible and useful to suppose a much less degree of acquaintance with many important theoriest of modern geometry, than that of which the possession was assumed, in several of the foregoing Sections. Indeed it is hoped that a very moderate amount of geometrical, algebraical, and trigonometrical preparation will be found sufficient to render the present Book, as well as the early parts of the preceding one, fully and easily intelligible to any attentive reader. 103. It may be proper to premise a few general principles respecting quotients of vectors, which are indeed suggested by algebra, but are here adopted by definition. And Ist, it is evident that the supposed operation of division (whatever its full geometrical import may afterwards be found to be), by which we here conceive ourselves to pass from a given divisorline a, and from a given dividend-line ẞ, to what we have called (provisionally) their geometric quotient, q, may (or rather must) be conceived to correspond to some converse act (as yet not fully known) of geometrical multiplication: in which new act the former quotient, q, becomes a FACTOR, and operates on the line a, so as to produce (or generate) the line 3. We shall therefore write, as in algebra, Bq.a, or simply, ẞ=qa, when ẞ: a=q; * More generally speaking, from every even multiple of a right angle. † Such as homology, homography, involution, and generally whatever depends on anharmonic ratio: although all that is needful to be known respecting such ratio, for the applications subsequently made, may be learned, without reference to any other treatise, from the definitions incidentally given, in Art. 25, &c. It was, perhaps, not strictly necessary to introduce any of these modern geometrical theories, in any part of the present work; but it was thought that it might interest one class, at least, of students, to see how they could be combined with that fundamental conception of the VECTOR, which the First Book was designed to develope. CHAP. I.] FIRST PRINCIPLES ADOPTED FROM ALGEBRA. 105 even if the two lines a and ẞ, or oa and oв, be supposed to be inclined to each other, as in Fig. 33. And this very simple and natural notation (comp. 16) will then allow us to treat as identities the two following formulæ : although we shall, for the present, abstain from writing also such formulæ as the following: where a, ẞ still denote two vectors, and q denotes their geometrical quotient: because we have not yet even begun to consider the multiplication of one vector by another, or the division of a quotient by a line. 104. As a IInd general principle, suggested by algebra, we shall next lay it down, that if or in words, and under a slightly varied form, that unequal vectors, divided by equal vectors, give unequal quotients. The importance of this very natural and obvious assumption will soon be seen in its applications. 105. As a IIIrd principle, which indeed may be considered to pervade the whole of mathematical language, and without adopting which we could not usefully speak, in any case, of EQUALITY as existing between any two geometrical quotients, we shall next assume that two such quotients can never be equal to the same third quotient, without being at the same time equal to each other: or in symbols, that if qq, and "=q, then q" = q'. = * It will be seen, however, at a later stage, that these two formulæ are permitted, and even required, in the development of the Quaternion System. + It is scarcely necessary to add, what is indeed included in this IIIrd principle, in virtue of the identity q=q, that if q'q, then q = q'; or in words, that we shall never admit that any two geometrical quotients, 9 and q', are equal to each other in one order, without at the same time admitting that they are equal, in the opposite order also. P |