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lating sphere, towards the point P of the given curve, that is, towards the point of osculation.

(51.) Again, if we only take account of s3, the deviation of P, from the osculating circle at P has been seen to be a vector tangential to the osculating sphere, which may be thus expressed (comp. 397, IX., LII.),

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if c, be the point on the circle, which is distant from the given point P by an are of that circles, with the same initial direction of motion, or of departure from P, represented by the common unit tangent r; the quantity of this deviation is therefore $3 R expressed by the scalar that is, by the deviation (comp. 397, (9.), (10.)) 6r2r from the osculating plane* at P, multiplied by the secant (r ́1R) of the inclination (P) of the radius (R) of spherical curvature, to the radius (r) of absolute curvature, and positive when this last deviation has the direction of the binormal v.


(52.) On the other hand (comp. (5.)) the small angle, which the small arc ss, of the cusp-edge (s) of the polar developable subtends at the point P, is ultimately expressed by the scalar,

LXXXVI... SPS, = (PSs - PS). R-1 cot P =

rR's nrs



(by XXXIII.),

this angle being treated as positive, when the corresponding rotation† round + from

Besides the nine expressions in 397, (42.) for the square r2 of the second curvature, the following may be remarked, as containing the law of the regression of the projection of a curve of double curvature on its own normal plane :

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K being still the centre of the osculating circle, and Q1, Q2, Q3 being still (as in 397, (10.)) the projections of a near point Q (or P,), on the tangent, the absolute normal (or inward radius of curvature PK), and the binormal at P. In fact, the principal terms of the three vector projections corresponding, of the small chord rq (or pr.), are (comp. LVIII.):

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+ Considered as a rotation, this small angle may be represented by the small vector, rp ̄`R'R-s7; and if the vector deviation LXXXV. from the osculating circle be multiplied by this, the quarter of the product is (comp. XXXV.) the vector deviation from the osculating sphere, under the form,

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Ps to PS, is positive and if we multiply this scalar, by that which has just been assigned (51.), as an expression for the deviation CP, from the osculating circle, we get, by XXXV., the product,

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(53.) Combining then the recent results (50.), (51.), (52.), we arrive at the following Theorem :

The deviation of a near point P, of a curve in space, from the osculating sphere at the given point P, is ultimately equal to the quarter of the deviation of the same near point from the osculating circle at P, multiplied by the sine of the small ungle which the arc sss, of the locus of centres of spherical curvature (s), or of the cusp-edge of the polar developable, subtends at the same point P; and this deviation (SP, -SP) from the sphere has an outward or an inward direction, according as the same arc ss, is concave or convex towards the same given point.

(54.) The vector of the centre ss, of the near osculating sphere at P., is (in the same order of approximation, comp. I.),

LXXXVIII... OSs = σ, = σ + so' + }s2o′′ + js3o” + 24810TM;

and although – p is already a function (by 397, IX., &c.) of 7, 7, 7", so that ☛ is (as in (2.) or (22.)) a function of r', r", r'", and o", σ"", σTM introduce respectively the new derived vectors TV, 7, 7", or D ̧3p, D ̧p, Dsp, which we are not at present employing (49.), yet we have seen, in (23.) and (24.), that some useful combinations of " and " can be expressed without TV, TV: and the following is another remarkable example of the same species of reduction, involving not only σ" and ☛"" but also o", but still admitting, like the former, of a simple geometrical interpretation.

(55.) Remembering (comp. (22.), and 397, XV.) that

LXXXIX... (o-p)2+ R2 = 0, and XC... Sr" ( − p) = r2 S = r2 + nr ̄1r ̄1, and reducing the successive derivatives of LXXXIX, with the help of the equations 397, XIX., and of their derivatives, we are conducted easily to the following system of equations, into which the derived vectors 7, 7', &c. do not expressly enter, but which involve σ', o", o'", o1, and R', R”, R””, R" :

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(56.) But, if Rs denote the radius of the near sphere, and if we still neglect s", we have,

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whence follows, by LXXXVIII., and by the recent equations, this very simple expression, from which (comp. (24.)) everything depending on rTM, 7, 7TM1 has disappeared,

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and which gives (within the same order of approximation, attending to XXXV.) the geometrical relation,

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(57.) This result might have been foreseen, from the following very simple consideration. When the coefficient S-1 of non-sphericity (395, (16.)), or of the deviation of a curve from a sphere, is positive, so that a near point P. of the curve is exterior to (what we may call) the given sphere, which osculates to that curve at P, by an amount which is ultimately proportional to the fourth power of the arc, s, of the curve, then the given point P must be, for the same reason, exterior to the near sphere, which osculates at the point P.; and the two deviations, PS, P.Ss, and SP, SP, which have been found by calculation to be equal (C.), if s5 be neglected, must in fact bear to each other an ultimate ratio of equality, because the two arcs, +s and - s, from P to Ps, and from P, back to P, are equally long, although oppositely directed; or because (+s)a1=(-s)1. And precisely the same reasoning applies, when the coefficient S-1 is negative, so that the deviations, equated in the formula C., are both inwards.

(58.) As regards the deviation (51.) of the near point P, of the curve from the osculating circle at P, we may generalize and render more exact the expression LXXXV., by considering a point c of that circle, which is distant by a circular are =t from the given point P; and of which the vector is, rigorously, by 896, (18.),

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(59.) In this way we shall have (comp. (34.)) the vector deviation,

CIII... CPs Ps w1 = XT + Yrr' + Zrv,

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or, neglecting s5 and t5, and attending to the expressions LVIII. and LVIII'.,

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in which r, r', r, p, and have the same significations as before.


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which differs (as we see) by only a quantity of the fourth order from the arcs of the curve, we shall have, to the same order of approximation, the expressions,

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the deviation at P, from the circle being here measured in a direction parallel to the normal plane at P; and if s1 be neglected (although the expressions enable us to take account of it), this deviation is also parallel (as before) to the tangent r(o− p) to the osculating sphere in that plane while it is represented in quantity by Rr1zs, which agrees with the result in (51.).

(61.) The expressions CVII. give also, without neglecting s1,

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such then is the component of the deviation from the osculating circle, which is parallel to the normal PS to the sphere at P; and we see that it only differs in sign (because it is positive when its direction is that of the inward normal, or inward radius PS), from the expression XXXV. (comp. C.), for the outward deviation SP, — SP of the near point Ps, from the same osculating sphere at the given point P.

(62.) This latter component (61.) is small, even as compared with the former small component (60.); and the small quotient, of the latter divided by the former, is ultimately (by LXXXVI.),

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where the small angle srs, is positive or negative, according to the rule stated in (52.), and may be replaced by its sine, or by its tangent.

(63.) Instead of cutting the given osculating circle, as in (60.), by a plane which is parallel to the given normal plane at P, we may propose to cut that circle by the near normal plane at Ps, or to satisfy this new condition,

CX...0=STs (ps-wi), or CX'. . . 0 = XSTT, + YSrt'ts + ZSrvts; which is easily found to give by CV. the values (s and t being still supposed to be small, and being still neglected):


r'st 24r3'


r'84 CXII. . . X= 6r3'

Y = &c., Z= &c., as in CVII.;

so that in passing to this new near point Ct of the circle, we only change X from zero to a small quantity of the fourth order, and make no change in the values of Y and Z.

(64.) The new deviation C¿P, from the given circle may be decomposed into two partial deviations, in the near normal plane, of which one has the direction of the unit-tangent R ̄1rs (σs - ps) to the near sphere at Ps, and the other has that of the unit-normal R ̧1(σ-ps) to the same sphere at the same point (or the opposites of these two directions); and the scalar coefficients of these two vector units, if we attend only to principal terms, are easily found to be,

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CXV... Deviation of near point P, from given osculating circle,
measured in the near normal plane to the curve at P„

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in which it may be observed, that the second scalar coefficient is equal to three times the scalar deviation SP. - SP (XXXV. or C.), of the near point P, of the curve, from the given osculating sphere (at P).

(66.) But we may also interpret the new coefficient last mentioned, as representing a new deviation; namely, that of the point Ct of the given circle, from the near osculating sphere at P, considered as positive when that new point C, is exterior to that near sphere; or as denoting the difference of distances, S.Ct S&P. We have therefore (comp. (56.)) this new geometrical relation, of an extremely simple kind: CXVI... S、Ct — S、P's = 3(SP, — SP) = 3 (S‚P – S‚P;) ;


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(67.) Supposing, then, at first, that the coefficient of non-sphericity S-1 is posi tive (comp. 395, (16.)), if we conceive a point to move backwards, upon the curve, from P, to P, and then forwards, upon the circle which osculates at P, to the new point c (63.), we see that it will first attain (at P) a position exterior to the sphere which osculates at Ps, or will have an amount, determined in (56.), of outward deviation, with respect to that near osculating sphere; and that it will afterwards attain (at the new point C1) a deviation of the same character (namely outwards, if S>1), from the same near sphere, but one of which the amount will be threefold the former : this last relation holding also when S<1, or when both deviations are inwards.

(68.) It is easy also to infer from (65.), (comp. (57.)), that if we go back from Ps, on the near circle which osculates at that near point, through an are (t) of that circle, which will only differ by a small quantity of the fourth order (comp. (60.)) from the arc (s) of the curve, so as to arrive at a point, which for the moment we shall simply denote by c, and in which (as well as in another point of section, not necessary here to be considered) the near osculating circle is cut by the given normal plane at P, the vector deviation of this new point c of the new circle, from the given point P of the curve, must be, nearly:

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the coefficients being formed from those of the formula CXV., by first changing s to and then changing the signs of the results:. while the relation CXVI. or CXVI. takes now the form,


.. SC-SP 3 (SP-SP), or

CXVIII'... SC = 3 SPs - 2SP.

(69.) Accordingly if, after going from P to P, along the curve, we go forward or backward, through any positive or negative arc, t, of the circle which osculates at that point Ps, we shall arrive at a point which we may here denote by Cst; and the vector (comp. again 396, (18.)) of this near point (more general than any of those hitherto considered) will be, rigorously,

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