CHAP. III.] LOCUS OF OSCULATING CIRCLE.

599

And if we develope this new expression to the accuracy of the fourth order inclusive, we find that we satisfy the new condition (comp. (63.)),

CXX... ST (ws, t-p) = 0, when

CXXI... t=-8

r's4

24,3 ;

and that then the expression CXIX. agrees with CXVII., within the order of approximation here considered.

(70.) A geometrical connexion can be shown to exist, between the two equivalents which have been found above, one for the quadruple (LXXXVII., comp. (53.)), and the other for the triple (CXVIII.), of the deviation SPs SP of a near point Ps of the curve, from the sphere which osculates at the given point P: in such a manner that if either of those two expressions be regarded as known, the other can be inferred from it.

(71.) In fact if we draw, in the normal plane, perpendiculars PD and PE to the lines PS and PS,, and determine points D and E upon them by drawing a parallel to Ps through the point c of (68.), letting fall also a perpendicular CF on PS, the two small lines PD and DC will ultimately represent the two terms or components CXVII. of PC; and the small angle DPC will ultimately be equal to three quarters of the small angle SPS,, and will correspond to the same direction of rotation round 7, be

Fig. 83.

But the line cr is ultimately the trace, on the given normal plane, of the tangent plane at c to the near osculating sphere; the small line FP (or CE) represents therefore the deviation 3,P — S、P, of the given point P from that near sphere, or the equal deviation (57.), SP-SP; its ultimate quadruple, DE, represents the product mentioned in (52.); and the ultimate triple, DC, of the same small line CE, is a geometrical representation of that other deviation SC-SP, which has been more recently considered.

(72.) When the two scalars, s and t, are supposed capable of receiving any values, the point Cs, t in (69.) may be any point of the Locus (8.) of the Osculating Circle to the given curve of double curvature; and if we seek the direction of the normal to this superficial locus, at this point, on the plan of Art. 372, writing first the equation of the surface under the slightly simplified, but equally rigorous form,

In Figs. 81, 82, the little arc near s is to be conceived as terminating there, or as being a preceding arc of the curve which is the locus of s, if r', r, n, and therefore also and 71, be positive (comp. the second Note to page 574). In the new Figure 83, the triangle PDE is to be conceived as being in fact much smaller than PKS, though magnified to exhibit angular and other relations.

so that u is here a new scalar variable, representing the angle subtended at the centre K, of the osculating circle at Ps, by the are, t, of that circle, we are led, after a few reductions, to the expression,

CXXVII. . . V(Duws, u⋅ Dsws, u) = rsrs ̈1 (@s, u — σs) vers u ;

which proves, by quaternions, what was to be expected from geometrical* considerations, that the locus of the osculating circle is also (as stated in (8.) and (22.)) the Envelope of the Osculating Sphere.

(73.) The normal to this locus, at any proposed point Cs,t of any one osculating circle, is thus the radius of the sphere to which that circle belongs, or which has the same point of osculation P, with the given curve, whether the arc (s) of that curve, and the arc (t) of the circle, be small or large. We must therefore consider the tangent plane to the locus, at the given point P of the curve, as coinciding with the tangent plane to the osculating sphere at that point; and in fact, while this latter plane (PS) contains the tangent r to the curve, which is at the same time a tangent to the locus, it contains also the tangent (p) to the sphere, which is by CXVII. another tangent to the locus, as being the tangent at P to the section of that surface, which is made by the normal plane to the curve.

(74.) But when we come to examine, with the help of the same equation CXVII., what is the law of the deviation DC (comp. Fig. 83) of that normal section of the locus, considered as a new curve (c), from its own tangent PD, we find that this law is ultimately expressed (comp. (71.)) by the formula,

hence DC varies ultimately as the power of PD, which has the fraction for its exponent; the limit of PD2: DC is therefore null, and the curvature of the section is infinite

at P.

(75.) It follows that this point P is a singular point of the curve (c), in which the locus (8.) is cut (73.), by the normal plane to the given curve at that point; but it is not a cusp on that section, because the tangential component PD of the vector chord PC is ultimately proportional to an odd power (namely to the cube, by CXVII., comp. (71.)) of the scalar variable, s, and therefore has its direction reversed, when that variable changes sign: whereas the normal component DC of the same chord PC is proportional to an even power (namely the fourth, by the same equation CXVII.) of the same arc, s, of the given curve, and therefore retains its direction unchanged, when we pass from a near point Ps, on one side of the given point P, to a near point P., on the other side of it.

(76.) To illustrate this by a contrasted case, let G be the point in which the tangent to the given curve at P, is cut by the normal plane at P ; or a point of the section, by that plane, of the developable surface of tangents. We shall then have

In the language of infinitesimals, two consecutive osculating spheres, to any curve in space, intersect each other in an osculating circle to that curve.

CHAP. III.] ENVELOPE OF SPHERE WITH VARYING RADIUS. 601

with the significations 397, (10.) of q2 and Q3; hence the point P of the curve is (as is well known) a cusp of the section (G) of the developable surface of tangents (comp. 397, (15.)), because the tangential component (− PQ2) of the vector chord (PG) has here a fixed direction, namely that of the outward radius (KP prolonged) of the circle of curvature at P: while it is now the normal component (-2PQ3) which changes direction, when the arcs of the curve changes sign. At the same time we see* that the equation of this last section (G) may ultimately be thus expressed:

comparing which with the equation CXXVIII., we see that although, in each case, the curvature of the section is infinite, at the point P of the curve, yet the normal component (or co-ordinate) varies (ultimately) as the power of the tangential component, for the section (G) of the Surface of Tangents: whereas the former component varies by (74.) as the power of the latter, for the corresponding section (c) of the Locus of the Osculating Circle.

(77.) It follows also that the curve (P) itself, although it is not a cusp-edge of the last-mentioned locus (8.), while it is such on the surface of tangents, is yet a Singular Line upon that locus likewise: the nature and origin of which line will perhaps be seen more clearly, by reverting to the view (8.), (22.), (72.), according to which that Locus of a Circle is at the same time the Envelope of a Sphere.

(78.) In general, if we suppose that σ and R are any two real functions, of the vector and scalar kinds, of any one real and scalar variable, t, and that o', R', and o", R", &c. denote their successive derivatives, taken with respect to it, then o may be conceived to be the variable vector of a point s of a curve in space, and R to be the variable radius of a sphere, which has its centre at that point s, but alters generally its magnitude, at the same time that it alters its position, by the motion of its centre along the curve (s).

(79.) Passing from one such sphere, with centre s and radius R, considered as given, and represented by the scalar equation,†

in which p is now conceived to be the vector of a variable point P upon its surface, to a near sphere of the same system, for which σ, s, and R are replaced by σt, St, and Rt, where t is supposed to be small, we easily infer (comp. 386, (4.)) that the equation,

So' (o-p) + RR' = 0,

XCI., which is formed from LXXXIX. by once derivating σ and R with respect to t, but

* Compare the first Note to page 594.

†This equation, and a few others which we shall require, occurred before in this series, but in a connexion so different, that it appears convenient to repeat them here.

treating p as constant, represents the real plane (comp. 282, (12.)) of the (real or imaginary) circle, which is the ultimate intersection of the near sphere with the given one; the radius of this circle, which we shall call r, being found by the following formula,

CXXXI... r2′2 = R2 ( R22 + σ′), or

and being therefore real when

CXXXI'. . . raTo'2 = R2 (To22 — R ́3),

CXXXII... R'2 + σ22 < 0, or CXXXII'... R22 <To22 ;

while the centre, say K, of the circle is always real, and its vector is,

CXXXI"... OK = x=σ + RR'o');

and the plane XCI. of the same circle is parallel to the normal plane of the curve (s).

(80.) With the condition CXXXII., the two scalar equations, LXXXIX. and XCI., represent then jointly a real circle; and the locus of all such circles (comp. 386, (6.)) is easily proved to be also the envelope of all the spheres, of which one is represented by the equation LXXXIX. alone; each such sphere touching this locus, in the whole extent of the corresponding circle of the system.

(81.) The plane XCI., considered as varying with t, has a developable surface for its envelope; and the real right line, or generatrix, along which one touches the other, is represented (comp. again 386, (6.)) by the system of the two scalar equations, XCI. and

where p is now the variable vector of the line of contact, although it has been treated as constant (comp. 386, (4.)), in the process by which we are here conceived to pass, by a second derivation, from LXXXIX. through XCI. to XCIII.

(82.) This real right line (81.) meets generally the sphere, and also the circle (as being in its plane), in two (real or imaginary) points, say P1, P2; and the curvilinear locus of all such points forms generally a species of singular line,* upon the superficial locus (or envelope) recently considered (80.); or rather it forms in general two branches (real or imaginary) of such a line: which generally two-branched line (or curve) is the (real or imaginary) envelope (comp. 386, (8.)), of all the circles of the system.

(83.) The equation,

So'o" (o-p) = 0,

XCII.,

which now represents (comp. 376, V.) the osculating plane to the curve (s), shows

* Called by Monge an arête de rebroussement, except in the case to which we shall next proceed, when its two branches coincide. The envelope (80.) of a varying sphere has been considered in two distinct Sections, § XXII. and § XXVI., of the Application de l'Analyse à la Géométrie; but the author of that great work does not appear to have perceived the interpretation which will soon be pointed out, of the condition of such coincidence. Meantime it may be mentioned, in passing, that quaternions are found to confirm the geometrical result, that when the two branches (P1) (P2) are distinct, then each is a cusp edge of the surface; but that when they are coincident, the singular line (P) in which they merge has then a different character.

CHAP. III. CONDITION OF COINCIDENCE OF CUSP-EDGES. 603

that this plane through the centre s of the sphere is perpendicular to the right line (81.), and consequently contains the perpendicular let fall from that centre on that line: the foot P of this last perpendicular is therefore found by combining the three linear and scalar equations, XCI., XCII., XCIII., and its vector is,

if

CXXXIV... g = — σ′a — R′2 — RR" = Tσ'2 — (RR′).

(84.) The condition of contact of the right line (81.) with the sphere (78.), or with the circle (79.), or the condition of contact between two consecutive* circles of the system (80.), or finally the condition of coincidence of the two branches (82.) of that singular line upon the surface which is touched by all those circles, is at the same time the condition of coexistence of the four scalar equations, LXXXIX., XCI., XCII., XCIII.; it is therefore expressed by the equation (comp. CXXXIII.), CXXXV... R2(Vo ́o′′)2 = (go' + RR′0′′)2 ;

which may also be thus written,†

or thus,

CXXXVI... (RSo'o” — R′g)2 = (R′2 + o′2) (R2o'2+go),
CXXXVII... R2 (R′2 +σ12) (Voʻo′′)2 = (go"2 + RR'So'o")2;

the scalar variable t (78.), with respect to which the derivations are performed, remaining still entirely arbitrary, but the point P, which is determined by the formula CXXXIII., being now situated on both the sphere and the circle: and its curvilinear locus, which we may call the curve (P), being now the singular line itself, in its re

Compare the Note to page 581.

+ In page 372 of Liouville's Edition already cited, or in page 325 of the Fourth Edition (Paris, 1809), of the Application de l'Analyse, &c., it will be found that this condition is assigned by Monge, as that of the evanescence of a certain radical, under the form (an accidentally omitted exponent of " in the second part of the first member being here restored):

[a(p'p” + ¥'¥”+ π'π") − h2]2 + h2 [a2(p′′2 + ¥”2 + π”2) − h1] = 0 ;

in which he writes, for abridgment,

and,, are the three rectangular co-ordinates of the centre of a moving sphere, considered as functions of its radius a. Accordingly, if we change R to a, and ☛ to ip+jų+ kπ, supposing also that R' = a' = 1, and R′′ = a" =0, whereby g is changed to − h2, and R′2 + o' to h2, in the condition CXXXVI., that condition takes, by the rules of quaternions, the exact form of the equation cited in this Note: which, for the sake of reference, we shall call, for the present, the Equation of Monge, although it does not appear to have been either interpreted or integrated by that illustrious author. Indeed, if Monge had not hastened over this case of coincident branches, on which he seems to have designed to return in a subsequent Memoir (unhappily not written, or not published), he would scarcely have chosen such a symbol as h2 (instead of — h2), to denote a quantity which is essentially negative, whenever (as here) the envelope of the sphere is real.