CONTENTS. Cases are discussed; and general forms (called cyclic, rectangular, focal, bifocal, &c., from their chief geometrical uses) are assigned, for the vector and scalar functions op and Spøp: one useful pair of such (ryelie) forms being, with real and constant values of g, λ, μ, And finally it is shown (pp. 491, 492) that if fq be a linear and qua- This Chapter, like the one preceding it, may be omitted in a first perusal of the Volume, as has indeed been already remarked. 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 The analysis, however condensed, of this long Section (III. iii. 6), 531-630 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 a (see Fig. 77, p. 511). Deduction and interpretation of general expressions for the vector K of the centre K of the circle so defined. The reciprocal of the radius Examples: curvatures of helix, ellipse, hyperbola, logarithmic spiral; ARTICLE 391.-Centre determined by three scalar equations; Polar Axis, Polar Developable, 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 curve 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 7, 7',.. for Dsp, D2p, &c.; properties of a curve depending on the square (s2) of its arc, measured from a given point P; T = unit-tangent, r' vector of curvature, r1= Tr' = cur- vature (or first curvature, comp. Art. 397), v = TT' = binormal; the 531-535 535, 536 537 538, 539 539-541 541-549 549-553 * ▲ Table of initial Pages of all the Articles will be elsewhere given, which will tire planes, respectively perpendicular to 7, r', v, are the normal ARTICLE 397.-Properties depending on the cube (s3) of the are; Radius r (denoted here, for distinction, by a roman letter), and Vector rr, of Second Curvature; this radius r may be either positive or ne- gatire (whereas the radius r of first curvature is always treated as positive), and its reciprocal r1 may be thus expressed (pp. 563, 559), the independent variable being the are in (T′), while it is arbitrary in (T): but quaternions supply a vast variety of other expressions for this important scalar (see, for instance, the Table in pp. 574, 575). We = projection of vector (7′) of (simple or first) curvature, on radius (R) of osculating sphere: and if p and P denote the linear and angular elevations, of the centre (s) of this sphere above the osculating plane, this line A may be called the Rectifying Vector; and if H denote the inclination (considered first by Lancret), of this rectifying line (^) to Krown right cone with rectifying line for its axis, and with H for its semiangle, which osculates at P to the developable locus of tangents to the curve (or by p. 568 to the cone of parallels already mentioned); new right cone, with a new semiangle, C, connected with H by the 554-559 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, "Sur 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. to other points a 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, r1) be constant, the cylinder is right, and therefore the curve a helix, Pages. . . 559-578 ARTICLE 398.-Properties of a curve in space, depending on the fourth and fifth powers (84, 85) of its are (8), 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), 578-612 with expressions (p. 588) for the coefficients (or co-ordinates) Xs, Ys, ≈s, in terms of r', r', r", r, r', and s. If 85 be taken into account, it becomes necessary to add to the expression (W) the term, $57; with corresponding additions to the scalar coefficients in (W ́), introducing "" 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 r”, v", k”, X′, o', and p', R′, P', H', to serve in questions in which $5 is neglected, are assigned (in p. 579); 7'' v′, k', λ, o, and p, R, P, H, having been previously expressed (in Series 397); while r', v", k”", X", ", &c. enter into investigations which take account of $5: the arc s being treated as the independent variable in all these derivations. (c). One of the chief results of the present Series (398), is the introduction (p. 581, &c.) of a new auxiliary 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 s3 (including s1 and s2). In fact, while tan H is represented by the expressions (V'), whereof one is - tan P, tan J admits (with many transformations) of the following analogous expression (p. 581), CONTENTS. where E' depends by (b) on s1, while and P depend (397) on no (4). To give a more distinct geometrical meaning to this new angle elope of all these planes is determined, which envelope (for reasons φ (e). If the recent line & be measured from the given point P, in To= R1cosec J; (X) this last being an expression for the velocity of rotation of the plane xvij * In other words, the calculation of and P introduces no differentials |