« PreviousContinue »
(20.) The first of the values XXIV., for the auxiliary scalar r, gives the expres sion (if v=pp, as it is for a central surface of the second order),
XLII... σ = p+r-1v = (1 + r ̄1p) p = r1 (p + r) p ;
whence, by inversion, and operation with 4,
XLIII...p=r(p+r) ̄1o ;
and therefore, because Spv = 1,
XLIV... v=r (p+r) ̄1po;
XLV... r2 = $((p + r) ̄1 o. (p + r) ·1po) = S. o (p+r)-200.
(21.) The following is a quite different way of arriving at this result, which is also useful for other purposes. Considering σ as the vector os of a point s on the Surface of Centres, that is, on the locus of all the centres of curvature of principal normal sections, the vector (say v) of the Reciprocal Surface is connected with σ (comp. 373, (21.)) by the equations of reciprocity,*
where is, as before, a tangent to the line of curvature: so that, if a denote the variable vector of the normal plane to this last curve, the equation of that plane (comp. 369, IV.) may be thus written,
L... Sv (w-p)=0.
This normal plane, to the line of curvature at P, is therefore at the same time the tangent plane to the surface of centres at s, as indeed it is known to be, from simple geometrical considerations, independently of the form of the given surface, which remains here entirely arbitrary.
(22.) The expression XLIX. for v gives generally the relation,
LI... Spv = 1;
giving also, by 410, V. and VI., these two other equations,
* It is understood that do and du, in the differential equations XLVII, XLVIII., are in general only obliged to have directions tangential to the surface of centres, and to its reciprocal, at corresponding points: so that the equations might be in some respects more clearly written thus, Svdo = 0, Sadv = 0, the mark d being reserved to indicate changes which arise from motion along a given line of curvature, while & should have a more general signification. Accordingly if, in particular, we write dp vdp, for a variation answering to motion along the other line, and denote the two radii of curvature for the two directions dp and do by R1 and R2, we shall have by II., R1 dp + dUv = 0, R1⁄2 ̄1êp+¿Uv=0, and therefore by L.,
do = dR1. Uv, do=dp+d(R1Uv) = (1 − R1R1⁄2 ̄1) vdp + ¿R1. Uv;
so that we have both Sdpdσ = 0, and Sdpdσ = 0, and therefore the tangent dp or r to the given line of curvature has the direction of the normal v to the corresponding sheet of the surface of centres, as is otherwise visible from geometry. And when we have thus found an equation of the form tur, operation with S. gives by XLVI. the value Spr, as in XLIX., because o pl|v + T.
CHAP. III.] RECIPROCAL of surface of centres.
LII... Svv = 0, and LIII... Svvpv = 0,
which are still independent of the form of the given surface.
(23.) But if that surface be a central quadric,* then the equation LI. may be thus written,
combining which with LII. and LIII., we derive the expressions:
wherein fu= Supv, and Fu= Sup-1v, as usual.
(24.) Operating with S. on this last expression for p, and attending to LII. and LIV., we find the following quaternion forms of the Equation of the Reciprocal of the Surface of Centres :
whereof the second, when translated into co-ordinates, is found to agree perfectly with a known† equation of the same reciprocal surface. (25.) Differentiating the form LX., and observing that
d. v=4Sv3dv, dfv = 2Spvdu, dFv = 2Sp-1udv,
we find, by comparison with XLVI. and XLVIII., the expression:
or finally by XLIX., with the recent signification XXIV. of r,
LXIV... σ = r2 (p+r)2 p ́1v,
and, for the same reason, the equation
because LXV. . . r =ƒÜr =ƒUv:
LX. of the reciprocal surface may be thus
LXVI. . . Fu+rly2 = 1, while LXVI... fv + rv2 = 0.
(26.) Inverting the last form for ø, and using again the relation XLVI., we first find for u the expression,
LXVII... v = r2 (p + r) ̄2 po ;
and then are conducted anew to the equation XLV., or to the following,
LXVIII. . . 1 = S.σ(1+r ̄1p) 2po.
Compare the last note to page 672; see also the use made of this known name "quadric," for a surface of the second order (or degree), in the sub-articles to 399 (pages 614, &c.).
The equation alluded to, which is one of the fourth degree, appears to have been first assigned by Dr. Booth, in a Tract on Tangential Co-ordinates (1840), cited in page 163 of Dr. Salmon's Treatise. See also the Abstract of a Paper by Dr. Booth, in the Proceedings of the Royal Society for April, 1858.
(27.) This last equation may also be thus written,
LXIX... 1 S. o(1+r-1p)-3 (p+r·1p2) o ;
but by combining XLIII. LI. LXVII. we have,
LXX. . . 1 = =(Sv) S. o (1+r ̄1p) ̈3¢o ;
a result which may be otherwise and more directly deduced, under the form Svv=0 (LII.), from the expressions XLIV. LXVII. for v and v.
(28.) If we write,
LXXII... r = Udp, r'=U(vdp), and therefore LXXIII. . . rr' = Uv,
and being thus unit-tangents to the lines of curvature, the equation III. gives, generally,
LXXIV... 0=Vrd(rr′) = − dr'+TSr'dr, whence LXXIV'... dr' | r;
of which general parallelism of dr' to r, the geometrical reason is (comp. again III.) that a line of curvature on an arbitrary surface is, at the same time, a line of curvature on the developable normal surface which rests upon that line, and to which the vectors or vdo are normals.
(29.) The same substitution LXXIII. for Uv gives by II., if we denote by s the are of a line of curvature, measured from any fixed point thereof, so that (by 380, (7.), &c.),
LXXV... Tdp=ds, dp = rds, D1 = 7,
the following general expression for the curvature of the given surface, in the direction of the given line, which by LXXIV. is also that of dr':
LXXVI... R-1=S. TD,(TT') = - S. rr'D ̧r = S (Uv-1‚ D,3p);
but D.2p is (by 389, (4.)) what we have called the vector of curvature of the line of curvature, considered as a curve in space, and R-1Uv is the corresponding vector of curvature of the normal section of the given surface, which has the same tangent at the given point: hence the latter vector of curvature is (generally) the projection of the former, on the normal v to the given surface.
(30.) In like manner, if we denote for a moment by R-1 the curvature of the developable normal surface (28.), for the same direction r, the general formula II. gives, by LXXIV.,
LXXVII... R-1 TD,7'-ST'Dsr = S. 7'-1D,'p;
the vector R1 of this new curvature is therefore the projection on the new normal 7', of the vector of curvature D2p of the given line of curvature. But we shall soon see that these two last results are included in one more general,* respecting all plane sections of an arbitrary surface.
(31.) The general parallelism LXXIV'. conducts easily, for the case of a central quadric, to a known and important theorem, which may be thus investigated. Writing, for such a surface,
LXXVIII. . . r =ft, r'=ft',
Namely in Meusnier's Theorem, which can be proved generally by quaternions with about the same ease as the two foregoing cases of it.
CHAP. III.] SURFACE OF CENTREs as an envelope.
so that r retains here its recent signification LXV., and r' is the analogous scalar for the other direction of curvature, we have by LXXIV. the differential,
LXXIX... dr' = 2Spr'dr' = 2Srpr'Sr'dr = 0,
because Sror' = 0, by 410, XI.
(32.) We have then the relation,
LXXX...ƒU (vdp) = fr' = r' = const.;
that is to say, the square (r'-1) of the scalar semidiameter (D') of the surface, which is parallel to the second tangent (r′), is constant for any one line of curvature (T); and accordingly (comp. XXII., and the expression 407, LXXI. for ƒ Uv1), the value of this square is,
LXXXI... (ƒUvdp)-1 =r'-1 = a2 — a'2 = b2 — b12 = c2 — c22,
if a', b', c' be the scalar semiaxes of the confocal, which cuts the given quadric (abc) along the line of curvature, whereof the variable tangent is T.
(33.) This constancy of fUvdo may be proved in other ways; for instance, the general equation Sydvdp = 0 gives, for a line of curvature on an arbitrary surface,
LXXXII... dv = vSv-1dv + dpS
LXXXIV... S. dpp (vdp) = 0, because dv = pdp;
while for a central quadric (fp = 1, ppv) it is easy to show that we have also,
LXXXV. . . 4 (vdp) = Vpdpf (vUdp);
hence, for such a surface, if we suppose for simplicity that ds or Tdp is constant, which gives Vvd2p || dp, we have,
LXXXVI... df(vdp) = 28(p(vdp).d(vdp)) = 2Sv-1dv.ƒ(vdp),
a differential equation of the second order, of which a first integral is evidently, LXXXVII. . . f(vdp)=Cv2dp2, or LXXXVII'.. . ƒ U (vdp) = C′ = const. (34.) But we see that the lines of curvature on a central quadric are thus included in a more general system of curves on the same surface, represented by the differential equation LXXXVI., of which the complete integral would involve two constants and which expresses that the semidiameters parallel to those tangents to the surface, which cross any one such curve at right angles, have a common square, and therefore (if real) a common length, so that (in this case) they terminate on a sphero-conic.
(35.) Admitting however, as a case of this property, the constancy LXXX. of the scalar lately called r', namely the second root of the quadratic XXXIV. or XXXV., of which the coefficients and the first root r vary, in passing from one point to another of what we may call for the moment a line of first curvature, we have only to conceiver and v to be accented in the equations LXVI. LXVI′., in order to perceive this theorem, which perhaps is new:
* Compare the sub-articles (6.) (7.) (8.) to 219, in page 231.
The Curve on the Reciprocal (24.) of the Surface of Centres of curvature of a central quadric, which answers to the second curvature of that given surface for all the points of a given line of first curvature, or which is itself in a known sense the reciprocal (with respect to the given centre) of the developable normal surface (28.) which rests upon that line, is the intersection of two quadrics; whereof one (LXVI.) is a cone, concyclic with the given surface (fp=1); while the other (LXVI.) is a surface concyclic with the reciprocal of that given quadric (Fv = 1).
(36.) Again, the scalar Equation of the Surface of Centres (21.) may be said to be the result of the elimination of r-1 between the equations LXVIII. and LXXL, whereof the latter is the derivativet of the former with respect to that scalar; we have therefore this theorem:
An Auxiliary Quadric (LXVIII. or XLV.) touches the Second Sheet of the Surface of Centres of a given quadric, along a Quartic Curve, which is the locus of the centres of Second Curvature for all the points of a Line of First Curvature (35.); and (for the same reason) the same auxiliary quadric is circumscribed, along the same quartic, by the Developable Normal Surface (28.), which rests on that first line: with permission, of course, to interchange the words first and second, in this enunciation.
(37.) When the arbitrary constant r is thus allowed to take successively all values, corresponding to both systems of lines of curvature, the Surface of Centres is therefore at once the Envelope of the Auxiliary Quadric LXVIII., and the Locus of the Quartic Curve (36.), in which one or other of its two sheets is touched, by that auxiliary quadric in one of its successive states, and also by one of the developable surfaces of normals to the given surface.
(38.) To obtain the vector equation of that envelope or locus, we may proceed
The variable vector of this curve is easily seen (comp. XLIX.) to be,
and the reciprocal surface (21.) or (24.) is by (25.) the locus of this quartie (35.). The analogous relation, between the co-ordinate forms of the equations, was perhaps thought too obvious to be mentioned, in page 161 of Dr. Salmon's Treatise; or possibly it may have escaped notice, since the quartic curve (36.) is only mentioned there as an intersection of two quadrics, which is on the surface of centres, and answers to points of a line of curvature upon the given surface. But as regards the possible novelty, even in part, of any such geometrical deductions as those given in the text from the quaternion analysis employed, the writer wishes to be understood as expressing himself with the utmost diffidence, and as most willing to be corrected, if necessary. The power of derivating (or differentiating) any symbolical expression of the form LXVIII., or of any analogous form, with respect to any scalar which it involves explicitly, as if the expression were algebraical, is an important but an easy consequence from the principles of the Section III. ii. 6, which has been so often referred to.
Compare the Note immediately preceding.