ratrix to the tangent: a result which agrees with, and includes, the known and fundamental property (397, XVII.) of the angle H, in connexion with the Rectifying Developable (396); and also the analogous property of the newer angle J, connected (8.) with what it has been above proposed to call the Circumscribed Developable.

(22.) We shall soon return briefly on the theory of that new developable surface (8.), and of the new locus (of the osculating circle, or envelope of the osculating sphere) to which it has been said to be circumscribed: but may here observe, that if we write for abridgment (comp. VIII. and XXIII.),

XXXIII. . . n =—= =p' + cot H = cot J sec P,

עיז

RK'
P

then what has been called the coefficient of non-sphericity (comp. 395, (14.) and (16.)) is easily seen to have by XIV. the values,

whence also the deviation of a near point P, of the curve, from the osculating sphere at P, is ultimately (by 395, XXVII.).

and accordingly, the square of the vector p.σ is given now (comp. I.) by the expression,

in which

(po)=(po)3 — {r3S (σ - p)r"" -1},
12ra

r2S (σ-p)+"=S=1+ nrr1= &c., as above.

(23.) The same auxiliary scalar n enters into the following expressions for the arc, and for the scalar radii of the first and second curvatures, of the locus of the centre s of the osculating sphere, or of the cusp-edge of the polar developable (comp. 391, (6.), and 395, (2.)) :

XXXVI. . . ±√ nds = Arc of that Cusp-Edge (or of locus of s);

XXXVI... T1 = nr=r+p'r=

RR
ř

=

(Scalar) Radius of Curvature of same edge;

XXXVI"... r1 = nr = o'v ̄1 = ( Scalar) Radius of Second Curvature of same curve; these two latter being here called scalar radii, because the first as well as the second (comp. 397, V.) is conceived to have an algebraic sign. In fact, if we denote by K1 the centre of the osculating circle to the cusp-edge in question, its vector is (by the general formula 389, IV.),

0'3 Vo'o'

XXXVII... OK1 = K1 = σ + · = σ - nrrr' = p-p'rrr' + prv=σ- T1rt',

with the signification XXXVI. of ri; because by XXXIII. (comp. 397, XI'.), XXXVIII... o' = nrv, o" = n'rv + n(rv)' = n'rv — nrr-'r',

and therefore

XXXIX... o2 = - n2, Vo'o" = n2r ̄1T.

CHAP. III.] CURVATURES OF CUSP-EDGE OF POLAR SURFACE. 585

We may also observe that the relation σv gives (by 397, IV.),

according as the cusp-edge turns its concavity or its convexity towards the given

curve at P.

(24.) The radius of (first) curvature of that cusp-edge, when regarded as a positive quantity, is therefore represented by the tensor,

and as regards the scalar radius XXXVI". of second curvature of the same cuspedge, its expression follows by XXXVIII. from the general formula 397, XXVII., which gives here,

the two scalar derivatives, n' and n", which would have introduced the derived vectors and r, or D,5p and Dp, of the fifth and sixth orders, thus disappearing from the expressions of the two curvatures of the locus of the centre s of the osculating sphere, as was to be expected from geometrical* considerations.

(25.) For the helix, the formula XXXVII. gives k1 = p, or K1 =P; we have then thus, as a verification, the known result, that the given point P of this curve is itself the centre of curvature K1 of that other helix (comp. 389, (3.), and 395, (8.)), which is in this case the common locus of the two coincident centres, K and s. It is scarcely necessary to observe that for the helix we have also J= H.

(26.) In general, the rectifying plane of the locus of s is parallel to the rectifying plane of the given curve, because the radii of their osculating circles are parallel; the rectifying lines for these two curves are therefore not only parallel but equal; and accordingly we have here the formula,

which will be found to agree with this other expression (comp. 397, XVII.),

the upper or lower sign being taken, according as the new curve is concave (as in Figs. 81, 82), or is convex at s (comp. (23.)), towards the old (or given) curve at P: and the new angle H1 being measured in the new rectifying plane, from the new

In fact, n represents here the velocity of motion of the point s along its own locus, while r1 and 1 represent respectively the velocities of rotation of the tangent and binormal to that curve: so that nr and nr must be, as above, the radii of its two curvatures.

tangent o' or nrv, to the new rectifying line X1, and in the direction from that new tangent to the new binormal vi, or (comp. XL.) to a line from s which is equal to the vector of second curvature r-r of the given curve, multiplied by a positive scalar, namely by Tn-1, or by the coefficient n-1 taken positively.

(27.) The former rectifying line λ touches the cusp-edge of the rectifying developable (396) of the given curve, in a new point R (comp. Fig. 81), of which by 397, (45.), and by XV., the vector from the given point is, generally,

with the verification that this expression becomes infinite (comp. 397, (49.), (50.)), when the curve is a geodetic on a cylinder.

(28.) In general, the vector OR of the point of contact R, which vector we shall here denote by v, may be thus expressed,

XLVII... v=OR=p+lUX, if XLVIII... l =

and because (r)'= (rr-1)'r, by VII'., its first derivative is,

XLIX. . . v=rλ

sin H

- TX

=

H'

(rr-1)' i

=UX cosec H ( sin H)' = UX (l′ + cos H);

in which however the new derived scalar l' involves H", and so depends on 1: while the scalar coefficient l itself represents the portion (+PR) of the rectifying line, intercepted between the given curve, and the cusp-edge (27.) of the rectifying developable, and considered as positive when the direction of this intercept PR coincides with that of the line +λ, but as negative in the contrary case.

(29.) For abridgment of discourse, the cusp-edge last considered, namely that of the rectifying developable, as being the locus of a point which we have denoted by the letter R, may be called simply "the curve (R);" while the former cusp-edge (23.), or that of the polar developable, may be called in like manner “the curve (s);" the locus of the centre K of (absolute) curvature may be called "the curve (K)" and the given curve itself (comp. again Figs. 81, 82) may be called, on the same plan, "the curve (P)."

(30.) The arc RR,, of the curve (R), is (by XLIX., comp. XXXVI.),

this are being treated as positive, when the direction of motion along it coincides with that of +X.

(31.) The expression VII. for X', combined with the former expression 397, XVI. for A, gives easily by the general formula 389, IV., LI... Vector of Centre of Curvature of the Curve (R)

(32.) We see, at the same time, that the angular velocity of the rectifying line

A, or of the tangent to this curve (R), is represented by H'; or that the small

CHAP. III.] CURVATURES OF EDGE OF RECTIFYING SURFACE. 587

angle between two such near lines, λ and As, is nearly equal to sH', or to H,- H: while the vector axis (VX'X-1) of rotation of the rectifying line, set off from the point R, has H'Ur', or – H'rr', for its expression.

(33.) As regards the second curvature of the same curve (R), we may observe that the expression (comp. VII. and LI.),

LIII... A"= (r-1)′′r + (r−1)"rv + r·1 (rr ̄1)'7′ = (r-1)"7 + (r ̄1)"rv + VAX', combined with the parallelism (XLIX.) of v' to λ, gives, by the general formula 397, XXVII.,

LIV... Radius of Second Curvature of Curve (R)

with the verification, that while l'+ cos H represents, by (30.), the velocity of motion along this curve (R), Tλ represents, by 397, (3.), the velocity of rotation of its osculating plane, namely the rectifying plane of the given curve (P): and it is worth observing, that although each of these two radii of curvature, LII. and LIV., depends on through l′ (28.), yet neither of them depends on TM (comp. (24.)). As another verification, it can be shown that the plane of the two lines and from P, namely the plane,

LIV'... Sr'λ(w - p) = 0,

which is the normal plane to the rectifying developable along the rectifying line, and contains the absolute normal to the given curve (P), touches its own developable envelope along the line RH, if H be the point determined by the formula 397, XXXVIII., or the point of nearest approach of a radius of curvature (r) of that given curve to its consecutive (comp. (6.); this line RH must therefore be the rectifying line of the curve (R): and accordingly (comp. 397, XVII.), the trigonometric tangent of its inclination to the tangent RP to this last curve has for expression (abstracting from sign),

LIV"... tan PRH = PH: PR=+l-'r sin2 H=+rH' sin H = T\-1H'

=

Radius (LIV.) of Second Curvature of Curve (R). Radius (LII.) of First Curvature of same Curve (34.) Without even introducing, we can assign as follows a twisted cubic (comp. 397, (34.)), which shall have contact of the fourth order with the given curve at P; or rather an indefinite variety of such cubics, or gauche curves of the third degree. Writing, for abridgment,

* A result substantially equivalent to this is deduced, by an entirely different analysis, in the above cited Memoir of M. de Saint-Venant, and is illustrated by geometrical considerations: which also lead to expressions for the two curvatures (or, as he calls them, the courbure and cambrure), of the cusp-edge of the rectifying developable; and to a determination of the rectifying line of that cusp-edge.

in which e is an arbitrary but scalar constant, represents evidently, by its form, a cone of the second order, with its vertex at the given point P; and this cone can be proved to have contact of the fourth order with the curve at that point: or of the third order with the cone of chords from it (comp. 397, (31.), (32.)). In fact the coefficients will be found to have been so determined, that the difference of the two members of this equation LVII. contains so as a factor, when we change to ps, as given by the formula I., or when we substitute for xyz their approximate values for the curve, as functions of the arc s; namely, by the expressions IV. for "", and 397, VI. for r",

where the terms set down are more than sufficient for the purpose of the proof.

24

It

may be added that the coefficient of in ys, which is the only one at all complex here, may be transformed as follows:

LVIII'... Srr'"" = — (r-1)" — r-1\2 = r−3 S + p (r¬3r-1)' ;

S being that scalar for which (or more immediately for its excess over unity) several expressions have lately been assigned (22.), and which had occurred in an earlier investigation (395, (14.), &c.).

(35.) With the same significations LV. of the three scalars xyz, this other equation,

or

LIX... 18ry (3x − r'y)2 = (9 + r22 - 3rr" – 3r2r-2) y2,

LIX'... 2ry-(x-'y)2 = (1 − fr$ (r3)” – žr2x2) y2,

will be found to be satisfied when we substitute for x and y the values LVIII. of and ys, and neglect or suppress s5; it therefore represents an elliptic (or hyperbolic) cylinder, which is cut perpendicularly, by the osculating plane to the given curve at P, in an ellipse (or hyperbola), having contact of the fourth order with the projection (comp. 397, (9.)), of that given curve upon that osculating plane: and the cylinder itself has contact of the same (fourth) order with the curve in space, at the

In the language of infinitesimals, the cone LVII. contains five consecutive points of the curve, or has five-point contact therewith: but it contains only four consecutive sides of the cone of chords from the given point, or has only four-side contact with that cone, except for one particular value of the constant, e, which we shall presently assign. It may be observed that xyz form here a (scalar) system of three rectangular co-ordinates, of the usual kind, with their origin at the point P of the curve, and with their positive semiaxes in the directions of the tangent 7, the vector of curvature r', and the binormal v.

It might have been observed, in addition to the eight forms XXXIV., that we have also,

XXXIV'... S-1 Rr1 cot J=n cot H

=

(9, 10).