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CHAP. III.] ELIMINATION OF ARBITRARY FUNCTIONS.

XV... p = p(x, y) = ¥x+yʊ¥'x;

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the sign of a versor being here introduced, for the sake of facilitating the passage, at a certain limit, to a cone (comp. 386, (12.)).

(9.) And the scalar equation of the same arbitrary developable may be represented as the result of the elimination of t, between the two equations,

XVI... Spxt +1=0, Spx't=0;

in which Xt is an arbitrary vector function of t.

(10.) The envelope of a plane with two arbitrary and scalar parameters, t and u, is generally a curved but undevelopable surface, which may be represented by the system of the three scalar equations,

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where -x denotes the reciprocal of the perpendicular from the origin on the tangent plane to the surface, at what may be called the point (t, u).

388. It remains, on the plan lately stated (380), to consider briefly surfaces of revolution, and to investigate the geodetic lines, on this additional family of surfaces; of which the equation, analogous to those marked I. II. III. IV. in 387, for spheres, cylinders, cones, and developables, is of the form,

I... Sapv = 0,

if a be a given line in the direction of the axis of revolution, supposed for simplicity to pass through the origin; but which may also be represented by either of these two other equations, not involving the normal v,

II... Tp=f(Sap), or III... TV ap= F(Sap),

where ƒ and Fare used as characteristics of two arbitrary but scalar functions: between which Sap may be conceived to be eliminated, and so a third form of the same sort obtained.

(1.) In fact, the equation I. expresses that v ||| a, p, or that the normal to the surface intersects the axis; while II. expresses that the distance from a fixed point upon that axis is a function of its own projection on the same fixed line, or that the sections made by planes perpendicular to the axis are circles; and the same circularity of these sections is otherwise expressed by III., since that equation signifies that the distance from the axis depends on the position of the cutting plane, and is constant or variable with it: while the two last forms are connected with each other in calculation, by means of the general relation (comp. 204, XXI.),

IV... (Tap)2 = (Sap)2 + (TVap)2.

(2.) The equation I. is analogous, in quaternions, to a partial differential equation of the first order, and either of the two other equations, II. and III., is analogous to the integral of that equation, in the usual differential calculus of scalars.

(3.) To accomplish the corresponding integration in quaternions, or to pass from the form I. to II., whence III. can be deduced by IV., we may observe that the equation I. allows us to write (because Svdp = 0),

V... v=xα +yp, VI... Sado+ySpdp = 0,

so that the two scalars Sap and Tp are together constant, or together variable, and must therefore be functions of each other.

(4.) Conversely, to eliminate the arbitrary function from the form II., quaternion differentiation gives,

hence

VII... 0 = S (Up.dp)+ƒ′ (Sap). Sadp= S. (Up + aƒ'Sap)dp ;
VIII. . . v || Up + af'Sap, and IX... v ||| a, p, as before;

so that we can return in this way to the equation I., the functional sign ƒ disappearing.

(5.) We have thus the germs of a Calculus of Partial Differentials in Quaternions,* analogous to that employed by Monge, in his researches respecting families of surfaces: but we cannot attempt to pursue the subject farther here.

(6.) But as regards the geodetic lines upon a surface of revolution, we have only to substitnte for v, in the recent formula I., by 380, IV., the expression dUdp, which gives at once the differential equation,

X... 0 SapdUdp = d. SapUdp (because S(adp.Udp)=– SaTdp = 0); whence, by a first integration, c being a scalar constant,

XI... c = SapUdp = TVap. SU (Vap.dp).

(7.) The characteristic property of the sought curves is, therefore, that for each of them the perpendicular distance from the axis of revolution varies inversely as the cosinet of the angle, at which the geodetic crosses a parallel, or circular section of the surface because, if Ta= 1, the line Vap has the length of the perpendicular let fall from a point of the curve on the axis, and has the direction of a tangent to the parallel.

:

The same remark was made in page 574 of the Lectures, in which also was given the elimination of the arbitrary function from an equation of the recent form III. It was also observed, in page 578, that geodetics furnish a very simple example of what may be called the Calculus of Variations in Quaternions; since we may write,

d § ds = ò √ Tdp = ƒ dTdp = − ƒ S (Udp.ddp)
=-ƒS(Udp.ddp) = − AS (Udp.dp) + ƒ S (dUdp.dp),

and therefore dUdp || v, or VvdUdp = 0, as in 380, IV., in order that the expression under the last integral sign may vanish for all variations do consistent with the equation of the surface: while the evanescence of the part which is outside that sign f supplies the equations of limits, or shows that the shortest line between two curves on a given surface is perpendicular to both, as usual.

+ Unless it happen that this cosine is constantly zero, in which case c = 0, and the geodetic is a meridian of the surface.

CHAP. III.] OSCULATING CIRCLES TO CURVES IN SPACE.

(8.) The equation XI. may also be thus written,

=

XII... cTp' Sapp', where p' = Dip;

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and if the independent variable t be supposed to denote the time, while the geodetic is conceived to be a curve described by a moving point, then while To' evidently represents the linear velocity of that point, as being = ds: dt, if s denote the arc (comp. 100, (5.), and 380, (7.), (11.)), it is easy to prove that Sapp' represents the double areal velocity, projected on a plane perpendicular to the axis; the one of these two velocities varies therefore directly as the other: and in fact, it is known from mechanics, that each velocity would be constant,* if the point were to describe the curve, subject only to the normal reaction of the surface, and undisturbed by any other force.

(9.) As regards the analysis, it is to be observed that the differential equation X. is satisfied, not only by the geodetics on the surface of revolution, but also by the parallels on that surface: which fact of calculation is connected with the obvious geometrical property, that every normal plane to such a parallel contains the axis of revolution.

(10.) In fact if we draw the normal plane to any curve on the surface, at a point where it crosses a parallel, this plane will intersect the axis, in the point where that axis is met by the normal to the surface, drawn at the same point of crossing; but this construction fails to determine that normal, if the curve coincide with, or even touch a parallel, at the point where its normal plane is drawn.

SECTION 6.- On Osculating Circles and Spheres, to Curves in Space; with some connected Constructions.

389. Resuming the expression 376, I. for p, and referring again to Fig. 77, we see that if a circle PQD be described, so as to touch a given curve PQR, or its tangent PT, at a given point P, and to cut the curve at a near point Q, and if PN be the projection of the chord PQ on the diameter PD, or on the radius CP, then because we have, rigorously,

I. we have also

and

..

with PQ = tp' + t'up",

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u =

= 1

for t=0,

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PC PD PQ2 (p' + tup'')2 p

Conceiving then that the near point a approaches indefinitely to the given point P, in which case the ultimate state or limiting position of

This remark is virtually made in page 443 of Professor De Morgan's Differential and Integral Calculus (London, 1842), which was alluded to in page 578 of the Lectures on Quaternions.

the circle PQD is said to be that of the osculating circle to the curve at that point P, we see that while the plane of this last circle is the osculating plane (376), the vector к of its centre K, or of the limiting position of the point c, is rigorously expressed by the formula:

p/3 IV... * = P + √"p Vp'p'

which may however be in many ways transformed, by the rules of the present Calculus.

(1.) Thus, we may write, as transformations of the expression IV., the following: To'

V...k=p + √pp"
p'

Tp'

ρ

= p

-1 Vp"p-1.Up' (Up')' ;

or introducing differentials instead of derivatives, but leaving still the independent variable arbitrary,

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dp3
do
Tdo
Vdodap Vd2pdp-1 dUp'

ds

P

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if s be the arc of the curve; so that the last expression gives this very simple formula, for the reciprocal of the radius of curvature, or for the ultimate value of 1: CP,

VII. . . (p − x)-1 =D,Up', where Up' = Udp, as before.

-

(2.) To interpret this result, we may employ again that auxiliary and spherical curve, upon the cone of parallels to tangents, which has already served us to construct, in 379, (6.) and (7.), the osculating plane, the absolute normal, and the binormal, to the given curve in space. And thus we see, that while the semidiameter PC has ultimately the direction of dUp', and therefore that of the absolute normal (379, II.) at P, the length of the same radius is ultimately equal to the arc PQ (or As) of the given curve, divided by the corresponding are of the auxiliary curve; or that the radius of curvature, or radius of the osculating circle at P, is equal to the ultimate quotient of the arc PQ, divided by the angle between the tangents, PT and (say) QU, to that arc PQ itself at P, and to its prolongation QR at Q, although these two tangents are generally in different planes, and have no common point, so long as PQ remains finite: because we suppose that the given curve is in general one of double curvature, although the formule, and the construction, above given, are applicable to plane curves also.

(3.) For the helix, the formula IV. gives, by values already assigned for p, p', p”, and a, the expression,

VIII... K = ctα- a'ẞ cot2 a, whence IX... p − = a1ß cosec2 a,

a being the inclination of the given helix to the axis; the locus of the centre of the osculating circle is therefore in this case a second helix, on the same cylinder, if

π

a= but otherwise on a co-axal cylinder, of which the radius = the given radius 4'

Tẞ, multiplied by the square of the cotangent of a; and the radius of curvature =T(p − x) = Tẞ x cosec2 a, so that this radius always exceeds the radius of the cylinder, and is cut perpendicularly (without being prolonged) by the axis.

CHAP. III. VECTOR OF curvature, examples.

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(4.) In general, if Tp'=const., and therefore Sp'p"=0 (comp. 379, (2.)), the expression IV. becomes,*

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X.. ..k=p+ ; whence, XI... k = p-p"-1, if Tp'=1,

that is, if the arc be taken as the independent variable (380, (12.)). Under this last condition, then, the formula VII. reduces itself to the following,

XII... (p −k)-1 = p′′ = Ds2p = ultimate reciprocal of radius CP;

so that p" (for Tp'= 1) may be called the Vector of Curvature, because its tensor Tp" is a numerical measure for what is usually called the curvature† at the point P, and its versor Up" represents the ultimate direction of the semidiameter PC, of the circle constructed as above.

(5.) As an example of the application (2.) of the formula IV. for к, to a plane curve, let us take the ellipse,

XIII...p=Vaß, Ta=1, Saß>

<

0,

337, (2.),

considered as an oblique section (314, (4.)) of a right cylinder. The expressions 376, (5.) for the derivatives of p, combined with the expression XIII. for that vector itself, give here the relations,

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so that p1 is the conjugate semidiameter of the ellipse (comp. 369, (4.)), and Vpp': p' is the perpendicular from the centre of that curve on the tangent. We recover then, by this simple analysis, the known result, that the radius of curvature of an ellipse is equal to the square of the conjugate semidiameter, divided by the perpendicular. (7.) We may also write the equation XVI. under the form,

XIX... =p

013 Vppi'

where XX... Vpp1 =ẞy = const.;

The expressions X. XI. may also be easily deduced by limits, from the construction in 383, (2.).

+ It may be remarked that the quantity z, or T", in the investigation (382) respecting geodetics on a developable, represents thus the curvature of the cusp-edge, for any proposed value of the arc, x, of that curve.

These values XV. might have been obtained without integrations, but this seemed to be the readiest way.

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