CHAP. III.] RULED CUBIC LOCUS, CUBIC CYLINDer.

619

(13.) Instead of thus introducing, as data, the derivatives of the two radii of curvature, r and r, taken with respect to the arc, s, it may be more convenient in many applications to treat the two co-ordinates y and z of the curve as functions of the third co-ordinate x, assumed as the independent variable: and so to write (comp. (6.)) these new developments,

2r

6

24 120'

6rr

+
24 120

;

and then the equation of the quadric cone XXXI. will be found to become (in xyz),

while the cubic surface XXIX. will also come to be represented by an equation of the same form as before, namely (in xyz) by the following,

XXXV. . . xz (y + hz) — rz2 = ay3 + by2z + cyz2 + ez3,

in which the coefficients are,

(14.) Whichever set of expressions for the coefficients we may adopt, some general consequences may be drawn from the mere forms of the equations, XXXI. and XXIX., or XXXIII. and XXXV., of the quadric cone and cubic surface, considered as two loci (12.) of the osculating twisted cubic to a given curve of double curvature. Thus, if we eliminate ac (comp. 398, (41.)) from XXIX. by XXXI., or zz by XXXIII. from XXXV., we get an equation between b, c, or between y, z, which rises no higher than the third degree, and is of the form,

XXXVII... 2rz2 = ay3 + b,y2z+cyz2 + e ̧z3,

with the same value of a as before; such then is the equation of the projection of the twisted cubic, on the normal plane to the curve; and we see that, as was to be expected, the plane cubic thus obtained has a cusp at the given point P, which (when we neglect s7 or x7) coincides with the corresponding cusp of the projection of the given curve of double curvature itself, on the same normal plane.

(15.) The equation XXXVII. may also be considered as representing a cubic cylinder, which is a third locus of the twisted cubic; and on which the tangent PT

to the curve is a cusp-edge, in such a manner that an arbitrary plane through this line, suppose the plane

XXXVIII... 3rz=vy,

where is any assumed constant, cuts the cylinder in that line twice, and a third time in a real and parallel right line, which intersects the quadric cone in a point at infinity (because the tangent PT is a side of that cone), and in another real point, which is on the twisted cubic, and may be made to be any point of that sought curve, by a suitable value of : in fact, the plane XXXVIII. touches both curves at P, and therefore intersects the cubic curve in one other real point. And thus may fractional expressions (comp. (8.)) for the co-ordinates of the osculating cubic be found generally, which we shall not here delay to write down.

(16.) Without introducing the cubic cylinder XXXVII., it is easy to see that any plane, such as XXXVIII., which is tangential to the given curve at P, cuts the cubic surface XXXV. in a section which may be said to consist of the tangent twice taken, and of a certain other right line, which varies with the .direction of this secant plane, so that the locus XXXV. or XXIX. is a Ruled Cubic Surface, with the given tangent PT for a singular* line, which is intersected by all the other right lines on that surface, determined as above: and if we set aside this line, the remaining part of the complete intersection of that cubic surface with the quadric cone XXXIII. or XXXI. is the twisted cubic sought. We may then consider ourselves to have completely and generally determined the Oscuculating Twisted Cubic to a curve of double curvature, without requiring (as in 398, (41.)), the solution of any cubic or other equation. †

(17.) As illustrations and verifications, it may be added that the general ruled cubic surface, and cubic cylinder, lately considered, take for the case of the helix (2.), the particular forms,‡

If the cubic surface be cut by a plane perpendicular to the tangent PT, at any point T distinct from the point P itself, the section is a plane cubic, which has T for a double point; and this point counts for three of the six common points, or points of intersection, of the plane cubic just mentioned with the plane conic in which the quadric cone is cut by the same secant plane, because one branch, or one tangent, of the plane cubic at T touches the plane conic at that point, in the osculating plane to the given curve at P, while the other branch, or the other tangent, cuts that plane conic there.

It may be remarked that, by equating the second member of XXXVII. to zero, and changing y, z to b, c, we obtain generally the cubic equation, referred to in 398, (41.); and that by suppressing the term rc2 in XXIX., or the term - rz2 in XXXV., we pass, in like manner generally, from the cubic surface of recent subarticles, to the earlier cubic cone (4.).

ry

By suppressing the term - rz2, dividing by, and transposing, we pass for the case of the helix from the equation XXXIX. of the cubic locus, to the equation IX'. in the last Note to page 614; namely to the equation of that quadric cone which forms (in this example) a separable part of the general cubic cone, the other part being here the tangent plane (y = 0) to the right cylinder.

CHAP. III.] INVOLUTES AND evolutes in space.

621

and that accordingly these two last equations are satisfied, independently of w, when the fractional expressions XXI. are substituted for xyz.

400. The general theory* of evolutes of curves in space may be briefly treated by quaternions, as follows: a second curve (in space, or in one plane) being defined to bear to a first curve the relation of evolute to involute, when the first cuts the tangents to the second at right angles.

(1.) Let p and ☛ be corresponding vectors, OP and os, of involute and evolute, and let p′, oʻ, p”, ☛" denote their first and second derivatives, taken with respect to a scalar variable t, on which they are both conceived to depend. Then the two fundamental equations, which express the relation between the two curves, as above defined, are the following:

which express, respectively, that the point s is in the normal plane to the involute at P, and that the latter point is on the tangent to the evolute at s: so that the locus of P (the involute) is a rectangular trajectory to all such tangents to the locus of s (the evolute).

(2.) Eliminating -p between the two preceding equations, and taking their derivatives, we find,

III... Sp'o'=0, IV...S(☛−p)p”-p2=0, V... V(o - p)o” – Vp'o' = 0 ; whence also, VI... Sp'o'o" = 0.

(3.) Interpreting these results, we see first, by IV. combined with I. (comp. 391, (5.)), that the point s of the evolute is on the polar axis of the involute at P, and therefore that the evolute itself is some curve on the polar developable of the involute; and second, by VI. (comp. 380, I.), that this curve is a geodetic line on that polar surface, because the osculating plane to the evolute at s contains the tangent to the involute at P, and therefore also the (parallel) normal to the locus of evolutes. (4.) The locus of centres of curvature (395, (6.)) of a curve in space is not generally an evolute of that curve, because the tangents† KK' to that locus do not generally intersect the curve at all; but a given plane involute has always the locus just

* Invented by Monge.

It might have been remarked, in connexion with a recent series of sub-artiticles (397), that this tangent KK' or ' is inclined to the rectifying line A, at an angle of which the cosine is,

- SUk'λ = ± R ̄1TX-1 = + sin H cos P;

upper or lower signs being taken, according as the second curvature r-1 is positive or negative, because Sx'λ=r1.

mentioned for one of its evolutes; and has, besides, indefinitely many others,* which are all geodetics on the cylinder which rests perpendicularly on that one plane evolute as its base.

(5.) An easy combination of the foregoing equations gives,

VII. . . (T(σ − p)) = − $ (U(o − p). (o' — p′)) = + So'Uo' = ± To',
VIII. . . dT( − P)=+Tảo ;

or with differentials,

whence by an immediate integration (comp. 380, XXII. and 397, LIV.),

IX. . . AT(σ − p) = ±ƒ Tdo = ± arc of the evolute :

this arc then, between two points such as s and s1 of the latter curve, is equal to the difference between the lengths of the two lines, PS and Pisi, intercepted between the two curves themselves.

(6.) Another quaternion combination of the same equations gives, after a few steps of reduction, the differential formula (comp. 335, VI.),

if then the involute be a curve on a given sphere, with its centre at the origin o, so that the evolute is a geodetic on a concentric cone, this differential X. vanishes, and we have the integrated equation,

XI... cos ors = const., or simply, XI'... OPS = const. ;

the tangents Ps to the evolute being thus inclined (in the case here considered) at a constant angle,† to the radii op of the sphere.

(7.) In general, if we denote by R the interval PS between two corresponding points of involute and evolute, we shall have the equation,

XII... (σ - p)2 + R2 = 0, or

XII'... T(☛ − p)= R ;

XIII'. . . D2R = ± TD0,

and the formula VII. may be replaced by the following,

XIII... R2+ σ2 = 0, or

in which the independent variable t is still left arbitrary.

(8.) But if we take for that variable the arc sost of the evolute, measured from some fixed point of that curve, we may then write,

XIV. . . t = f Tdo,

XV... dR1 = +dt,

XVI... DtRt = + 1;

Compare the first Note to page 534; from the formula of which page it now appears, that if the involute be an ellipse, with ẞ=OB and y = OC for its major and minor semiaxes, and therefore with the scalar equations,

(SB 1p)+(Sy 'p) = 1, Sẞyp=0,

the evolutes are geodetics on the cylinder of which the corresponding equation is,

(S3)+(Syo) = (32 — y2)§.

This property of the evolutes of a spherical curve was deduced by Professor De Morgan, in a Paper On the Connexion of Involute and Evolute in Space (Cambridge and Dublin Mathematical Journal for November, 1851); in which also a definition of involute and evolute was proposed, substantially the same as that above adopted.

CHAP. III.] INVOLUTES AS LIMITS OF ENVELOPES.

whence

XVII... Dt (Rtt)=0, and XVIII... Rt t = const. = Ro, the integral IX. being thus under a new form reproduced.

(9.) In this last mode of obtaining the result,

XIX... A PS = Rt-Ro=+t= + arc sost of evolute,

623

no use is made of infinitesimals,* or even of small differentials. We only infer, as in XVIII. (comp. 380, (9.)), that the quantity Rt is constant, because its derivative is null it having been previously proved (380, (8.)), as a consequence of our definition of differentials (320, 324) that if s be the arc and p the vector of any curve, then the equation ds = Tdp (380, XXII.) is rigorously satisfied, whatever the independent variable t may be, and whether the two connected and simultaneous differentials be small or large.

(10.) But when we employ the notation of integrals, and introduce, as above, the symbol fTds, we are then led to interpret that symbol as denoting the limit of a sum (comp. 345, (12.)); or to write, generally,

XX... Tdp = lim. ETAP, if lim. Ap = 0,

with analogous formulæ for other cases of integration in quaternions. Geometrically, the equation,

if s and t denote arcs of curves of which p and σ are vectors, comes thus to be interpreted as an expression of the well-known principle, that the perimeter of any curve (or of any part thereof) is the limit of the perimeter of an inscribed polygon (or of the corresponding portion of that polygon), when the number of the sides is indefinitely increased, and when their lengths are diminished indefinitely. (11.) The equations I. and XII. give,

XXII... So' (o − p) + RR' = 0,

the independent variable t being again arbitrary; but these equations XII. and XXII. coincide with the formulæ 398, LXXXIX. and XCI.; we may then, by 398, (79.) and (80.), consider the locus of the point P as the envelope of a variable sphere, namely of the sphere which has s for centre and R for radius, and is represented by the recent equation XII., if p=OP be the vector of a variable point thereon.

(12.) But whereas such an envelope has been seen to be generally a surface, which is real or imaginary (398. (79.)) according as R′2 + o'2 < or > 0, we have here by XIII. the intermediate or limiting case (comp. 398, CXXXI.), for which the circles

* In general, it may have been observed that we have hitherto abstained, at least in the text of this whole Chapter of Applications, from making any use of infinitesimals, although they have been often referred to in these Notes, and employed therein to assist the geometrical investigation or enunciation of results. But as regards the mechanism of calculation, it is at least as easy to use infinitesimals in quaternions as in any other system: as will perhaps be shown by a few examples, farther on.

† Compare the Note to page 516.