and may interpret it as expressing, that the radius of curvature is equal to the cube of the conjugate semidiameter, divided by the constant parallelogram under any two such conjugates; or by the rectangle under the major and minor semiaxes, which are here the vectors ẞ and y (comp. 314, (2.)).

(8.) The expression XVI. or XIX. for x is easily seen to vanish, as it ought to do, at the limit where the ellipse becomes a circle, by the cylinder being cut perpendicularly, or by the condition Saẞ= 0 being satisfied; and accordingly if we write, XXI... e = SUaß excentricity of ellipse, or XXII... y2 = (1 − e2)ẞ2,

=

(9.) And it may be remarked in passing, that the expression XVI., or its recent transformation XXV., for к as a function of t, may be considered as being in quaternions the vector equation (comp. 99, I., or 369, I.) of the evolute* of the ellipse, or the equation of the locus of centres of curvature of that plane curve; and that the last form gives, by elimination of t (comp.† 315, (1.), and 371, (5.)), the following system of two scalar equations for the same evolute,

or

XXVI'. . . (S,3x)3 + (Sy«)¤ = (eß)‡, &c. ;

which will be found to agree with known results.

(10.) As another example of application to a plane curve, we may consider the hyperbola,

XXVII... p = ta+t'ß,

comp. 371, II.,

with a and ẞ for asymptotes, and with its centre at the origin. In this case the derived vectors are,

XXVIII. . . p' = a-t-2ß, p" = 2t 3ß,

XXIX... Vp"p' = 2t3Vẞa=t¬Vpp',

whence

where ov is the perpendicular from the centre o on the tangent to the curve at P, and PT is the portion of that tangent, intercepted between the same point P and an asymptote (comp. (6.) and 371, (3.)).

* That is to say, of the plane evolute; for we shall soon have occasion to consider briefly those evolutes of double curvature, which have been shown by Monge to exist, even when the given curve is plane.

In lately referring (373, (1.)) to the formula 315, V., that formula was inadvertently printed as (a')2 + (at-1)2 = 1, the sign S. before each power being omitted.

CHAP. III.]

ABRIDGED GENERAL CALCULATION.

535

(11.) We may also interpret the denominator in XXX. as denoting the projection of the semidiameter OP on the normal, or as the line NP where N is the foot of the perpendicular from the curve on that normal line; if then к be the sought centre of the osculating circle, we have the geometrical equations,

whereof the last furnishes evidently an extremely simple construction for the centre of curvature of an hyperbola, which we shall soon find to admit of being extended, with little modification, to a spherical conic* and its cyclic arcs. (12.) The logarithmic spiral with its pole at the origin,

XXXIII...p=a2ß, Saß=0, Ta≥1,

comp. 314, (5.)

may be taken as a third example of a plane curve, for the application of the foregoing formulæ. A first derivation gives, by 333, VII.,

XXXIV... p' = (c+y)p = p(c− y), p′p ́1=c+y, if c=1Ta, and

π

= Ua;

the constant quaternion quotient, p': p, here showing that the prolonged vector OP makes with the tangent PT a constant angle, n, which is given by the formula,

2

XXXV... tan n = (TV: S) (p': p)=c 1Ty, or cot n=- ITait

and a second derivation gives next,

π

XXXVI...p" = (c+ y)2p, Vp"p' = (c2 - y2) p2y = p'ay.

The formula IV. becomes therefore, in this case,

XXXVII. . . K = p + p'y1=pcy-1=-cy-1p=

21Ta

at+B;

πΤα

the evolute is therefore a second spiral, of the same kind as the first, and the radius of curvature KP subtends a right angle at the common pole. But we cannot longer here delay on applications within the plane, and must resume the treatment by quaternions of curves of double curvature.

390. When the logic by which the expression 389, IV. was obtained, for the vector of the centre of the osculating circle, has once been fully understood, the process may be conveniently and safely abridged, as follows. Referring still to Fig. 77, we may write briefly,

It was in fact for the spherical curve that the geometrical construction alluded to was first perceived by the writer, soon after the invention of the quaternions, and as a consequence of calculation with them: but it has been thought that a sub-article or two might be devoted, as above, to the plane case, or hyperbolic limit, which may serve at least as a verification.

2

+ If r be radius vector, and polar angle, and if we suppose for simplicity that TB=1, the ordinary polar equation of the spiral becomes r = a0, with a = Tan, and cot n = la, as usual.

as equations which are all ultimately true, or true at the limit, in a sense which is supposed to be now distinctly seen:

I... FT = dp, TQ = d2p, PN = (part of PQ 1 PT =)

Vdpdp
2dp

as before: this last expression, in which Vd2pdp denotes briefly V(d2p.dp), being rigorous, and permitting the choice of any scalar, to be used as the independent variable. And then, by writing, III... dp = p'dt, dt = 0, d2p = p''dť3,

the factor de3 disappears, and we pass at once to the expression,

.

IV.. K- =
P

pla Vp"p"

389, IV.,

which had been otherwise found before.

(1.) When the arc of the curve is taken for the independent variable, then (comp. 380, (12.), &c.) the expresssion II. reduces itself to the following,

and accordingly the angle PTQ in Fig. 77 is then ultimately right (comp. 383, (5.)), so that we may at once write, with this choice of the scalar variable,

(2.) Suppose then that we have thus geometrically (and very simply) deduced the expression V. for p, for this particular choice of the scalar variable; and let us consider how we might thence pass, in calculation, to the more general formula II., in which that variable is left arbitrary. For this purpose, we may write, by principles already stated,

and the required transformation is accomplished.

(3.) And generally, if s denote the are of any curve of which vector, we may establish the symbolical equations,

is the variable

(4.) For example (comp. 389, XII.), the Vector of Curvature, D,3p, admits of being expressed generally under any one of the five last forms VII.

CHAP. III.] POLAR AXIS, POLAR DEVELOPABLE.

537

391. Instead of determining the vector of the centre of the osculating circle by one vector expression, such as 389, IV., or any of its transformations, we may determine it by a system of three scalar equations, such as the following,

=

I... S(k − p) p' = 0; II... S(k-p) p"-p"2 = 0;

K

III... S(k-p)p'p" =0,

of which it may be observed that the second is the derivative of the first, if be treated as constant (comp. 386, (4.)); and of which the first expresses (369, IV.) that the sought centre is in the normal plane to the curve, while the third expresses (376, V.) that it is in the osculating plane; and the second serves to fix its position on the absolute normal (379), in which those two planes intersect.

(1.) Using differentials instead of derivatives, but leaving still the independent variable arbitrary, we may establish this equivalent system of three equations, IV... S (x − p) dp = 0; V... S(x − p)&2p — dp2 = 0; VI... S(x − p)dpd2p = 0; of which the second is the differential of the first, if x be again treated as constant. (2.) It is also permitted (comp. 369, (2.), 376, (3.), and 380, (2.)), with the same supposition respecting «, to write these equations under the forms,

VII... dT (k-p)=0 ;

к

VIII...d2T(k-p) = 0; IX. . . dUV(x − p)dp = 0;

and to connect them with geometrical interpretations.

(3.) For instance, we may say that the centre of the osculating circle is the point, in which the osculating plane, III. or VI. or IX., is intersected by the axis of that circle; namely, by the right line which is drawn through its centre, at right angles to its plane and which is represented by the two scalar equations,

:

I. and II., or IV. and V., or VII. and VIII.

(4.) And we may observe (comp. 370, (8.)), that whereas for a point R taken arbitrarily in the normal plane to a curve at a given point P, we can only say in general, that if a chord PQ be called small of the first order, then the difference of distances, RQ-RP, is small of an order higher than the first; yet, if the point R be taken on the axis (3.) of the osculating circle, then this difference of distances is small, of an order higher than the second, in virtue of the equations VII. and VIII.

(5.) The right line I. II., or IV. V., or VII. VIII., as being the locus of points which may be called poles of the osculating circle, on all possible spheres passing through it, is also called the Polar Axis of the curve itself, corresponding to the given point of osculation.

(6.) And because the equation II. is (as above remarked) the derivative of I., the known theorem follows (comp. 386), that the locus of all such polar axes is a developable surface, namely that which is called the Polar Developable, or the envelope of the normal planes to the given curve; of which surface we shall soon have occasion to consider briefly the cusp-edge.

392. The following is an entirely different method of investigating, by quaternions, not merely the radius or the centre of the osculating circle to a curve in space, but the vector equation of that circle itself and in a way which is applicable alike, to plane curves, and to curves of double curvature.

(1.) In general, conceive that or=r is a given tangent to a circle, at a given point which is for the moment taken as the origin; and let PP' = p' represent a rariable tangent, drawn at the extremity of the variable chord Op=p: also let u be the intersection, OT PP', of these two tangents. Then the isosceles triangle oup, combined with the formula 324, XI. for the differential of a reciprocal, gives easily the equations,

if a be the vector OA of any second given point a of the circumference. (2.) The vector equation of the circle PQD (389) is therefore,

IV... V

2p' 2

= =V

=

2p'
w-p Pt-P

V. (1 + {tup'p'-1)-1 = − V. up”p'-1 (1 + {tup”p'~1)-' ;

whence, passing to the limit (t = 0, u= 1), the analogous equation of the osculating ciccle is at once found to be,

with the verification (comp. 296, (9.)), that when we suppose,

which agrees with recent results (389, IV., &c.).

(3.) Instead of conceiving that a circle is described (389), so as to touch a given curve (Fig. 77) at P, and to cut it at one near point q, we may conceive that a circle cuts the curve in the given point P, and also in two near points, Q and R, unconnected by any given law, but both tending together to coincidence with P: and may inquire what is the limiting position (if any) of the circle PQR, which thus intersects the curve in three near points, whereof one (P) is given.

(4.) In general, if a, ß, p be three co-initial chords, OA, OB, OP, of any one circle, their reciprocals a-1, ß-1, p ̄1, if still co-initial, are termino-collinear (260); applying which principle, we are led to investigate the condition for the three co-initial vectors,

with uo

=

X... (w− p) ́1, (sp'+{s2usp′′)-', (tp′+{t2up")-1,

1, thus ultimately terminating on one right line; or for our having ultimately a relation of the form,