right line B'C' is prolonged to c", in such a manner that c'c"C'D'; for then these two equally long lines from c' make equal angles with the line c'c, so that the one may be formed from the other by a rotation round that line as an axis; whence c"D′, which is evidently parallel to the bisector of B'C'D', is also perpendicular to c′c. (9.) In quaternions, if a and p be any two vectors, and if t be any scalar, we have the equation, VIII... S.a(a pa ̄t − p) = 0 ; which is, by 308, (8.), an expression for the geometrical principle last stated. 384. The recent analysis (382) enables us to deduce with ease, by quaternions, other known and important properties of developable surfaces: for instance, the property that each such surface may be considered as the envelope of a series of planes, involving only one scalar and arbitrary constant (or parameter) in their common equation; and that each plane of this series osculates to the cusp-edge of the developable. (1.) The equation of the developable surface being still, 1. p=4(x,y)=4x+Y¥'x = ¥ +y4', as in 382, I., its normal v is easily found to have as in 382, V., the direction of V↓↓”, whether the scalar variable z be, or be not, the arc of the cusp-edge, of which curve the equation is, II...p=4x (2.) Hence, by 373, VII., the equation of the tangent plane takes the form, III... Sw" S¥¥'¥", from which the second scalar variable y thus disappears: this common equation, of all the tangent planes to the developable, involves therefore, as above stated, only one variable and scalar parameter, namely r; and the envelope of all these planes is the developable surface itself. (3.) The plane III., for any given value of this parameter x, that is, for any given point of the cusp-edge, touches the surface along the whole extent of the generating line, which is the tangent to this last curve. (4.) And by comparing its equation III. with the formula 376, V., we see at once that this plane osculates to the same cusp-edge, at the point of contact of that curve with the same generatrix of the developable. 385. If the reciprocals of the perpendiculars, let fall from a given origin, on the tangent planes to a developable surface, be considered as being themselves vectors from that origin, they terminate on a curve, which is connected with the cusp-edge of the developable by some interesting relations of reciprocity (comp. 373, (21.)): in such a manner that if this new curve be made the cusp-edge of a new developable, we can return from it to the former surface, and to its cuspedge, by a similar process of construction. CHAP. III.] THEOREM OF RECIPROCAL CURVES IN SPACE. 525 (1.) In general, if and X, or briefly and x, be two vector functions of a scalar variable x, such that x may be deduced from by the three scalar equations, I... S&x=c, S'x=0, S4"x = 0, с in which Sex is written briefly for S(V.xx), and c is any scalar constant, we have then this reciprocal system of three such equations, (3.) But if p be the variable vector of a curve in space, and p', p" its first and ρ second derivatives with respect to any scalar variable, then, by the equation 376, V. of the osculating plane to the curve, we have the general expression, VI... Spp'p" = perpendicular from origin on osculating plane; so that if and x be considered as the vectors of two curves, each vector is ex the reciprocal of the perpendicular, thus let fall from a common point, on the osculating plane to the other. (4.) We have therefore this Theorem:— If, from any assumed point, o, there be drawn lines equal to the reciprocals of the perpendiculars from that point, on the osculating planes to a given curve of double curvature, or to those perpendiculars multiplied by any given and constant scalar; then the locus of the extremities of the lines so drawn will be a second curve, from which we can return to the first curve by a precisely similar process. 386. The theory of developable surfaces, considered as envelopes of planes with one scalar and variable parameter (384), may be additionally illustrated by connecting it with Taylor's Series, as follows. (1.) Let a denote any vector function of a scalar variable t, so that I... atao + tua'o = a + tua', with u1 = 1; or, by another step in the expansion, II... aao+ta'o+ t2via”o=a+ta' + {t2va", vo = 1; where u and v are generally quaternions, but ua' and va" are vectors. * The two curves may be said to be polar reciprocals, with respect to the (real or imaginary) sphere, p2 = c; and an analogous relation of reciprocity exists generally, when the points of one curve are the poles of the osculating planes of the other, with respect to any surface of the second order: corresponding tangents being then reciprocal polars. Compare the theory of developables reciprocal to curves, given in Salmon's Analytical Geometry of Three Dimensions, page 89; see also Chapter XI. (page 224, &c.), of the same excellent work. (2.) Then, as the rigorous equation of the variable plane, the reciprocal of the perpendicular on which from the origin is - at, we have either, or III... - 1 = Satp = Sap + tSua'p, IV... - 1 = Sap + tSa′p + ft2Sva”p, according as we adopt the expression I., or the equally but not more rigorous expression II., for the variable vector a¿. (3.) Hence, by the form III., the line of intersection of the two planes, which answer to the two values 0 and t of the scalar variable, or parameter, t, is rigorously represented by the system of the two scalar equations, V... Sap + 1 = 0, Sua'p = 0. (4.) And the limiting position of this right line V., which answers to the conceived indefinite approach of the second plane to the first, is given with equal rigour by the equations, VI... Sap+1=0, Sa'p=0; whereof it is seen that the second may be formed from the first, by derivating with respect to t, and treating p as constant: although no such rule of calculation had been previously laid down, for the comparatively geometrical process which is here supposed to be adopted. (5.) The locus of all the lines VI. is evidently some ruled surface; to determine the normal v to which, at the extremity of the vector p, we may consider that vector to be a function (372) of two independent and scalar variables, whereof one is t, and the other may be called for the moment w; and thus we shall have the two partial derivatives, VII... Sa Dip = 0, SaDup = 0, giving || a. (6.) Hence the line a has the direction of the required normal v; the plane Sap+1=0 touches the surface (comp. 384, (3.)) along the whole extent of the limiting line VI.; and the locus of all such lines is the envelope of all the planes, of the system recently considered. (7.) The line VI. cuts generally the plane IV., in a point which is rigorously determined by the three equations, VIII... Sap +1=0, Sa'p=0, Sva"p=0; and the limiting position of this intersection is, with equal rigour, the point determined by this other system of equations, IX... Sap +1=0, Sa'p=0, Sa"p=0; in which it may be remarked (comp. (4.)), that the third is the derivative of the second, if p be treated as constant. (8.) The locus of all these points IX. is generally some curve upon the surface (5.), which is the locus of the lines VI., and has been seen to be the envelope (6.) of the planes III. or IV.; and to find the tangent to this curve, at the point answering to a given value of t, or to a given line VI., we have by IX. the derived equations, X... Sap=0, Sa'p'=0, whence p' || Vaa'; comparing which with the equations VI. we see that the lines VI. touch the curve, which is thus their common envelope. CHAP. III.] FAMILIES OF SURFACES. 527 (9.) We see then, in a new way, that the envelope of the planes III., which have one scalar parameter (t) in their common equation, and may represent any system of planes subject to this condition, is a developable surface: because it is in general (comp. 382) the locus of the tangents to a curve in space, although this curve may reduce itself to a point, as we shall shortly see. (10.) We may add that if at in III. be considered as the vector of a given curve, this curve is the locus of the poles* of the tangent planes to the developable, taken with respect to the unit sphere; and conversely, that the developable surface is the envelope of the polar planes of the points of the same given curve, with respect to the same sphere. (11.) If then it happen that this given curve, with at for vector, is a plane one, so that we have this new condition, XI... Sẞat +1 = 0, ẞ being any constant vector, namely the vector of the pole of the supposed plane of the given curve, the variable plane III., or Spa1+1=0, of which the surface (5.) is the envelope, passes constantly through this fixed pole; so that the developable becomes in this case a cone, with ẞ for the vector of its vertex: the equations IX. giving now p= ß. (12.) The same degeneration, of a developable into a conical surface, may also be conceived to take place in another way, by the cusp-edge (or at least some finite portion thereof) tending to become indefinitely small, while yet the direction of its tangents does not tend to become constant. For example, with recent notations, the developable which is the locus of the tangents to the helix may have its equation written thus: 2 XII. . . p = p(x, y) = c(xa + − tan a. a*Uẞ) +ya (1 + tan a. a*Uẞ); which when the quarter-interval, c, between the spires, tends to zero, without their inclination a to the axis a being changed, tends to become a cone of revolution round that axis, with its semiangle = a. 387. So far, then, we may be said to have considered, in the present Section, and in connexion with geodetic lines, the four following families of surfaces (if the first of them may be so called). First, spherical surfaces, of which the characteristic property is expressed by the equation, I. . . V v(p − a) = 0, if a be vector of centre; second, cylindrical surfaces, with the property, II... Sva = 0, if a be parallel to the generating lines; third, conical surfaces, with the property, III... Sv (p − a)=0, if a be vector of vertex ; and fourth, developable surfaces, with the distinguishing property expressed by the more general equation, IV... Vvdv =0, if dp have the direction of a generatrix ; being in each the normal vector to the surface, so that V... Svdp = 0, for all tangential directions of dp; and the fourth family including the third, which in its turn includes the second. A few additional remarks on these equations may be here made. (1.) The geometrical signification of the equation I. (as regards the radii) is obvious; but on the side of calculation it may be useful to remark, that elimination of between I. and V. gives, for spheres, VI... Spa) dp = 0, or VII... T(p − a) = const. (2.) The equations II. and V. show that dp, and therefore Ap, may have the given direction of a; for an arbitrary cylinder, then, we have the vector equation (372), where VIII... p=4(x,y)=4x+ya, is an arbitrary vector function of x. (3.) From VIII. we can at once infer, that IX. S3p = SB, Syp = Sy if a= = VẞY; .. the scalar equation (373) of a cylindrical surface is therefore generally of the form (comp. 371, (6.), (7.)), X... 0= F(Sẞp, Syp); B and y being two constant vectors, and the generating lines being perpendicular to both. (4.) The equation III. may be thus written, XI... SvUa= Ta-'Svp; whence XII... SvUa = 0, if Ta= ∞; the equation for cones includes therefore that for cylinders, as was to be expected, and reduces itself thereto, when the vertex becomes infinitely distant (5.) The same equation III., when compared with V., shows that do may have the direction of p a, and therefore that pa may be multiplied by any scalar; the vector equation of a conical surface is therefore of the form, XIII. . . p = a + y, being an arbitrary vector function. (6.) The scalar equation of a cone may be said to be the result of the elimination of a scalar variable t, between two equations of the forms, XIV... S(pa) xt = 0, S(o-a) x't = 0, which express that the cone is the envelope (comp. 386, (11.)) of a variable plane, which passes through a fixed point, and involves only one scalar parameter in its equation with a new reduction to a cylinder, in a case on which we need not here delay. (7.) The equation IV. implies, that for each point of the surface there is a direction along which we may move, without changing the tangent plane; and therefore that the surface is an envelope of planes, &c., as in 386, and consequently that it is developable, in the sense of Art. 382. (8.) The vector equation of a general developable surface may be written under the form, |