CONTENTS. xiii Spop= Cp?+ C°, in which C and C are constants. (Q) laves are discussed ; and general forms (called cyclic, rectangular, feal, bifocal, &c., from their chief geometrical uses) are assigned, Iz the vector and scalar functions op and Spøp: one useful pair of ach (ydie) fortos being, with real and constant values of g, n, p, And finally it is shown (pp. 491, 492) that if fq be a linear and qua- 08 SOME ADDITIONAL APPLICATIONS OF QUATERNIONS, WITH This Chapter, like the one preceding it, may be omitted in a first SECTION 1.-Remarks Introductory to this concluding SECTION 2.-On Tangents and Normal Planes to Curves in SECTION 3.-On Normals and Tangent Planes to Surfaces, 501-510 SECTION 4.—On Osculating Planes, and Absolute Normals, to Curves of Double Curvature, Section 5.-On Geodetic Lines, and Families of Surfaces, 515-531 In these Sections, dp usually denotes a tangent to a curve, and v a normal to a surface. Some of the theorems or constructions may Perhaps be new; for instance, those connected with the cone of paral- lels (pp. 498, 513, &c.) to the tangents to a curve of double curvature ; and possibly the theorem (p. 525), respecting reciprocal curves in space : at least, the deductions here given of these results may serve as exemplifications of the Calculus employed. In treating of Families of Surfaces by quaternions, a sort of analogue (pp. 529, 530) to the for- mation and integration of Partial Differential Equations presents itself; as indeed it had done, on a similar occasion, in the Lectures SECTION 6.-On Osculating Circles and Spheres, to Curves in Space; with some connected Constructions, . . 531-630 The analysis, however condensed, of this long Section (III. iii. 6), being followed by several subarticles, which form with it a sort of ARTICLE 389.–Osculating Circle defined, as the limit of a circle, which touches a given curve (plane or of double curvature) at a given point P, and cuts the curve at a near point Q (see Fig. 77, p. 511). Deduction and interpretation of general expressions for the vector a of the centre k of the circle so defined. The reciprocal of the radius KP being called the vector of curvature, we have generally, and if the arc (s) of the curve be made the independent variable, then dap 531-535 535, 536 537 539-541 541-549 Vector of Curvature = p" =D,20 = ds2 ARTICLE 390.--Abridged general calculations; return from (8') ARTICLE 391.–Centre determined by three scalar equations ; Polar Axis, Polar Derelopable, ARTICLE 392.- Vector Equation of osculating circle, . Article 393.—Intersection (or intersections) of a circle with a ARTICLE 394.- Intersection (or intersections) of a spherical curre with a small circle osculating thereto; example, spherical conic ; con- structions for the spherical centre (or pole) of the circle osculating to such a curve, and for the point of intersection above mentioned, ARTICLE 395.—Osculating Sphere, to a curve of double curvature, defined as the limit of a sphere, which contains the osculating circle to ARTICLE 396.—Notations T, r', .. for Dso, D, p, &c.; properties 549-553 * A Table of initial Pages of all the Articles will be elsewhere given, which will CONTENTS. XV 554-559 three plants, respectively perpendicular to 7, r', v, are the normal ARTICLE 397.—Properties depending on the cube (83) of the are; Radius r (denoted here, for distinction, by a roman letter), and Vector I'7, of Secind Currature; this radius r may be either positive or ne- gatise (whereas the radius r of first curvature is always treated as Sound Currature*=r-l=S. (T), or, r-!=s (T) " Vector of Spherical Currature = spl= (2-0)-1 = &c., (U) = projection of vector (') of (simple or first) curvature, on radius (R) of osculating sphere : and if p and P denote the linear and angular elstations, of the centre (s) of this sphere above the osculating plane, Again (pp. 560, 561), if we write (comp. Art. 396), = Vector of Second Curvature plus Binormal, (V) this line 1 may be called the Rectifying Vector; and if H denote the inclination (considered first by Lancret), of this rectifying line (1) to the tangent () to the curve, then Kronen right cone with rectifying line for its axis, and with H for its Emiangle, which osculates at to the developable locus of tangents to the curve (or by p. 568 to the cone of parallels already mentioned); Het right cone, with a new semiangle, C, connected with H by the * In this article, or Series, 397, and indeed also in 396 and 398, several re- ferences are given to a very interesting Memoir by M. de Saint-Venant les lignes courbes non planes" in which, however, that able writer objects to such known phrases as second curvature, torsion, &c., and proposes in their stead a new name " cambrure,” which it has not been thought necessary here to adopt. Pages. 559-578 578-612 to other points of the given curve. Other osculating cones, cylinders, helix, and parabola ; this last being (pp. 562, 566) the parabola which osculates to the projection of the curve, on its own osculating plane. Deviation of curve, at any near point q, from the osculating circle at P, decomposed (p. 566) into two rectangular deviations, from osculating helix and parabola. Additional formulæ (p. 576), for the general theory of emanants (Art. 396); case of normally emanant lines, or of tangentially emanant planes. General auxiliary spherical curve (pp. 576-578, comp. p. 515); new proof of the second expression (V') for tan H, and of the theorem that if this ratio of curvatures be constant, the proposed curve is a geodetic on a cylinder : new proof that if each curvature (r 1, r1) be constant, the cylinder is right, and therefore the curve a helix, ARTICLE 398.- Properties of a curve in space, depending on the fourth and fifth powers (84, 85) of its arc (*), This Series 398 is so much longer than any other in the Volume, and is supposed to contain so much original matter, that it seems necessary here to subdivide the analysis under several separate heads, lettered as (a), (b), (c), &c. (a). Neglecting s5, we may write (p. 578, comp. Art. 396), OP,=po=+87+ }$2,' +1537" + 4847"; (W) or (comp. p. 587), Ps= p +257 + ysrt' + Esrv, (W) with expressions (p. 588) for the coefficients (or co-ordinates) Xs, Yo, ms, in terms of r', .', g", r, r', and s. If 85 be taken into account, it becomes necessary to add to the expression (W) the term, thos5p!!; with corresponding additions to the scalar coefficients in (W'), introducing r" and r": the laws for forming which additional terms, and for extending them to higher powers of the arc, are assigned in a subsequent Series (399, pp. 612, 617). (6). Analogous expressions for 7", v", k”, X', o', and p', R', P, H', to serve in questions in which s5 is neglected, are assigned (in p. 579); pre v', k', 1, o, and P, R, P, H, having been previously expressed (in Series 397); while 7", v", «", 1", o“, &c. enter into investigations which take account of $5: the arc 8 being treated as the independent variable in all these derivations. (©). One of the chief results of the present Series (398), is the introduction (p. 581, &c.) of a new auriliary angle, J, analogous in several respects to the known angle H (397), but belonging to a higher order of theorems, respecting curves in space : because the new angle J depends on the fourth (and lower) powers of the arc s, while Lancret's angle H depends only on 93 (including 81 and s2). In fact, while tan H is represented by the expressions (V), whereof one is g'-1 tan P, tan J admits (with many transformations) of the following analogous expression (p. 581), tan J= R-Itan P; CONTENTS. xvii a a There I depends* by (6) on s', while r' and P depend (397) on no (). To give a more distinct geometrical meaning to this new angle (). If the recent line o be measured from the given point p, in this last being an expression for the velocity of rotation of the plane In other words, the calculation of you and P introduces no differentials d |