SECTION 7.-On Differentials of Vectors. 100. The equation (99, I.), F = Q(t), in which pop is generally the vector of a point P of a curve in space, PQ..., gives evidently, for the vector oq of another point a of the same curve, an expression of the form P+Ap=(t+At); so that the chord PQ, regarded as being itself a vector, comes thus to be represented (4) by the finite difference, PQ = Ap=A¢(t) = ¢ (t + At) − q (t). Suppose now that the other finite difference, At, is the nth part of a new Р V W Fig. 32. T scalar, u; and that the chord Ap, or PQ, is in like manner (comp. Fig. 32), the nth part of a new vector, o, or PR; so that we may write, nAt=u, and n▲p=n. PQ = σn= Then, if we treat the two scalars, t and u, as constant, but the number n as variable (the form of the vector-function p, and the origin o, being given), the vector p and the point p will be fixed: but the two points Q and R, the two differences At and Ap, and the multiple vector nap, or σ, will (in general) vary together. And if this number n be indefinitely increased, or made to tend to infinity, then each of the two differences At, Ap will in general tend to zero; such being the common limit, of n ̄1u, and of ¢ (t+n ́1u) − 4 (t); so that the variable point a of the curve will tend to coincide with the fixed point P. But although the chord PQ will thus be indefinitely shortened, its n1 multiple, PR or o, will tend (generally) to a finite limit,* depending on the supposed continuity of the function (t); namely, to a certain definite vector, PT, or σ, or (say) т, which vector PT will evidently be (in general) tangential to the curve: or, in other words, the variable point R will tend to a fixed position T, on the tangent to that curve at P. We shall thus have a limiting equation, of the form T=PT=lim. PR=σ = lim. n^p(t), if n▲t=u; t and u being, as above, two given and (generally) finite scalars. And Compare Newton's Principia. CHAP. III.] DIFFERENTIALS OF VECTORS. 99 if we then agree to call the second of these two given scalars the differential of the first, and to denote it by the symbol dt, we shall define that the vector-limit, 7 or σ, is the (corresponding) differential of the vector p, and shall denote it by the corresponding symbol, dp; so as to have, under the supposed conditions, u=dt, and 7=dp. Or, eliminating the two symbols u and 7, and not necessarily supposing that P is a point of a curve, we may express our Definition* of the Differential of a Vector p, considered as a Function of a Scalar t, by the following General Formula: in which t and dt are two arbitrary and independent scalars, both generally finite; and dp is, in general, a new and finite vector, depending on those TWO scalars, according to a law expressed by the formula, and derived from that given law, whereby the old or former vector, p or (t), depends upon the single scalar, t. (1.) As an example, let the given vector-function have the form, น p=4(t)= ft2a, where a is a given vector. Then, making At=~, where u is any given scalar, and n is a variable whole number, we have a Ap = A¢(t) = :{( น t + n On=nAp: = au t+ and finally, writing dt and dp for u and σ, 2 -12 น 2n t2a ; σ = atu; a(ra) = dp=do(t)= d = atdt. (2.) In general, let $(t) = af (t), where a is still a given or constant vector, and f(t) denotes a scalar function of the scalar variable, t. Then because a is a common factor within the brackets {} of the recent general formula (100) for dp, we may write, provided that we now define that the differential of a scalar function, f(t), is a new scalar function of two independent scalars, t and dt, determined by the precisely similar formula: which can easily be proved to agree, in all its consequences, with the usual rules for differentiating functions of one variable. (3.) For example, if we write dt = nh, where h is a new variable scalar, namely, the nth part of the given and (generally) finite differential, dt, we shall thus have the equation, df (t) dt = lim. f(t + h) − f(t), h in which the first member is here considered as the actual quotient of two finite scalars, df (t): dt, and not merely as a differential coefficient. We may, however, as usual, consider this quotient, from the expression of which the differential dt disappears, as a derived function of the former variable, t; and may denote it, as such, by either of the two usual symbols, f'(t) and Dif(t). (4.) In like manner we may write, for the derivative of a vector-function,* $(t), the formula: dp dp (1) p' ='(t) = Dtp = Dip (t) = these two last forms denoting that actual and finite vector, p' or '(t), which is obtained, or derived, by dividing (comp. 16) the not less actual (or finite) vector, dp or do (t), by the finite scalar, dt. And if again we denote the nth part of this last scalar by h, we shall thus have the equally general formula : exactly as if the vector-function, p or p, were a scalar function, f. (5.) The particular value, dt = 1, gives thus dp = p'; so that the derived vector p' is (with our definitions) a particular but important case of the differential of a vector. In applications to mechanics, if t denote the time, and if the term P of the variable vector p be considered as a moving point, this derived vector p' may be called the Vector of Velocity: because its length represents the amount, and its direction is the direction of the velocity. And if, by setting off vectors ov=p' (comp. again Fig. 32) from one origin, to represent thus the velocities of a point moving in space according to any supposed law, expressed by the equation p = $(t), we construct a new curve vw.. of which the corresponding equation may be written as p′ = '(t), then this new curve has been defined to be the HODOGRAPH,† as the old curve PQ.. may be called the orbit of the motion, or of the moving point. In the theory of Differentials of Functions of Quaternions, a definition of the differential do (g) will be proposed, which is expressed by an equation of precisely the same form as those above assigned, for df (t), and for dø (t); but it will be found that, for quaternions, the quotient do(q): dq is not generally independent of dq ; and consequently that it cannot properly be called a derived function, such as ø′(q), of the quaternion q alone. (Compare again the Note to page 39.) The subject of the Hodograph will be resumed, at a subsequent stage of this work. In fact, it almost requires the assistance of Quaternions, to connect it, in what appears to be the best mode, with Newton's Law of Gravitation. CHAP. III.] DIFFERENTIALS OF VECTORS. 99 Or, eliminating the two symbols u and 7, and not necessarily supposing that P is a point of a curve, we may express our Definition* of the Differential of a Vector p, considered as a Function of a Scalar t, by the following General Formula : in which t and dt are two arbitrary and independent scalars, both generally finite; and dp is, in general, a new and finite vector, depending on those TWO scalars, according to a law expressed by the formula, and derived from that given law, whereby the old or former vector, p or (t), depends upon the single scalar, t. (1.) As an example, let the given vector-function have the form, p=4(t)=t2a, where a is a given vector. Then, making At= where u is any given scalar, and n is a variable whole number, we have (2.) In general, let (t) = af (t), where a is still a given or constant vector, and f(t) denotes a scalar function of the scalar variable, t. Then because a is a common factor within the brackets {} of the recent general formula (100) for dp, we may write, dp=do (t)=d. af (t) = adf (t) ; which can easily be proved to agree, in all its consequences, with the usual rules for differentiating functions of one variable. (3.) For example, if we write dt = nh, where h is a new variable scalar, namely, the nth part of the given and (generally) finite differential, dt, we shall thus bave the equation, in which the first member is here considered as the actual quotient of two finite scalars, df(1): dt, and not merely as a differential coefficient. We may, however, as usual, consider this quotient, from the expression of which the differential dt disappears, as a derived function of the former variable, t; and may denote it, as such, by either of the two usual symbols, f'(t) and Dif(t). (4.) In like manner we may write, for the derivative of a vector-function,* $(t), the formula: do do(t) p' = 4′(t) = D1p = Dip (t) = these two last forms denoting that actual and finite vector, p' or '(t), which is obtained, or derived, by dividing (comp. 16) the not less actual (or finite) vector, dp or do (t), by the finite scalar, dt. And if again we denote the nth part of this last scalar by h, we shall thus have the equally general formula: exactly as if the vector-function, p or p, were a scalar function, f. vector. (5.) The particular value, dt = 1, gives thus dp = p'; so that the derived vector` p' is (with our definitions) a particular but important case of the differential of a In applications to mechanics, if t denote the time, and if the term P of the variable vector p be considered as a moving point, this derived vector p' may be called the Vector of Velocity: because its length represents the amount, and its direction is the direction of the velocity. And if, by setting off vectors ov = p′ (comp. again Fig. 32) from one origin, to represent thus the velocities of a point moving in space according to any supposed law, expressed by the equation p = $(t), we construct a new curve vw.. of which the corresponding equation may be written as p''(t), then this new curve has been defined to be the HODOGRAPH,† as the old curve PQ.. may be called the orbit of the motion, or of the moving point. * In the theory of Differentials of Functions of Quaternions, a definition of the differential do (g) will be proposed, which is expressed by an equation of precisely the same form as those above assigned, for dƒ(t), and for dø (t); but it will be found that, for quaternions, the quotient do (q): dq is not generally independent of dq; and consequently that it cannot properly be called a derived function, such as p ́(g), of the quaternion q alone. (Compare again the Note to page 39.) The subject of the Hodograph will be resumed, at a subsequent stage of this work. In fact, it almost requires the assistance of Quaternions, to connect it, in what appears to be the best mode, with Newton's Law of Gravitation. |