CHAP. III.] CONNEXION WITH TAYLOR'S SERIES. 499 (3.) Conceiving next that w= OR = the vector of some new and arbitrary point B, we may let fall a perpendicular QM on the line PR, and so decompose the chord PQ into the two rectangular lines, PM and MQ; which, when divided by the same chord, give rigorously the two (generally) quaternion quotients, the variable t thus disappearing through the division, except so far as it enters into u, which tends as above to 1. (4.) Passing then to the limits, we have these other rigorous equations, by comparing which with 369, III. and IV., we see that those two equations represent respectively, as before stated, the tangent and the normal plane to the proposed curve at P; because, if Vp'(w - p) = 0, the chord PQ tends, by V. or VI., to coincide, both in length and in direction, with its projection PM on the line PR; while, on the other hand, if Sp’(w − p) = 0, that projection tends to ranish, even as compared with the chord PQ; which chord tends now to coincide with its other projection MQ, or with the perpendicular to the line PR, erected so as to reach the point q: whence PR must, in this last case, be a normal to the curve at P. (5.) We may also investigate an equation for the normal plane, by considering it as the limiting position of the plane which perpendicularly bisects the chord. If R be supposed to be a point of this last plane, then, with the recent notations, the vector w= OR must satisfy the condition, от VII... T(wpt) = T(w− po), or VIII... (w-p- utp')2 = (w − p)2, in which it may be noted that up' is a vector (in the direction of the chord, PQ), although u itself is generally a quaternion, as before: such then is the equation of the bisecting plane, with w for its variable vector, and its limit, X... Sp'(w-p) = 0, as before. (6.) The last process may also be presented under the form, XI...0= lim. t1{T(w− pt) – T(w− po}} = D¿T(w−Pt), when t=0 ; and thus the equation 369, VIII. may be obtained anew. (7.) Geometrically, if we set off on EQ a portion RS equal in length to RP, as in the annexed Figure 76, we shall have the limiting equation, XII... +SQ: PQ = (RQ- RP): PQ = (ultimately) — cos RPT ; which agrees with 369, IX. R Fig. 76. S T (8.) If then the point R be taken out of the normal plane at P, this limit of the quotient, RQ – RP divided by PQ, has a finite value, positive or negative; and if the chord PQ be called small of the first order, the difference of distances of its extremities from R may then be said to be small of the same (first) order. But if R be taken in the normal plane at P (and not coincident with that point P itself), this difference of dis tances may then be said to be small, of an order higher than the first: which answers to the evanescence of the first differential of the tensor, T(w− p) in XI., or T(v-p) in 369, VIII'. 371. A curve may occasionally be represented in quaternions, by an equation which is not of the form, 369, I., although it must always be conceived capable of reduction to that form: for instance, this new equation, I... Vap.V pa' = (Vaa'), with TVaa'>0, is not immediately of the form p = pt, but it is reducible to that form as follows, An equation such as I. may therefore have its differential or its derivative taken, with respect to the scalar variable t on which is thus conceived to depend, even if the exact law of such dependence be unknown: and dp, or p', may then be changed to the tangential vector wp to which it is parallel, in order to form an equation of the tangent, or a condition which the vector w of a point on that sought line must satisfy. (1.) To pass from I. to II., we may first operate with the sign V, which gives, III. pSaa'p=0, or simply, III'... Saa'p = 0; whence, t and t being scalars, we may write, IV. p = ta + t'a', Vap=t'Vaa', Vpa' = tVaa', tt' = 1, and the required reduction is effected while the return from II. to I., or the elimination of the scalar t, is an even easier operation. (2.) Under the form II., it is at once seen that p is the vector of a plane hyperbola, with the origin for centre, and the lines a, a' for asymptotes; and accordingly all the properties of such a curve may be deduced from the expression II., by the rules of the present Calculus. (3.) For example, since the derivative of that expression is, V... p'a-t2a', the tangent may (comp. 369, VII.) have its equation thus written: VI... w = · (t + x) a + t-2 (t − x) a'; it intersects therefore the lines a, a' in the points of which the vectors are 2ta, 2t-'a'; so that (as is well known) the intercept, upon the tangent, between the asymptotes, is bisected at the point of contact: and the intercepted area is constant, because V(ta.t'a') = Vaa', &c. (4.) But we may also operate immediately, as above remarked, on the form I.; and thus arrive (by substitution of wp for dp, &c.) at the equation of conjugation, VII... Vaw Vpa' + Vap Vwa' = 2 (Vaa')3, CHAP. III.] NORMALS AND TANGENT PLANES TO SURFACES. 501 which expresses (comp. 215, (13.), &c.) that if p=OP, and w= OR, as before, then either R is on the tangent to the curve, at the point P, or at least each of these two points is situated on the polar of the other, with respect to the same hyperbola. (5.) Again, it is frequently convenient to consider a curve as the intersection of two surfaces; and, in connexion with this conception, to represent it by a system of two scalar equations, not explicitly involving any scalar variable: in which case, both equations are to be differentiated, or derivated, with reference to such a variable understood, and dp or p' deduced, or replaced by w p as before. (6.) Thus we may substitute, for the equation I., the system of the two following (whereof the first had occurred as III'.); VIII... Saa'p=0, p2Saa' - SapSa'p = (Vaa')2; and the derivated equations corresponding are, IX... Saa'p' = 0, 2Saa'Spp'-Sap'Sa'p - SapSa'p' = 0; or, with the substitution of wp for p', &c., X.. Saa'w=0, 2Saa'Spw - SawSa’p – SapSa’'w = 2(Vaa')2 ; the last of which might also have been deduced from VII., by operating with S. (7.) And it may be remarked that the two equations VIII. represent respectively in general a plane and an hyperboloid, of which the intersection (5.) is the hyperbola I. or II.; or a plane and an hyperbolic cylinder, if Saa' = 0. SECTION 3.-On Normals and Tangent Planes to Surfaces. 372. It was early shown (100, (9.)), that when a curved surface is represented by an equation of the form, I... p = p(x, y), in which is a functional sign, and x, y are two independent and scalar variables, then either the two partial differentials, or the two partial derivatives, of the first order, represent two tangential vectors, or at least vectors parallel to two tangents to the surface, drawn at the extremity or term P of p; so that the plane of these two differential vectors, or of lines parallel to them, is (or is parallel to) the tangent plane at that point: and the principle has been since exemplified, in 100, (11.) and (12.), and in the sub-articles to 345, &c. It follows that any vector v, which is perpendicular to both of two such non-parallel differentials, or derivatives, must (comp. 345, (11.)) be a normal vector at P, or at least one having the direction of the normal to the surface at that point; so that each of the two vectors, IV... V.d.pd,p, if actual, represents such a normal. V... V. D, DP, (1.) As an additional example, let us take the case of the ruled paraboloid, on which a given gauche quadrilateral ABCD is superscribed. The expression for the vector p of a variable point P of this surface, considered as a function of two independent and scalar variables, æ and y, may be thus written (comp. 99, (9.)): VI... p = xya + (1 − x)yß + (1 − x) (1 − y) y + x(1 − y)d; where the supposition y = 1 places the point P on the line AB; x=0 places it on BC; y= 0, on CD; and x= = 1, on DA. (2.) We have here, by partial derivations, VII... Dxp=y(a− ẞ) + (1 − y) (d − y); Dyp=x(a−d) + (1 − x) (B − y) ; these then represent the directions of two distinct tangents to the paraboloid VI., at what may be called the point (x, y); whence it is easy to deduce the tangent plane and the normal at that point, by constructions on which we cannot here delay, except to remark that if (comp. Fig. 31, Art. 98) we draw two right lines, os and KT, through P, so as to cut the sides AB, BC, CD, DA of the quadrilateral in points Q, R, S, T, we shall have by VI. the vectors, so that these two tangents are simply the two generating lines of the surface, which pass through the proposed point P. (3.) For example, at the point (1, 1), or A, the tangents thus found are the sides BA, DA, and the tangent plane is that of the angle BAD, as indeed is evident from geometry. (4.) Again, the equation of the screw surface (comp. 314, XVI.), X... p = cxa+yaß, with Ta = 1, and Saß = 0, gives the two tangents, π XI... D2p = ca+ya+1ß, Dyp=a3ß, whereof the latter is perpendicular to the former, and to the axis a of the cylinder; so that the corresponding normal to the surface X. at the point (x, y) is represented by the product, XII. . . v = D ̧p.Dyp = ca2+1ß + Tyß2a. 373. Whenever a variable vector p is thus expressed or even conceived to be expressed, as a function of two scalar variables, x and y (or s and t, &c.), if we assume any three diplanar vectors, such as a, ß, y (or i, k, λ, &c.), the three scalar expressions, Sap, Spp, Syp (or Sip, Skp, Sap, &c.) will then be functions of the same two scalar variables; and will therefore be connected with each other by some one scalar equation, of the form, or briefly, I... F(Sap, Sẞßp, Syp) = 0, CHAP. III.] CONNEXION WITH QUATERNION DIFFERENTIALS. 503 II... fp = C; where C' is a scalar constant, introduced (instead of zero) for greater generality of expression; and F, ƒ are used as functional but scalar signs. If then (comp. 361, XIV.) we express the first differential of this scalar function fp under the form, III... dfp = 2Svde, in which is a certain derived vector, and is here considered as being (at least implicitly) a vector function (like p) of the two scalar variables above mentioned, we shall have the two equations, IV... Svd,p=0, Svd,p=0, or these two other and corresponding ones, ρ from which it follows (by 372) that has the direction of the normal to the surface I. or II., at the point P in which the vector terminates. Hence the equation of that normal (with w for its variable vector) may, under these conditions, be thus written: VI... Vv(we)=0; and the corresponding equation of the tangent plane at the same point P is, VII... Sv(w-p) = 0. (1.) For example, if we take the expression 308, XVIII., or 345, XII., namely VIII. . . p = rktjskj-sk ̄t, in which kj ̄s =jsk, &c., treating the scalar r as constant, but s and t as variable, we have then (comp. 345, XIV.), the equations, a denoting any unit-vector, IX... Sip=rS. a2tS. a2+1, Sjp=rS. a2-18.a2+1, Skp=rS. a2s+2; between which s and t can be eliminated, by simply adding their squares, because (at) + (at-1)2 = 1, by 315, V., if Ta = 1. In this manner then we arrive at equations of the forms I. and II., namely (comp. 357, VII., and 308, (10.) and (13.)), X... (Sip)2 + (Sjp)2 + (Skp)2 — r2 = 0, and XI. . . fp = p2 = — r2 = const., or XI'... Tp=r; which last results had indeed been otherwise obtained before. (2.) With this form XI. of fp, we have the differential expression of the first order, XII... dfp = 2Svdp = 2Spdp, whence XIII... v=p ; and if we still conceive that p is, as above, some vector function of two scalar variables, 8 and t, although the particular law VIII. of its dependence on them may now be supposed to be unknown (or to be forgotten), we may write also, = XIV... dfp = Svdp Spdp = Sp (d, + di) p = SpDsp. ds + SpDip. dt; if then the function fp have (as above) a value, - r2, which is constant, or is inde |