XXXI... rl=b1 sina, rib-1 sin a cosa;

while the common centre (395), of the osculating circle and sphere, has now for its vector (comp. 389, (3.)),

XXXII. . . * = σ = p-beatß cosec2 a = b cot a (at - eatß cot a);

b being here the radius of the cylinder, but a denoting still the constant inclination of the tangent (p′) to the axis (a).

(3.) The rectifying line (396), considered merely as to its position, being the line of contact of the rectifying plane (396, XIV.) with its own envelope, is represented by the equations,

XXXIII... 0 =ST' (w− p) = ST" (w−p), or XXXIII'. . . 0 = Vλ(w − p), with the signification XVI. of λ; and accordingly, if we treat the rectifying planes as emanants, or change n to r', we find the value 0 = Vr”7'-1 = λ, which shows also that in the passage from P to P, the rectifying plane turns (nearly) round the rectifying line, through a small angle=sTλ, or with a velocity of rotation represented by the tensor,

XXXIV... Tλ = √ (r-2 + r ̃2) = r−1 cosec Hr1 sec H;

so that what we have called the rectifying vector, A, coincides in fact (by the general theory of emanants) with the vector axis (396, (14.)) of this rotation of the rectifying plane: as the vector of second curvature (r-17) has been seen to be, in the same full sense (comp. (1.)), the vector axis of rotation of the osculating plane, when velocity, direction, and position are all taken into account.

(4.) When the derivative s′ of the arc is only constant, without being equal to unity (comp. 395, (12.)), the expression XVI. may be put under this slightly more general form,

"

XXXV... λ= V = V

d3p dsd2p

and accordingly for the helix (2.) we have thus the values,

XXXVI... λ = as'-1 = ab-1 sin a = ar-1 cosec a, Uλ= a;

the rectifying line is therefore, for this curve, parallel to the axis, and coincides with the generating line of the cylinder, as is otherwise evident from geometry. The value, Tλ=b1 sin a, of the velocity of rotation of the rectifying plane, which is here the tangent plane to the cylinder, when compared with a conceived velocity of motion along the curve, is also easily interpreted; and the formule XVII., XVIII. give, for the same helix (by XXXI.), the values,

XXXVII... '=0, H= a, P=0.

(5.) The normal (or the radius of curvature), as being perpendicular to the rectifying plane, revolves with the same velocity, and round a parallel line; to determine the position of which new line, or the point H in which it cuts the normal, we have only to change ʼn to ' in the formula 396, XXXII., which then becomes,

CHAP. III.] AXIS OF DISPLACEMENT OF NORMAL.

565

the vector of rotation (396, (9.)) of the normal is therefore a line || and =λ, which divides (internally) the radius (r) of curvature into the two segments,*

XXXIX... PH = r sin2 H, HK = r cos2 H;

*

namely, into segments which are proportional to the squares (r-2 and r2) of the first and second curvatures.

(6.) At the same time, what we have called generally the vector of translation of an emanant line becomes, for the normal (by 396, (10.), changing 0 to λ), the line

and the indefinite right line, or axis, through that point н,

XLI... 0 = Vλ(w-wo), or XLI'... 0=Vλ (w− p cos2 H− k sin2 H), along which axis the normal moves, through the small line si, while it turns round the same axis (as before) through the small angle sTX, may be called (comp. again 396, (10.)) the Axis of Displacement of the Normal (or of the radius of curvature). (7.) As a verification, for the helix (2.) we have thus the values,

XLII... PH=b, wo = Pt-beatẞ= bat cot a, i = a cos a;

so that the axis of displacement (6.) coincides with the axis (a) of the cylinder, as was of course to be expected.

(8.) When the given curve is not a helix, the values VI., XVI., XXXVIII., and XL., of 7", λ, wo, and i, enable us to put the expression I. for p, under the form,

the curve therefore generally deviates, by this last small vector of the third order, namely by that part of the term s3r" which has the direction of the normal r', or of -', and which depends on r', from the osculating helix,

XLIV...

· · · ws = wo + Si + εsλ(p − wo),

and from the osculating right cylinder,

XLV... TVA (ww。) = sin H,

whereon that helix is traced, and of which the rectifying line (XXXIII.) is a side, while its axis of revolution (comp. (7.)) is the axis of displacement (XLI.) of the normal.

(9.) Another general transformation, of the expression I. for the vector of the curve, is had by the substitution,

in which t is a new scalar variable; for this gives the new form,

This law of division of a radius of curvature into segments, by the common perpendicular to that radius and to its consecutive, has been otherwise deduced by M. de Saint Venant, in the Memoir already referred to.

and therefore shows that the curve deviates, by this other small vector of the third order,

XLVIII... ft3r-1v = Js3r-177',

that is, by the part of the term ¿s3-” which has the direction of the binormal v, and which depends on r, from what we propose to call the Osculating Parabola, namely that new auxiliary curve of which the equation is,

or from the parabola which osculates at the given point P, to the projection of the given curve on its own osculating plane.

(10.) And because the small deviation XLVIII. of the curve from the parabola is also the deviation of the same curve from this last plane, if we conceive that a near point of the curve is projected into three new points Q1, Q2, Q3, on the tangent, normal, and binormal respectively, we shall have the limiting equation,

the sign of this scalar quotient being determined by the rules of quaternions.

(11.) But we may also (comp. 396, (17.), (18.)) employ this third general transformation of I., analogous to the forms XLIII. and XLVII.,

with the value XI. of v'; in which the sum of the two first terms gives the vector of the point of the osculating circle, which is distant from the given point PP, by an are of that circle equal to the arc s of the given curve; and the third term,

LII... Js3v'7 = js3 (7′′ + p-27) = − Js3r-17'7′ + £s3r ̄1v,

which represents the deviation from the same circle, measured in a direction (comp. IX. or X.) tangential to the osculating sphere, is (as we see) the vector sum of two rectangular components, which represent respectively the deviations of the curve, from the osculating helix (8.), and from the osculating parabola (9.).

(12.) It follows, then, that although neither helix nor parabola has in general complete contact of the third order with a given curve in space, since the deviation from each is generally a small vector of that (third) order, yet each of these two auxiliary curves, one on a right cylinder XLV., and the other on the osculating plane, approaches in general more closely to the given curve, than does the osculating circle while circle, helix, and parabola have, all three, complete contact of the second* order with the curve, and with each other.

* It appears then that we may say that the helix and parabola have each a contact with the curve in space, which is intermediate between the second and third orders or that the exponent of the order of each contact is the fractional index, 2ț. But it must be left to mathematicians to judge, whether this phraseology can properly be adopted.

CHAP. III.] SECTION OF SURFACE OF TANGENTS.

567

(13.) As regards the geometrical signification of the new variable scalar, t, in the equation XLIX. of the parabola, that equation gives,

r't
3r

r't

12

=1+

3r 2,2

LIII... Tw't = T<
T{( 1 + 1 ) + + &'} =

and therefore (to the present order of approximation),

LIV... Arc of Osculating Parabola (from wo to wt)

if then an arcs be thus set off upon the parabola, with the same initial point P, and the same initial direction, and if this parabolic arc, or itз chord w; — wo, be obliquely projected on the initial tangent 7, by drawing a diameter of the parabola through its final point, the oblique tangential projection so obtained will be = tr by XLIX.; and its length, or the ordinate to that diameter, will be the scalar t.

(14.) And as regards the direction of the diameter of the osculating parabola, drawn as we may suppose from P, if we denote for a moment by D its inclination to the normal + r', regarded as positive when towards the tangent +7, we have (by XLIX. and XVIII.) the formula,

which is an instance of the reducibility, above mentioned, of all affections of the curve depending on s3, to a dependence on the two angles, H and P.

(15.) Some of these affections, besides the direction of the rectifying line λ, can be deduced from the angle H alone. As an example, we may observe that the vector equation of the surface of tangents is of the form,

LVI... ws, t = ps+tp's = ps+tTs,

in which s and t are two independent and scalar variables, and

+ terms depending on s4 in ps. If then we cut this developable LVI. by the plane,

LVIII... ST(w− p)=-c= any given scalar constant,

which is, relatively to the surface, a normal plane at the extremity of the tangential vector cr from P, while this tangent is also a generating line, we get thus a principal* normal section, of which the variable vector has for its approximate expression,

LIX... ws = (p + cr) + (cs +. .) 7' + (}cs2r-1 + . .) v ;

the terms suppressed being of higher orders than the terms retained, and having no influence on the curvature of the section. We find then thus, that the vector of the centre of the osculating circle to this normal section of the surface of tangents to the given curve is, rigorously,

* Some general acquaintance with the known theory of sections of surfaces is here supposed, although that subject will soon be briefly treated by quaternions.

so that the locus of all such centres is the rectifying line XXXIII'. And if, in particular, we make c=r, or cut the developable at the extremity of the tangential vector rr, the expression LX. becomes then p+r7+rUv; which expresses that the radius of the circle of curvature of this normal section of the surface is precisely what has been called the Radius (r) of Second Curvature, of the given curve in space. But this radius (r=rtan H) depends only on the angle H, when the radius (r) of (absolute) curvature is given, or has been previously determined. (16.) The cone of the second order, represented by the quaternion equation, LXI... 0=2rST (w− p) Sv (w− p) + (Vr (w − p))2,

has its vertex at the given point P, and rests upon the circle last determined; it is then the locus of all the circles lately mentioned (15.), and is therefore (in a known sense) an osculating oblique cone to the developable surface of tangents: its cyclic normals (comp. 357, &c.) being 7 and 7+2rv, or 7 and rr + 2rUv. But, by 394, (30.), the osculating right cone to this cone LXI., and therefore also (in a sense likewise known) to the surface of tangents itself, is one which has the recent locus of centres (15.), namely the rectifying line (A), for its axis of revolution, while the tangent (7) to the curve is one of its sides: its semiangle is therefore = H, and a form of the quaternion equation of this osculating right cone is the following (comp. XLV.),

LXII... TVUA (w− p) = sin H.

(17.) The right cone LXII., which thus osculates to the developable surface of tangents LVI., along the given tangent 7, osculates also along that tangential line to the cone of parallels to tangents, which has its vertex at the given point P; as is at once seen (comp. 394, (30.)), by changing p' and p" to r' and r", in the general expression Vp'p" (393, (6.), or 394, (6.)), for a line in the direction of the axis of the osculating circle to a curve upon a sphere. And the axis of the right cone thus determined, namely (again) the rectifying line (^), intersects the plane of the great circle of the osculating sphere, which is parallel to the osculating plane, in a point L of which the vector is,

LXIII... OL= p + rpλ = p + rr'z +rpv.

(18.) We have thus, in general, a gauche quadrilateral, PKSL, right-angled except at L, with the help of which one figure all affections of the curve, not depending on s1, can be geometrically represented or constructed: although it must be observed that when r = 0, which happens for the helix (XXXVII.), the osculating circle is then itself a great circle of the osculating sphere, and the points P and L, like the points K and s, coincide.

(19.) In the general case, it may assist the conceptions to suppose lines set off, from the given point P, on the tangent and binormal, as follows:

LXIV... PT = BL=rr'T; PB =TL=KS=rpv;

for thus we shall have a right triangular prism, with the two right-angled triangles, TPK and LBS, in the osculating plane and in the parallel plane (17.), for two of its faces, while the three others are the rectangles, PKSB, PBLT, KSLT, whereof the two first are situated respectively in the normal and rectifying planes.