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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,

pgp + Vapu, Spop=gp2 + Spup.


And finally it is shown (pp. 491, 492) that if fq be a linear and qua-
tersion function of a quaternion, 9, then the Symbol of Operation, f,
satisfies a certain Symbolic and Biquadratic Equation, analogous to the
salic equation in R, and capable of similar applications.

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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),
cannot conveniently be performed otherwise than under the heads of
• the respective Articles (389-401) which compose it: each Article



535, 536

538, 539



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Vector of Curvature = p" =D,20 =


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 (8')

to (S), · ·

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
plane curve to which it osculates; example, hyperbola,

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
the curve at a given point P, and cuts the same curve at a near point
Q (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, o = os,
and for the radius, R = SP; R-' 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 T, r', .. for Dso, D, p, &c.; properties
of a curve depending on the square (s?) of its arc, measured from a
given point P; r= unit-tangent, 1' = vector of curvature, 4-1= Tr' = cur-
vature (or first curvature, comp. Art. 397), v = 7T' = binormal ; the


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




three plants, respectively perpendicular to 7, r', v, are the normal
pisut, the rectifying plans, and the osculating plane; general theory
donant lines and planes, vector of rotation, axis of displacement, oscu-
liting #Trw surface ; condition of developability of surface of emanants,

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
paritice), and its reciprocal r' may be thus expressed (pp. 563, 559),


Sound Currature*=r-l=S. (T), or, r-!=s (T)

the independent variable being the arc 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
bare also (by p. 560, comp. Arts. 389, 395, 396),

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,

then (ty same page 560),

p=r tan P= R sin P=r'r=rDr.


Again (pp. 560, 561), if we write (comp. Art. 396),

1= v=r's+r+'=

= 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

tan H =ql-1 tan P=ylr.


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


tan C= -tan H,



whieh osculates to the cone of chords, drawn from the given point P




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;

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There I depends* by (6) on s', while r' and P depend (397) on no
higher power than 13.

(). To give a more distinct geometrical meaning to this new angle
, than can be easily gathered from such a formula as (X), respecting
všich it may be observed, in passing, that J is in general more simply
deined 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,
atch osculates (395) to the curve at that point; and that then the
sticpe of all these planes is determined, which envelope (for reasons
afterwards more fully explained) is called here (p. 581) the “Cir-
anscribed Developable :" being a surface analogous to the “ Rectifying
Detelspable" of Lancret, but belonging (e) to a higher order of ques-
tions. And then, as the known angle I denotes (397) the inclina-
tien, suitably measured, of the rectifying line (^), which is a genera-
tris of the rectifying developable, to the tangent (1) to the curve; so
the new angle J represents the inclination of a generating line (ø), of
what has just been called the circumscribed developable, to the same
tangent (1), 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, so J
is right when the curve is spherical. For the helix (p. 585), the an-
gles H and I are equal ; and the rectifying and circumscribed deve-
Lojalles coincide, with each other and with the right cylinder, on
which the helix is a geodetic line.

(). If the recent line o 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
Rotation of the Tangent Plane (d) to the Osculating Sphere, and then
it satisfies, among others, the equations (pp. 579, 581, comp. (V)),

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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 you 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-
tire 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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