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And finally it is shown (pp. 491, 492) that if fq be a linear and qua-
ternion function of a quaternion, q, then the Symbol of Operation, f,
satisfies a certain Symbolic and Biquadratic Equation, analogous to the
cubic equation in ø, and capable of similar applications.

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The analysis, however condensed, of this long Section (III. iii. 6),
cannot conveniently be performed otherwise than under the heads of
the respective Articles (389-401) which compose it: each Article


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Examples: curvatures of helix, ellipse, hyperbola, logarithmic spiral;
locus of centres of curvature of helix, plane evolute of plane ellipse,
ARTICLE 390.-Abridged general calculations; return from (S')
to (S), .

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
plane curve to which it osculates; example, hyperbola,

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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
the curve at a given point P, and cuts the same curve at a near point
a (comp. Art. 389). The centre s, of the sphere so found, is (as usual)
the point in which the polar axis (Art. 391) touches the cusp-edge of
the polar developable. Other general construction for the same centre
(p. 551, comp. p. 573). General expressions for the vector, σ = 08,
and for the radius, R = SP; R1 is the spherical curvature (comp. Art.
397). Condition of Sphericity (S= 1), and Coefficient of Non-sphericity
(S-1), for a curve in space. When this last coefficient is positive
(as it is for the helix), the curve lies outside the sphere, at least in the
neighbourhood of the point of osculation,

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


535, 536


538, 539




* ▲ Table of initial Pages of all the Articles will be elsewhere given, which will
much facilitate reference.

tire planes, respectively perpendicular to 7, r', v, are the normal
plane, the rectifying plane, and the osculating plane; general theory
of manant lines and planes, vector of rotation, axis of displacement, oscu-
lating screw surface; condition of developability of surface of emanants,

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),

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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,


. . 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),


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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),

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where E' depends by (b) on s1, while and P depend (397) on no
higher power than 83.

(4). To give a more distinct geometrical meaning to this new angle
J, than can be easily gathered from such a formula as (X), respecting
which it may be observed, in passing, that J is in general more simply
defined by expressions for its cotangent (pp. 581, 588), than for its
tangent, we are to conceive that, at each point P of any proposed
curve of double curvature, there is drawn a tangent plane to the sphere,
which occulates (395) to the curve at that point; and that then the

elope of all these planes is determined, which envelope (for reasons
afterwards more fully explained) is called here (p. 581) the "Cir-
cunscribed Developable being a surface analogous to the "Rectifying
Developable" of Lancret, but belonging (c) to a higher order of ques-
tions. And then, as the known angle H denotes (397) the inclina-
ties, suitably measured, of the rectifying line (A), which is a genera-
trix of the rectifying developable, to the tangent (7) to the curve; so
the new angle J represents the inclination of a generating line (p), of
what has just been called the circumscribed developable, to the same
tangent (7), measured likewise in a defined direction (p. 581), but
in the tangent plane to the sphere. It may be noted as another ana-
logy (p. 582), that while H is a right angle for a plane curve, 80 J
is right when the curve is spherical. For the helix (p. 585), the an-
gles H and J are equal; and the rectifying and circumscribed deve-
lopables coincide, with each other and with the right cylinder, on
which the helix is a geodetic line.


(e). If the recent line & be measured from the given point P, in
a suitable direction (as contrasted with the opposite), and with a suit-
able length, it becomes what may be called (comp. 396) the Vector of
Retation of the Tangent Plane (d) to the Osculating Sphere; and then
it satisfies, among others, the equations (pp. 579, 581, comp. (V)),
Ø = V

To= R1cosec J;


this last being an expression for the velocity of rotation of the plane
just mentioned, or of its normal, namely the spherical radius R, if the
gieen curve be conceived to be described by a point moving with a con-


* In other words, the calculation of and P introduces no differentials
higher than the third order; but that of R' requires the fourth order of differen-
tials. In the language of modern geometry, the former can be determined by
the consideration of four consecutive points of the curve, or by that of two consecu-
tive osculating circles; but the latter requires the consideration of two consecu-
tive osculating spheres, and therefore of five consecutive points of the curve (sup-
posed to be one of double curvature). Other investigations, in the present and
immediately following Series (398, 399), especially those connected with what
we shall shortly call the Osculating Twisted Cubic, will be found to involve the
consideration of six consecutive points of a curve.

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