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(9,4,=) considered as functions of a fourth (a), namely the co-ordinates of the centre of the sphere, regarded as varying with the radiaz, but which does not appear to have been either integrated or interpreted by that illustrious analyst. The general integral here fund presents itself at first in a quaternion form (p. 609), but is easily translated (p. 610) into the usual language of analysis. A less geseral integral is also assigned, and its geometrical signification exhibited, as answering to a case for which the singular line lately considered reduces itself to a singular point (pp. 610, 611).

(). Among the verifications (q) of this whole theory, it is shown (PP. 608, 609) that although, when the two branches (o) of the general curvilinear envelope of the circles of the system are real and distinct, each branch is a cusp-edge (or arête de rebroussement, as Monge perceived it to be), upon the superficial envelope of the spheres, yet in the case of fusion (p) this cuspidal character is lost (as was likewise seen by Monge): and that then a section of the surface, made by a normal plane to the singular line, has precisely the form (m), expressed by the equation (F). In short, the result is in many ways confirmed, by calculation and by geometry, that when the condition of coincidence (p) is satisfied, the Surface is, as in (n), at once the Envelope of the osculating Sphere and the Locus of the osculating Circle, to itself, into which by (g) the two branches (0)

that Singular Line on

of its general cusp-edge are.



xxiii Pages.

(t). Other applications of preceding formulæ might be given; for instance, the formula for " enables us to assign general exlates at K to the locus of the centre of the osculating circle, to a given pressions (p. 611) for the centre and radius of the circle, which oscue in space with an elementary verification, for the case of the plane evolute of the plane evolute of a plane curve. But it is time to conclude this long analysis, which however could scarcely have been much abridged, of the results of Series 398, and to pass to a more brief account of the investigations in the following Series. gauche curve of the third order (or degree), which has been above ARTICLE 399.-Additional general investigations, respecting that called an Osculating Twisted Cubic (398, (h)), to any proposed curve of double curvature; with applications to the case, where the given

curve is a helix, .

nodal side of the cubic cone 398, (h); one tangent plane to that cone (a). In general (p. 614), the tangent PT to the given curve is a (C), along that side, being the osculating plane (P) to the curve, and therefore touching also, along the same side, the osculating oblique cone (C) of the second order, to the cone of chords (397) from P; while the other tangent plane to the cubic cone (C3) crosses that first plane (P),

or the quadric


(C2), at an angle of which the trigonometric cotan

Compare the first Note to p. 609 of these Elements,


gent (r) is equal to half the differential of the radius (r) of second curvature, divided by the differential of the are (s). And the three common sides, PE, PE, PE", of these two cones, which remain when the tangent PT is excluded, and of which one at least must be real, are the parallels through the given point P to the three asymptotes (398, (i)) to the gauche curve sought; being also sides of three quadric cylinders, say (L2), (L′2), (L′′2), which contain those asymptotes as other sides (or generating lines): and of which each contains the twisted cubic sought, and is cut in it by the quadric cone (C2).

(b). On applying this First Method to the case of a given helix, it is found (p. 614) that the general cubic cone (C3) breaks up into the system of a new quadric cone, (C2), and a new plane (P'); which latter is the rectifying plane (396) of the helix, or the tangent plane at P to the right cylinder, whereon that given curve is traced. The two quadric cones, (C2) and (C2′), touch each other and the plane (P) along the tangent PT, and have no other real common side: whence two of the sought asymptotes, and two of the corresponding cylinders (a), are in this case imaginary, although they can still be used in calculation (pp. 614, 615, 617). But the plane (P) cuts the cone (C2), not only in the tangent PT, but also in a second real side PE, to which the real asymptote is parallel (a); and which is at the same time a side of a real quadric cylinder (L2), which has that asymptote for another side (p. 617), and contains the twisted cubic: this gauche curve being thus the curvilinear part (p. 615) of the intersection of the real cone (C2), with the real cylinder (L2).

(c). Transformations and verifications of this result; fractional expressions (p. 616), for the co-ordinates of the twisted cubic; expression (p. 615) for the deviation of the helix from that osculating curve, which deviation is directed inwards, and is of the sixth order: the least distance, between the tangent PT and the real asymptote, is a right line PB, which is cut internally (p. 617) by the axis of the right cylinder (b), in a point A such that PA is to AB as three to seven.

(d). The First Method (a), which had been established in the preceding Series (398), succeeds then for the case of the helix, with a facility which arises chiefly from the circumstance (6), that for this case the general cubic cone (C3) breaks up into two separate loci, whereof one is a plane (P'). But usually the foregoing method requires, as in 398, (h)), the solution of a cubic equation: an inconvenience which is completely avoided, by the employment of a Second General Method, as follows.

(e). This Second Method consists in taking, for a second locus of the gauche osculatrix sought, a certain Cubic Surface (S3), of which every point is the vertex of a quadric cone, having six-point con


* It is known that the locus of the vertex of a quadric cone, which passes through six given points of space, A, B, C, D, E, F, whereof no four are in one

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in which p and σ are corresponding vectors of involute and evolute;
together with a theorem of Prof. De Morgan (p. 622), respecting the
case when the involute is a spherical curve.

(b). An involute in space is generally the only real part (p. 624) of

the envelope of a certain variable sphere (comp. 398), which has its

centre on the evolute, while its radius R is the variable intercept be-

tween the two curves: but because we have here the relation (p. 622,

comp. p. 602),

R2 + o'2 = 0,

the circles of contact (398, (o)) reduce themselves each to a point (or
rather to a pair of imaginary right lines, intersecting in a real point),
and the preceding theory (398), of envelopes of spheres with one
varying parameter, undergoes important modifications in its results,
the conditions of the application being different. In particular, the
involute is indeed, as the equations (H1) express, an orthogonal tra-
jectory to the tangents of the evolute; but not to the osculating planes


of that curve, as the singular line (398, (q)) of the former envelope was, to those of the curve which was the locus of the centres of the spheres before considered, when a certain condition of coincidence (or of fusion, 398, (p)) was satisfied.


(c). Curvature of hodograph of evolute (p. 625); if r, P1, P2, .. and S, S1, S2, be corresponding points of involute and evolute, and if we draw right lines ST1, ST2, in the directions of S1P1, S2P2, . . and with a common length = SP, the spherical curve PT1T2 .. will have contact of the second order at P, with the involute PPIP2 (pp. 625, 626).


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ARTICLE 401.-Calculations abridged, by the treatment of quaternion differentials (which have hitherto been finite, comp. p. xi.) as infinitesimals;* new deductions of osculating plane, circle, and sphere, with the vector equation (392) of the circle; and of the first and second curvatures of a curve in space, .

SECTION 7.-On Surfaces of the Second Order; and on
Curvatures of Surfaces,

ARTICLE 402.-References to some equations of Surfaces, in earlier parts of the Volume,

ARTICLE 403.-Quaternion equations of the Sphere (p2 = 1, &c.), In some of these equations, the notation N for norm is employed (comp. the Section II. i. 6).

ARTICLE 404.-Quaternion equations of the Ellipsoid,

One of the simplest of these forms is (pp. 307, 635) the equation,
T (ip + pk) = x2 – 12,

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Although, for the sake of brevity, and even of clearness, some phrases have been used in the foregoing analysis of the Series 398 and 399, such as four-side or five-side contact between cones, and five-point or six-point contact between curves, or between a curve and a surface, which are borrowed from the doctrine of consecutive points and lines, and therefore from that of infinitesimals; with a few other expressions of modern geometry, such as the plane at infinity, &c.; yet the reasonings in the text of these Elements have all been rigorously reduced, so far, or are all obviously reducible, to the fundamental conception of Limits : compare the definitions of the osculating circle and sphere, assigned in Articles 389, 395. The object of Art. 401 is to make it visible how, without abandoning such ultimate reference to limits, it is possible to abridge calculation, in several cases, by treating (at this stage) the differential symbols, dp, d3p, &c., as if they represented infinitely small differences, Ap, A2p, &c.; without taking the trouble to write these latter symbols first, as denoting finite differences, in the rigorous statement of a problem, of which statement it is not always easy to assign the proper form, for the case of points, &c., at finite distances: and then having the additional trouble of reducing the complex expressions so found to simpler forms, in which differentials shall finally appear. In short, it is shown that in Quaternions, as in other parts of Analysis, the rigour of limits can be combined with the facility of infinitesimals.


in which and are real and constant vectors, in the directions of the cyclic normals. This form (I1) is intimately connected with, and indeed served to suggest, that Construction of the Ellipsoid (II. i. 13), by means of a Diacentric Sphere and a Point (p. 227, comp. Fig. 53, p226), which was among the earliest geometrical results of the Quaterations. The three semiaxes, a, b, c, are expressed (comp. p. 230) in terms of 4, as follows:

xxvii Pages.

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ARTICLE 405.-General Central Surface of the Second Order (or central quadric), Spøp = fp = 1,


ARTICLE 406.-General Cone of the Second Order (or quadric cone), Sppp = fp=0,


ARTICLE 407.-Bifocal Form of the equation of a central but nonconical surface of the second order: with some quaternion formulæ, relating to Confocal Surfaces,

(a). The bifocal form here adopted (comp. the Section III. ii. 6) is the equation,

in which,

Cfp=(Sap)2-2eSapSa'p+ (Sa'p)2 + (1-e) p2 = C, (J1)
C=(-1) (e + Saa') l2.


a, a' are two (real) focal unit-lines, common to the whole system of confocals; the (real and positive) scalar 7 is also constant for that system: but the scalar e varies, in passing from surface to surface, and may be regarded as a parameter, of which the value serves to distinguish one confocal, say (e), from another (pp. 643, 644).

(6). The squares (p. 644) of the three scalar semiaxes (real or imaginary), arranged in algebraically descending order, are,

a2 = (e + 1)2, b2= (e + Saa') 12, c2 = (e− 1) 12;


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and the three vector semiaxes corresponding are,

aU (a + a'), bUVaa', cU (a-a').


(c). Rectangular, unifocal, and cyclic forms (pp. 644, 648, 650), of the scalar function fp, to each of which corresponds a form of the vector function op; deduction, by a new analysis, of several known theorems* (pp. 644, 645, 648, 652, 653) respecting confocal surfaces,

For example, it is proved by quaternions (pp. 652, 653), that the focal lines of the focal cone, which has any proposed point P for vertex, and rests on the focal hyperbola, are generating lines of the single-sheeted hyperboloid (of the given confocal system), which passes through that point: and an extension of this result, to the focal lines of any cone circumscribed to a confocal, is deduced by a similar analysis, in a subsequent Series (408, p. 656). But such known theorems respecting confocals can only be alluded to, in these Contents.

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