duced and one-branched state. And the last form CXXXVII. shows, what was to be expected from geometry, that when this condition of coincidence is satisfied, the earlier condition of reality CXXXII. is satisfied also: together with this other inequality, CXXXVIII... R2 o”2 + g2 < 0, which then results from the form CXXXVI. (85.) The equations CXXXI., CXXXIV., and the general formula 389, IV., give the expressions, go'2 + RR'So'o" – σι (Vo'o')2. = where r is still the radius of the circle of contact of the sphere with its envelope, and r is the radius of curvature of the locus of the centre s of the same variable sphere; whence it is easy to infer, that the condition CXXXV. may be reduced to the following very simple form (comp. XXXVI′. and XLII.) : CXLI... (r'r1)2 = (RR)2; or CXLI'. . . r1dr = ± RdR ; the independent variable being still arbitrary. (86.) If the arc of the curve (s) be taken as that variable t, the form CXXXVI. of the same condition is easily reduced to the following, CXLII... R2= (RR') + gr2, with CXLIII...g=1−(RR′)'; derivating then, and dividing by 2g, we have this new differential equation, which is of linear form with respect to RR', whereas the condition itself may be considered as a differential equation of the second degree, as well as of the second order,* if CXLIV... RR = r1(gr1)'; or CXLV... r12u" + r1rı' (u' − 1) + u = 0, CXLVI. . ."u = RR' = RD,R, and therefore CXLVII. . . u2 = R2 — r3‚ by CXXXI. or CXXXI'., because we have now, CXLVIII... σ2 = — 1, or To' = 1, or dt = Tdo: so that the new scalar variable, RR', or u, with respect to which the linear equation CXLIV. or CXLV. is only of the second order, represents the perpendicular height† of the centre s of the sphere, above the plane of the circle, considered as a function of the arc (t) of the curve (s), and as positive when the radius R of the sphere increases, for positive motion along that curve, or for an increusing value of its are. (87.) If the curve (s) be given, or even if we only know the law according to which its radius of curvature (ri) depends on its arc (t), the coefficients of the linear equation CXLV. are known; and if we succeed in integrating that equation, so as to * We shall soon assign the complete integral of the differential equation in quaternions (84.), and also that of the corresponding Equation of Monge, cited in the preceding Note. It will be found that this new scalar u, if we abstract from sign, corresponds precisely to the p of earlier sub-articles, although presenting itself in a different connexion for the sphere (78.), and the circle (79.), under the condition (84.), will soon be shown to be the osculating sphere and circle to the recent curve (P), or to the singular line (84.) upon the surface at present considered, that is, on the locus or envelope (80.). CHAP. III.] DETERMINATION OF THE SINGULAR LINE. 605 find an expression for the perpendicular u as a function of that arc t, we shall then be able to express also, as functions of the same arc, the radii R and r of the sphere and circle, by the formulæ, CXLIX... + r = gr1 = r1(1-u), and CL... R2 = 2 § udt = u2 + r12 (1 − u′)2 ; the third scalar constant, which the integral 2 fudt would otherwise introduce into the expression for R2, being in this manner determined, by means of the other two, which arise from the integration of the equation above mentioned. (88.) For example, it may happen that the locus of the centres of the sphere has a constant curvature, or that r1 = const.; and then the complete integral of the linear equation CXLV. is at once seen to be of the form, CLI... ua sin (r1-1t + b), a and b being two arbitrary (but scalar) constants; after which we may write, by (87.), CLII...+r=r1 − a cos (r1 ̄1t+b); CLIII... R2= r12 - 2ar1 cos (r1 ́1t+b) + a2; so that, in this case, both the radii, r and R, of circle and sphere, are periodical functions of the arc of the curve (s). (89.) In general, if that curve (s) be completely given, so that the vector σ is a known function of a scalar variable, and if an expression have been found (or given) for the scalar R which satisfies any one of the forms of the condition (84.), we can then determine also the vector p, by the formula CXXXIII., as a function of the same variable; and so can assign the point P of the singular line (84.), which corresponds to any given position of the centre s of the sphere. For this purpose we have, when the arc of the curve (s) is taken, as in (86.), for the independent variable t, the formula, CLIV. p =σ - uo' - (1 - u′) σ′′-1 = к1 — uơ′ — r1au'o”, ... if 1 be the vector of the centre, say K1, of the osculating circle at s to that given curve, so that (comp. 389, XI.) it has the value, CLV... OK1=k1 = 0 − 6′′-1 = 0 + ro", with CLV'.. If then we denote by v the distance of the point P from this centre K1, and attend to the linear equation CXLV., we see that CLVI... v=K1P = T(p − x1) = √(u2 + r12u ́2), CLVI. . . vv′ = r1riu', with To' = 1; CLVII. . . vv'sı' = r1r'iu', and or more generally, if CLVII'... u = while .. u = RR's, and CLVII". . . . 81= JTdo, so that si denotes the arc of the curve (s), when the independent variable t is again left arbitrary. This distance, v, is therefore constant (= a) in the case (88.), namely when the radius of curvature r1 of that curve is itself a constant quantity. (90.) When si To'= 1, as in CXLVIII., the part o uo' of the first expression CLIV. for p becomes к, by CXXXI”. and CXLVI.; attending then to CLV., = the independent variable t being again arbitrary. Accordingly, if we combine the general expression CXXXIII. for p, with the expression CXXXI". for к, and with the following for x1 (comp. 389, IV.), (91.) It has then been fully shown, how to determine the vector p as a function of the scalar t, when σ and R are two known functions of that variable, which satisfy any one of the forms of the condition (84.). It must then be possible to determine also the derived vectors, p', p", &c., as functions of the same variable; and accordingly this can be done, by derivating any three of the four scalar equations, LXXXIX. XCI. XCII. XCIII., of which that condition (84.) expresses the coexistence. Now if we derivate a first time the two first of these, and then reduce by the second and fourth, we get the equations, CLX... Sp' (o-p)=0, Sp'o' = 0, whence CLX'. . . p' || Vo'(o−p); and although this last formula only determines the direction of the tangent to the singular line at P, namely that of the common tangent at that point to two consecutive circles (84.), yet it enables us to infer, by the remaining equation XCII., that CLXI... p'o", p' Vo'o", and CLXI... Sp'o"= 0; reducing by which the derivative of XCIII., we find, So" (o-p) + 3So'σ" + (RR′)" =0, XCIV., the scalar variable being still arbitrary. And conversely, the system* of the four equations LXXXIX. XCI, XCIII. XCIV. gives the three equations CLX. CLXI., and so conducts to the equation XCII., and thence to the condition (84.); unless we suppose that p is a constant vector a, or that the variable sphere passes through a fixed point A, a case which we do not here consider, because in it the singular line (P) would reduce itself to that one point. (92.) Derivating the two equations CLX., and reducing with the help of CLXI., we find these new equations, whence CLXII... Sp"(o-p)-p2 = 0, Sp❞o' = 0; In the language of infinitesimals, this system of equations expresses that four consecutive spheres intersect, in one common point P. When that point happens to be a fixed one, the condition (84.) requires that we should have the relation So'o" (-a)=0; or geometrically, that the curve (s) should be in a plane through the fixed point, which is then a singular point of the envelope. CHAP. III.] ENVELOPE OF OSCULATING SPHERE. 607 We are led then, by elimination of the derivatives of o, to the system of the three equations 395, VII.; and we conclude, that the point s is the centre, and the radius R is the radius, of the osculating sphere to the singular line (P): whence it is easy to infer also, that the plane of contact (79.) of the sphere with its envelope is the osculating plane, and that the circle of contact (80.) is the osculating circle (comp. (72.)), to the same curve (P), at the point where two consecutive circles touch one another (84.). (93.) In general, and even without the condition (84.), the tangent to a branch (82.) of the curvilinear envelope of the circles of the system, at any point P1 of that branch, has the direction represented by the vector Vo'(o- p1), of the tangent to the circle at that point; but when that condition is satisfied, so that the two branches of the singular line coincide, the point P of that line is in the osculating plane (83.) to the curve (s): and then the equation XCII. shows that the tangent p', or Vo' (op), to the line, is perpendicular to o", or parallel to Vo'o" (comp. CLXI.), and therefore that the singular line crosses that plane at right angles. (94.) It follows that, with the condition (84.), the singular line (P) is an orthogonal trajectory to the system of osculating planes to the curve (s); and whereas, when this last curve is given, there ought to be one such trajectory for every point of a given osculating plane, this circumstance is analytically represented, in our recent calculations, by the biordinal form of the differential equation CXLV., of which the complete integral must be conceived (87.) to involve generally, as in the case (88.), two arbitrary constants. (95.) It follows also that, with the same condition of coincidence of branches, the singular line (P) must have the curve (s) for the cusp-edge of its polar developable; or that the sphere, with s for centre, and with R for radius, must be the osculating sphere to the curve (P), as otherwise found by calculation in (92.): while the circle (80.) must be, as before, the osculating circle to that curve. (96.) Accordingly, all equations, and inequalities, which have been stated in the recent sub-articles (79.), &c., respecting the envelope of a moving sphere with variable radius, under that condition (84.), and without any special selection of the independent variable, admit of being verified, by means of the earlier formulæ for the osculating circle and sphere to a curve (P) treated as a given one, when the arc (s) of that curve is taken as such a variable. (97.) For example, we had lately the two inequalities, R'2 + σ'2 < 0, CXXXII., and R2♂''2 +g2 <0, CXXXVIII. And accordingly the earlier sub-articles (22.), (23.) give, for those two combinations, the essentially negative values, CLXIV... R'2 + o'2 = − p-2r2 R′2 ; * In the language of infinitesimals (comp. the preceding Note), if every four consecutive spheres of a system intersect in one point of a curve, then each sphere passes through four consecutive points of that curve. Simple as this geometrical reasoning is, the writer is not aware that it has been anticipated; and indeed he is at present led to suppose that this whole theory, of the Locus of the Osculating Circle, as the Envelope of the Osculating Sphere, is new. Monge had however considered, but rejected (page 374 of Liouville's Edition), the case of a system of circles having each a simple contact with a curve in space. in obtaining which last, the following transformations have been employed: CLXVI... "2 = — n'2 — n2r-9; CLXVII. . . 9= =— n'p + nrr-1, (98.) As regards the verification of the equations, it may be sufficient to give one example; and we shall take for it the last general form CLVII. of the differential equation of condition (84.). For this purpose we may now write, by (22.) and (23.), CLXVIII. . . s1' =±n, u=±p, u'=±p', r1u\'si'-'=p'r\n`' = p't ; and have only to observe that CLXIX... (p2 + p′2r3)' = p′r (r+p'r)', because p=r'r. (99.) If we denote by c1, c2, c3 the first members of the equations XCI., XCIII., XCIV., then besides the equation LXXXIX., which may be regarded as a mere definition of the radius R, we have c1 = 0 for the whole of the superficial locus or envelope (80.); but we have not also c2 =0, except for a point on one or other of the two (generally distinct) branches of the singular line (82.) upon that locus. And if, at any other and ordinary point, we cut the surface by a plane perpendicular to the circle at that point, we find, by a process of the same kind as some which have been already employed, expressions for the tangential and normal components of the vector chord, whereof the principal terms involve the scalar c2 as a factor, while the latter varies (ultimately) as the square of the former, so that the curvature of the section is finite and known, but tends to become infinite when c tends to zero. = (100.) If the condition of coincidence (84.) be not satisfied, so that the two branches of the singular line (82.) remain distinct, and that thus c2 = 0, but not c30 (comp. (91.)), for any ordinary point on one of those two branches, then if we cut the surface at that point by a plane perpendicular to the branch, or to the circle which touches it there, we find an ultimate expression for the vector chord which involves the scalar c3 as a factor, and of which the normal component varies as the sesquiplicate power of the tangential one: so that we have here the case of a semicubical cusp, and each branch of the singular line is a cusp-edge* of the surface, exactly in the same known sense (comp. (76.)) as that in which a curve of double curvature is generally such, on the developable locus of its tangents. (101.) But when the condition (84.) is satisfied, so that the two branches coincide, and that thus (comp. again (91.)) we have at once the three equations, CLXX... c1 = 0, c2 = 0, C3 = 0, then the terms, which were lately the principal ones (100.), disappear: and a new expression arises, for the vector chord of a section of the surface, made by a plane perpendicular to the singular line, which (when we take t=s, as in (96.)) is found to admit of being identified with the formula CXVII., and of course conducts to precisely the same system of consequences; the tangential component now varying ultimately as the cube, and the normal component as the fourth power of a small variable, so that the cuspidal property of the point P of the section no longer exists although the curvature at that point is still infinite, as in (74.): and the Singular Line, reduced now to a single branch, to which all the circles of the system osculate, * Compare the Note to page 602. |