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Ap represents a small and indefinitely decreasing chord rq of the surface, drawn from the extremity P of p, so that

V... Afp=f(p+Ap) -fp=0, and lim. TAp=0,

the equation IV. becomes simply,

VI... lim. SvUAp=0;

and thus proves, in a new way, that is normal to the surface at the proposed point P, by proving that it is ultimately perpendicular to all the chords PQ from that point, when those chords become indefinitely small, or tend indefinitely to vanish.

(1.) For example, if

VII... fp = p2, v=p, then VIII... R= Ap2, and R: TAp=-TAP; thus, for every point of space, we have rigorously, with this form of fp,

IX... Afp: TAρ = 2SpUAp - TAp ;

and for every point Q of the spheric surface, fp = const., we have with equal rigour, X... 2SpUAp=TAp, or XI... PQ 20P.cos OPQ;


in fact, either of these two last formulæ expresses simply, that the projection of a diameter of a sphere, on a conterminous chord, is equal to that chord itself, and of course diminishes with it.

(2.) Passing then to the limit, or conceiving the point of the surface to approach indefinitely to P, we derive the limiting equations,

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either of which shows, in a new way, that the radii of a sphere are its normals ; with the analogous result for other surfaces, that the vector v in I. has a normal direction, as before: because its projection on a chord PQ tends indefinitely to diminish with that chord.

(3.) We may also interpret the differential equation I. as expressing, through II. and III., that the plane 373, VII., which is drawn through the point P in a direction perpendicular to v, is the tangent plane to the surface: because the projection of the chord Ap on the normal v to that plane, or the perpendicular distance,

XIV... - S (Uv. ▲p) = {R.Tv ̄1,

of a near point Q from the plane thus drawn through P, is small of an order higher than the first (comp. 370, (8.)), if the chord rq itself be considered as small of the first order.

375. This occasion may be taken (comp. 374, I. II. III.), to give a new Enunciation of Taylor's Theorem, in a form adapted to Quaternions, which has some advantages over that given (342) in the preceding Chapter. We shall therefore now express that important Theorem as follows:

"If none of the m + 1 functions,

I... fq, dfq, d2fq,... d" fq, in which d3q = 0,

become infinite in the immediate vicinity of a given quaternion q, then the

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can be made to tend indefinitely to zero, for any ultimate value of the versor Udq, by indefinitely diminishing the tensor Tdq.”

(1.) The proof of the theorem, as thus enunciated, can easily be supplied by an attentive reader of Articles 341, 342, and their sub-articles; a few hints may however here be given.

(2.) We do not now suppose, as in 342, that dmfg must be different from zero; we only assume that it is not infinite: and we add, to the expression 342, VI. for Fx, the term,

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(3.) Hence each of the expressions 342, VII, for the successive derivatives of Fx, receives an additional term; the last of them thus becoming,

IV... Dm Fx = F(m}x = d1ƒ (q+xdx) − dmfq;

so that we have now (comp. 342, X.) the values

V... F0=0, F′0, F"0=0,... F(m-1)0=0, F(m)0 = 0.

(4.) Assuming therefore now (comp. 342, XII.) the new auxiliary function,

which gives,

VI... x=

xmdqm 2.3...1

with Tdq > 0,

VII... 400, 400, 4′′0=0,.. (m1)0=0, (m)0=dqTM,

we find (by 341, (8.), (9.), comp. again 342, XII.) that

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(5.) But these two new functions, Fr and r, are formed from the dividend and the divisor of the quotient Q in II., by changing dq to rdq; and (comp. 342, (3.)) instead of thus multiplying a given quaternim differential dq, by a small and indefinitely decreasing scalar, x, we may indefinitely diminish the tensor, Tdq, without changing the versor, Udq.

(6.) And even if Udq be changed, while the differential dq is thus made to tend to zero, we can always conceive that it tends to some limit; which limiting or ultimate value of that versor Udq may then be treated as if it were a constant one, without affecting the limit of the quotient Q.

(7.) The theorem, as above enunciated, is therefore fully proved; and we are at liberty to choose, in any application, between the two forms of statement, 342 and 375, of which one is more convenient at one time, and the other at another.


SECTION 4.- On Osculating Planes, and Absolute Normals, to Curves of Double Curvature.

376. The variable vector p, of a curve in space may in general be thus expressed, with the help of Taylor's Series (comp. 370, (1.)):

I... P= p + tp' + up", with u=1;

P, p', p'', u being here abridged symbols for Po, po, p''o, ut; and the product up" being a vector, although the factor u is generally a quaternion (comp. 370, (5.)). And the different terms of this expres

sion I. may be thus constructed (compare the annexed Figure 77):

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the line TQ, or the term 'up", being thus what may be called the deflexion of the curve PQR, at Q, from its tangent PT at P, measured in a direction which depends on the law according to which varies with t, and on the distance of a from P. The equation of the plane of the triangle PTQ is rigorously (by II.) the following, with w for its variable vector,

IV... 0 = Sup''p' (w - p);


Fig. 77.

this plane IV. then touches the curve at P, and (generally) cuts it at a; so that if the point a be conceived to approach indefinitely to P, the resulting formula,

V... 0 Sp'p' (w-p), or


V'... 0 Sp'p" (w-p),


is the equation of the plane PTQ in that limiting position, in which it is called the osculating plane, or is said to osculate to the curve PQR, at the point P.

(1.) If the variable vector p be immediately given as a function p, of a variable scalur, s, which is itself a function of the former scalar variable t, we shall then have (comp. 331) the expressions, = s'Daps_p"t=s"Daps+s2D2ps, with s'= Dts, s′′ = Di3s; thus the vector p” may change, even in direction, when we change the independent scalar variable; but p" will always be a line, either in or parallel to the osculating plane; while p' will always represent a tangent, whatever scalar variable may be selected.

(2.) As an example, let us take the equation 314, XV., or 369, XIII., of the

helix. With the independent variable t of that equation, we have (comp. 369, XIV.) the derived expressions,

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p" has therefore here (comp. 369, (8.)) the direction of the normal to the cylinder ; and consequently, the osculating plane to the helix is a normal plane to the cylinder of revolution, on which that curve is traced: a result well known, and which will soon be greatly extended.

(3.) When a curve of double curvature degenerates into a plane curve, its osculating plane becomes constant, and reciprocally. The condition of planarity of a curve in space may therefore be expressed by the equation,

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(4.) Accordingly, for a plane curve, if λ be a given normal to its plane, we have the three equations,

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(5.) For example, if we had not otherwise known that the equation 337, (2.) represented a plane ellipse, we might have perceived that it was the equation of some plane curve, because it gives the three successive derivatives,

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which are complanar lines, the third having a direction opposite to the first.

(6.) And generally, the formula X. enables us to assign, on any curve of double curvature, for which p is expressed as a function of t, the points* at which it most resembles a plane curve, or approaches most closely to its own osculating plane,

377. An important and characteristic property of the osculating plane to a curve of double curvature, is that the perpendiculars let fall on it, from points of the curve near to the point of osculation, are small of an order higher than the second, if their distances from that point be considered as small of the first order.

(1.) To exhibit this by quaternions, let us begin by considering an arbitrary plane,

*Namely, in a modern phraseology, the places of four-point contact with a plane. The equation, Vp'p"= 0, indicates in like manner the places, if any, at which a curve has three-point contact with a right line. For curves of double curvature, these are also called points of simple and double inflexion.


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drawn through a point P of the curve. Using the expression 376, I., for the vector OQ, or Pt, of another point q of the same curve, we have, for the perpendicular distance of o from the plane I., this other rigorous expression,

II... Sλ(pt - p)=tS\p′ + ¥t2S\up";

which represents, in general, a small quantity of the first order, if t be assumed to be such.

(2.) The expression II. represents however, generally, a small quantity of the second order, if the direction of λ satisfy the condition,

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be also satisfied by λ, then, but not otherwise, the expression II. tends to bear an evanescent ratio to t2, or is small of an order higher than the second.

(4.) But the combination of the two conditions, III. and IV., conducts to the expression,


comparing which with 376, V., we see that the property above stated is one which belongs to the osculating plane, and to no other.

378. Another remarkable property of the osculating plane to a curve is, that it is the tangent plane to the cone of parallels to tangents (369, (6.)), which has its vertex at the point of osculation.

(1.) In general, if p=px be (comp. 369, I.) the equation of a curve in space, the equation of the cone which has its vertex at the origin, and passes through this curve, is of the form,


in which x and y are two independent and scalar variables.

(2.) We have thus the two partial derivatives,

II... Dxp=yo'x, Dyp=px;

and the tangent plane along the side (x) has for equation,

III... 0=S(w.px.q'x); or briefly, III'. . . 0 = Swpp'.

(3.) Changing then x, o, p', w to t, p', p", w-p, we see that the equation 376, V., of the osculating plane to the curve 376, I., is also that of the tangent plane to the cone of parallels, &c., as asserted.

379. Among all the normals to a curve, at any one point, there are two which deserve special attention; namely the one which is in

* The writer does not remember seeing this property in print; but of course it is an easy consequence from the doctrine of infinitesimals, which doctrine however it has not been thought convenient to adopt, as the basis of the present exposition.

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