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396. When the arc (s) of the curve is made the independent variable, the calculations (as we have seen) become considerably simplified, while no essential generality is lost, because the transformations requisite for the introduction of an arbitrary scalar variable (t) follow a simple and uniform law (395, (11.), &c.). Adopting then the expression (comp. 395, IV.),

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III... 2+1=0, STT=0, STT" + T2 = 0,

we shall proceed to deduce some other affections of the curve, besides its spherical curvature (395, (18.)), which do not involve the consideration of the fourth power of the arc (or chord). In particular, we shall determine expressions for that known Second Curvature (or torsion), which depends on the change of the osculating plane, and is measured by the ultimate ratio of that change, expressed as an angle, to the arc of the curve itself; and shall assign the quaternion equations of the known Rectifying Plane, and Rectifying Line, which are respectively the tangent plane, and the generating line, of that known Rectifying Developable, whereon the proposed curve is a geodetic (382): so that it would become a right line, by the unfolding of this last surface into a plane. But first it may be well to express, in this new notation, the principal affections or properties of the curve, which depend only on the three first terms of the expansion I., or on the three initial vectors p, 7, 7', or rather on the two last of these; and which include, as we shall see, the rectifying plane, but not the rectifying line: nor what has been called above the second* curvature.

(1.) Using then first, instead of I., this less expanded but still rigorous expression (comp. 376, I.),

IV... ps = p + 8T+82u,T', with 0 = 1,

In a Note to a very able and interesting Memoir, "Sur les lignes courbes non planes" (referred to by Dr. Salmon in the Note to page 277 of his already cited Treatise, and published in Cahier XXX. of the Journal de l'Ecole Polytechnique), M. de Saint-Venant brings forward several objections to the use of this appellation, and also to the phrases torsion, flexion, &c., instead of which he proposes to introduce the new name, "cambrure." but the expression "second curvature" may serve us for the present, as being at least not unusual, and appearing to be suffieiently suggestive




and with the relations II. and III., we have at once the following system of three rectangular lines, which are conceived to be all drawn from the given point P of the


V... T unit tangent;

VI... r' vector of curvature (389, (4.);

and VII... v = TT' — — T'T = T'T1 = binormal (comp. 379, (4.));

7 being a line drawn in the direction of a conceived motion along the curve, in virtue of which the arc (s) increases; while r' is directed towards the centre of curvature, or of the osculating circle, of which centre K the vector is now,

VIII... OK = K = p − 7'-1 = p + r2y' = p+rUr',

if IX... r1=Tr' = curvature at P, or IX'... r = Tr'-1= radius of curvature; and the third line v (which is normal at P to the surface of tangents to the curve) has the same length (Tv = r1) as r', and is directed so that the rotation round it from 7 to r' is positive.

(2.) At the same time, we have evidently a system of three rectangular vector units from the same point P, which may be called respectively the tangent unit, the normal unit, and the binormal unit, namely the three lines,

X... Ut=t, Ur'=rr', Uv=rTT′ ;

the normal unit being thus directed (like r') towards the centre of curvature.

(3.) The vector equation (comp. 392, (2.)) of the circle of curvature takes now the form,

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with the verification that it is satisfied by the value,

XII... w = μ = 2k-p=p2r'1,

in which μ (comp. 395. (6.)) is the vector oм of the extremity of the diameter of

curvature PM.

(4.) The normal plane, the rectifying plane, and the osculating plane, to the curve at the given point, form a rectangular system of planes (comp. 379, (5.)), perpendicular respectively to the three lines (1.); so that their scalar equations are, in the present notation,

XIII... ST (w - p) = 0 ; XIV. . . Sr' (w - p) = 0 ;

XV... Sv (w-p)=0;

by pairing which we can represent the tangent, normal, and binormal to the curve, regarded as indefinite right lines; or by the three vector equations,

XVI... Vr (w-p)=0; XVII... Vr' (w− p)=0; XVIII... Vv (w− p) = 0. (5.) In general, if the two vector equations,

XIX... Vn (w− p)=0, and XIX'. . . Vns (ws − ps) = 0,

represent two right lines, PH and P,Hs, which are conceived to emanate according to any given law from any given curve in space, the identical formula,*

It is obvious that we have thus an easy quaternion solution of the problem, to draw a common perpendicular to any two right lines in space.

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shows that the common perpendicular to these two emanants, which as a vector is represented by either member of this formula XX., intersects the two lines in the two points of which the vectors are,

XXI... w = p +nS

(Ps − p)ns.

(P.- p)n

XXI'... w1 = ps + NoS ;

(6.) In general also, the passage of a right line from any one given position in space to any other may be conceived to be accomplished by a sort of screw motion, with the common perpendicular for the axis of the screw, and with two proportional velocities, of translation along, and of rotation round that axis: the locus of the two given and of all the intermediate positions of the line (when thus interpolated) being a Screw Surface, such as that of which the vector equation was assigned in 314, (11.), and was used in 372, (4.).

(7.) Again, for any quaternion, q, we have (by 316, XX. and XXIII.*) the two equations,

XXII. . . lUq = Lq.UVq,

comparing which we see that

XXII'. . . VUq=sin £q.UVq;

XXIII. . . VUq : lUq = sin / q : ≤ q = (very nearly) 1,

if the angle of the quaternion be small; so that the logarithm and the vector of the versor of a small-angled quaternion are very nearly equal to each other, and we may write the following general approximate formula for such a versor:

XXIV... Uq = (¿1Uq =) &VUq, nearly, if ≤ q be small ;

the error of this last formula being in fact small of the third order, if the angle be small of the first.

(8.) And thus or otherwise (comp. 334, XIII. and XV.), we may perceive that if the quaternion q have the form (comp. (5.)),

XXV... q = nsn1, with XXVI. . . n, = n + sn2 + • •,

and if we write for abridgment,

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in which last binomial, the first (or exponential) term alone influences the direction of the near emanant line (5.).

Although the expression XXII'. for VUq is here deduced from 316, XXIII., yet it might have been introduced at a much earlier stage of these Elements; for instance, in connexion with the formula 204, XIX., namely TV Uq=sin q.


(9.) At the same time, by supposing & to tend to 0, the formula XXI. gives, as a limit,

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for the vector of the point, say H, on the given emanant PH, in which that given line is ultimately intersected by the common perpendicular (5.), or by the axis of the screw rotation (6.); but the direction of that axis is represented by the versor U0, and the angular velocity of that rotation is represented by the tensor TO, if the velocity of motion (1.) along the given curve be taken as unity: we may therefore say that the vector 0 itself, or the factor which multiplies the arc, s, in the exponential term XXXI., if set off from the point н determined by XXXII., is the Vector of Rotation of the Emanant, whatever the law (5.) of the emanation may be.

(10.) And as regards the screw translation (6.), its linear velocity is in like manner represented, in length and in direction, by the following expression (obtained by limits from XX.),



(set off from H) = Vector of Translation of Emanant,

= projection of unit-tangent on screw-axis (or of 7 on ◊).

And the indefinite right line through the point H, of which this line is a part, may be called the Axis of Displacement of the Emanant.

(11.) It is easy in this manner to assign what may be called the Osculating Screw Surface to the (generally gauche) Surface of Emanants, or indeed to any proposed skew surface; namely, the screw surface which has the given emanant (or other) line for one of its generatrices, and touches the skew surface in the whole extent of that right line.

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(12.) It is however more important here to observe, that in the case when the surface of emanants is developable, the vector of translation vanishes; and that conversely this vector cannot be constantly zero, if that surface be undevelopable. The Condition of Developability of the Surface of Emanants is therefore expressed by the equation,

XXXIV. . . . =

0, or Sr0=0, or

XXXIV'... Snn'T = 0;

and accordingly this condition is satisfied (as was to be expected) when ŋ= 7, that is, for the surface of tangents.

(13.) In the same case, of ŋ: or || 7, the vector 0 of rotation becomes equal (by XXVII. and VII.) to the binormal v; and the expression XXXII., for the vector wo of the foot H of the axis reduces itself to p; and thus we might be led to see (what indeed is otherwise evident), that the passage from a given tangent to a near one may be approximately made, by a rotation round the binormal, through the small angle, sTv=sr ̄1 = arc divided by radius of curvature.

(14.) Instead of emanating lines, we may consider a system of emanating planes, which are respectively perpendicular to those lines, and pass through the same points of the given curve. It may be sufficient here to remark, that the passage from one to another of two such near emanant planes, represented by the equations,

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may be conceived to be made by a rotation through an angle = sTO, round the right line,


XXXVI... Sn (w− p) = 0, Sn'(w− p) — SNT= 0,
XXXVI. . . V0 (w− p) + n ̄1Snr = 0,

in which the plane XXXV. touches its developable envelope, and which is parallel to the recent vector 0, or to the vector of rotation (9.) of the emanant line; so that if an equal vector be set off on this new line XXXVI., it may be said to be the Vector Axis of Rotation of the Emanant Plane.

(15.) For example, if we again make ŋ=7, so that the equation XXXV. represents now the normal plane to the curve, we are led to combine the equation XIII. of that plane with its derived equation, and so to form the system of the two scalar equations,

XXXVII... ST (w− p) = 0, ST' (w − p) + 1 = 0,

whereof the second represents a plane parallel to the rectifying plane XIV., and drawn through the centre of curvature VIII.; and which jointly represent the polar axis (391, (5.)), considered as an indefinite right line, which is represented otherwise by the one vector equation,

XXXVIII... Vv (w−x) = 0, or XXXVIII'... Vv (w− p) = − T.

(16.) And if, on this indefinite line, we set off a portion equal to the binormal v, such portion (which may conveniently be measured from the centre K) may be said, by (14.), to be the Vector Axis of Rotation of the Normal Plane; or briefly, the Polar Axis, considered as representing not only the direction but also the velocity of that rotation, which velocity = Tv=r1 = the curvature (IX.) of the given curve: while another portion Uv = the binormal unit (2.), set off on the same axis from the same centre of curvature, may be called the Polar Unit.


(17.) This suggests a new way of representing the osculating circle by a vector equation (comp. (3.), and 316), as follows:

XXXIX. . . w2 = k + εsv (p − k) = p + (εsv − 1) p'-1

=P+ST+ (ε3 − 1 − sv) 7′-1

= P+ST + 1827′ + (e3v − 1 − sv − } s2x2) 7'−1;

which agrees, as we see, with the expression I. or IV., if s3 be neglected; and of which, when the expansion is continued, the next term is,

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(18.) The complete expansion of the exponential form XXXIX., for the variable vector of the osculating circle, may be briefly summed up in the following trigonometric (but vector) expression:

in which,

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XLII... p - k=-r2r', and Uv. (p −k)=rvr'-\ = TT ;

so that we may also write, neglecting no power of s,

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and if this be subtracted from the full expression for the vector ps, the remainder may be called the deviation of the given curve in space, from its own circle of curvature : which deviation, as we already see, is small of the third order, and will soon be de

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