CHAP. III.] DIFFERENTIAL AND SCALAR EQUATION. 519 381. In general, if we conceive (comp. 372, I.) that the vector p of a given surface is expressed as a given function of two scalar variables, x and y, whereof one, suppose y, is regarded at first as an unknown function of the other, so that we have again, I. p= p(x, y), but now with II. y=fx, where the form of is known, but that of f is sought; we may then' regard p as being implicitly a function of the single (or independent) scalar variable, x, and may consider the equation, III. . . p = p(x, fx), as being that of some curve on the given surface, to be determined by assigned conditions. Denoting then the unknown total derivative DO(x, fx) by p', but the known partial derivatives of the same first order by D. and D,, with analogous notations for orders higher than the first, we have (comp. 376, VI.) the expressions, IV.. p' = D ̧+y'1‚‚ p" =D ̧2+ 2y1Ð ̧Ð ̧ + y22Ð ̧2 + y''Ð ̧Þ, &c.; in which y=D,y=f'x, y" =D2y=fx, &c. Hence, writing for the normal v to the surface the expression, V... v=V(D,.,) =V.D ̧ØÐ ̧Ø, comp. 372, V., or this vector multiplied by any scalar, the equation 380, I. of a geodetic line takes this new form, VI... 0 Svp'p" =S(V.D,OD,P.Vp'p"); = or, by a general transformation which has been often employed already (comp. 352, XXXI., &c.), VII... 0 Sp'D,P.Sp'D ̧-Sp'D2. Sp''D; and thus, by substituting the expressions IV. for p' and p", we obtain an ordinary (or scalar) differential equation, of the second order, in x and y, which is satisfied by all the geodetics on the given surface, and of which the complete integral (when found) expresses, with two arbitrary and scalar constants, the form of the scalar function fin II., or the law of the dependence of y on x, for the geodetic curves in question. (1.) As an example, let us take the equation, VIII. . . p = (x, y) = y¥x, comp. 378, I., of a cone with its vertex at the origin; which cone becomes a known one, when the form of the vector function is given, that is, when we know a guiding curve p = ↓x, through which the sides of the cone all pass. We have here the partial derivatives, and IX. . . Ð1⁄2ß =уÐ ̧¥x=y¥', Dyp=¥x = 4, the expressions IV. become, then, XI. . . p′ = y↓' + y'¥, _p"=y¥"+2y'¥' + y”¥ ; comp. 378, II. ; and since only the direction of the normal is important, we may divide V. by — y, and write, XII. . . v=V¥Y". (2.) The expressions XI. and XII. give (comp. VI. and VII.) for the geodetics on the cone VIII., the differential equation of the second order, in which XIII... 0=S(V¥¥'.Vp'p”) = Sp′′¥Sp'V' — Sp′′¥'Sp'¥ and 2 are abridged symbols for (x)2 and (x)2; but this equation in x and y may be greatly simplified, by some permitted suppositions. (3.) Thus, we are allowed to suppose that the guiding curve (1.) is the intersection of the cone with the concentric unit sphere, so that XIV... Tx = 1, 42=-1, S¥¥'=0, S¥¥"+ ¥22 = 0; and if we further assume that the arc of this spherical curve is taken as the independent variable, x, we have then, by 380, (12.), combined with the last equation XIV., XV... T¥'x= 1, 2=-1, S¥¥"=0, $¥¥"= — 4'2 = 1. (4.) With these simplifications, the differential equation XIII. becomes, XVI... 0-(y- y′′) (− y) − (− 2y') (− y') = yy” — 2y'2 — y2 ; and its complete integral is found by ordinary methods to be, XVII... y = b sec (x + c), in which b and c are two arbitrary but scalar constants. (5.) To interpret now this integrated and scalar equation in x and y, of the geodetics on an arbitrary cone, we may observe that, by the suppositions (3.), y represents the distance, Tp or UP, from the vertex o, and x + c represents the angle AOP, in the developed state of cone and curve, from some fixed line oa in the plane, to the variable line OP; the projection of this new OP on that fixed line OA is therefore constant (being = b, by XVII.), and the developed geodetic is again found to be a right line, as before. A B' 382. Let ABCDE... (see the annexed Figure 79) be any given series of points in space. Draw the successive right lines, AB, BC, CD, DE, . . and prolong them to points B', C', D', E',... the lengths of these prolongations being arbitrary; join also B'C', C'D', D'E',... We shall thus have a series of plane triangles, B'BC', c'cD', D'DE',... all generally in different planes; so that BCD'C'B', CDE'D'C', ... are generally gauche pentagons, while BCDE'D'C'B' is a gauche heptagon, &c. But we B C D E Fig. 79. CHAP. III.] DEVELOPABLE SURFACE, CUSP-EDGE. 521 can conceive the first triangle B'BC' to turn round its side BCC', till it comes into the plane of the second triangle, c'cD'; which will transform the first gauche pentagon into a plane one, denoted still by BCD'C'B'. We can then conceive this plane figure to turn round its side CDD', till it comes into the plane of the third triangle, D'DE'; whereby the first gauche heptagon will have become a plane one, denoted as before by BCDE'D'C'B': and so we can proceed indefinitely. Passing then to the limit, at which the points ABCDE... are conceived to be each indefinitely near to the one which precedes or follows it in the series, we conclude as usual (comp. 98, (12.)) that the locus of the tangents to a curve of double curvature is a developable surface: or that it admits of being unfolded (like a cone or cylinder) into a plane, without any breach of continuity. It is now proposed to translate these conceptions into the language of quaternions, and to draw from them some of their consequences: especially as regards the determination of the geodetic lines, on such a developable surface. (1.) Let, or simply, denote the variable vector of a point upon the curve, or cusp-edge, or edge of regression of the developable, to which curve the generating lines of that surface are thus tangents, considered as a function & of its arc, x, measured from some fixed point ▲ upon it; so that while the equation of the surface will be of the form (comp. 100, (8.)), y being a second scalar variable, we shall have the relations (comp. 381, XV.), II... T¥'z = 1, _Y2=-1, S¥¥"=0, $¥′4′′"=-4′′2=z2, if z=T¥”. (2.) Hence III... DxO ='+y¥",_Dyp=4' ; and IV. . . p' = (1 + y')¥′+y¥", p′′= y′¥′ + (1 + 2y′) 4"+y¥"" ; (3.) The differential equation of the geodetics may therefore be thus written (comp. 381, XIII.), VI...0 S(V".Vpp") = Sp¥"Sp" - Sp""Sp''; in which, by (1.) and (2.), VII... = Sp""(1+2y) 22-yzz', Sp'' = (1+y'); the equation becomes therefore, after division by - z, or simply, VIII... 0 = z{(1+ y′)2 + (y2)2 } + (1 + y′) (yz)' — y′′yz, IX. . . z + v′ = 0, or IX'... Tdy' + dv = 0, if X... tan v= yz 1+ y (4.) To interpret now this very simple equation IX. or IX'., we may observe that z, or T", or Td': dx, expresses the limiting ratio, which the angle between two near tangents &' and ' + A', to the cusp-edge (1.), bears to the small are Ax of that curve which is intercepted between their points of contact; while v is, by IV., that other angle, at which such a variable tangent, or generating line of the developable, crosses the geodetic on that surface; and therefore its derivative, v'or dr: dz, represents the limiting ratio, which the change Av of this last angle, in passing from one generating line to another, bears to the same small are Ax of the curve which those lines touch. (5.) Referring then to Figure 79, in which, instead of two continuous curves, there were two gauche polygons, or at least two systems of successive right lines, connected by prolongations of the lines of the first system, we see that the recent formula IX. or IX'. is equivalent to this limiting equation, but these three angles remain unaltered, in the development of the surface: the bent line B'C'D' for space becomes therefore ultimately a straight line in the plane, and similarly for all other portions of the original polygon, or twisted line, B'C'D'B'..., of which B'C'D' was a part. (6.) Returning then to curves and surfaces in space, the quaternion analysis (3.) is found, by this simple reasoning,* to conduct to an expression for the known and characteristic property of the geodetics on a developable: namely that they become right lines, as those on cylinders (380, (4.)), and on cones (380, (6.) and (10.), or 381, (5.)), were lately seen to do, when the surface on which they are thus traced is unfolded into a plane. 383. This known result, respecting geodetics on developables, may be very simply verified, by means of a new determination of the absolute† normal (379) to a curve in space, as follows. (1.) The arc s of any curve being taken for the independent variable, we may write (comp. 376, I.), by Taylor's Series, the following rigorous expressions, I. . . p-, = p − sp' + js2u_sp′′, Po=P, ps = p + sp' + {s2u«p”, with u=1, for the vectors of three near points, P., Po, P„ on the curve, whereof the second bisects the arc, 2s, intercepted between the first and third. (2.) If then we conceive the parallelogram P_POP,R, to be completed, we shall have, for the two diagonals of this new figure these other rigorous expressions, In the Lectures (page 581), nearly the same analysis was employed, for geodetics on a developable; but the interpretation of the result was made to depend on an equation which, with the recent significations of and v, may be thus written, as the integral of IX'., v + STdy' = const.; where fTdy' represents the finite angle between the extreme tangents to the finite arc (Tdy, or Ax, of the cusp-edge, when that curve is developed into a plane one. + Called also, and perhaps more usually, the principal normal. (3.) But the length P-P, of what may be called the long diagonal, or the chord of the double arc, 28, is ultimately equal to that double arc; we have therefore, by IV., the equation, = VI... Tp'=1, if p' Dp, and if a denote the arc, considered as the scalar variable on which the vector p depends: a result agreeing with what was otherwise found in 380, (12.). (4.) At the same time, since the ultimate direction of the same long diagonal is evidently that of the tangent at Po, we see anew that the same first derived vector p' represents what may be called the unit-tangent♣ to the curve at that point. (5.) And because the lengths of the two sides P-Po and POPs, considered as chords of the two successive and equal arcs, s and s, are ultimately equal to them and to each other, it follows that the parallelogram (2.) is ultimately equilateral, and therefore that its diagonals are ultimately rectangular; but these diagonals, by IV. and V., have ultimately the directions of p' and p"; we find therefore anew the equation, VII... Sp'p"= 0, if the arc be the independent variable, which had been otherwise deduced before, in 380, (12.). (6.) But under the same condition, we saw (379, (2.)) that the second derived vector p" has the direction of the absolute normal to the curve; such then is by V. the ultimate direction of what we may call the short diagonal PoRs, constructed as in (2.); or, ultimately, the direction of the bisector of the (obtuse) angle P_¿POPs, between the two near and nearly equal chords from the point Po: while the plane of the parallelogram becomes ultimately the osculating plane. (7.) All this is quite independent of the consideration of any surface, on which the curve may be conceived to be traced. But if we now conceive that this curve is formed from a right line B'C'D' ... (comp. Fig. 79), by wrapping round a developable surface a plane on which the line had been drawn, and if the successive portions B'C', C'D',.. of that line be supposed to have been equal, then because the two right lines c'B' and ɗ'v' originally made supplementary angles with any other line c'c in the plane, the two chords c'B' and C'D' of the curve on the developable tend to make supplementary angles with the generatrix c'c of that surface; on which account the bisector (6.) of their angle B'C'D' tends to be perpendicular to that generating line c'c, as well as to the chord B'D', or ultimately to the tangent to the curve at c', when chords and arcs diminish together. The absolute normal (6.) to the curve thus formed is therefore perpendicular to two distinct tangents to the surface at c', and is consequently (comp. 372) the normal to that surface at that point; whence, by the definition (380), the curve is, as before, a geodetic on the developable. (8.) As regards the asserted rectangularity (7.), of the bisector of the angle B'C'D' to the line c'c, when the angles cc'B' and cc'D' are supposed to be supplementary, but not in one plane, a simple proof may be given by conceiving that the Compare the Note to page 152. |