and therefore from (14), 16), (18), and (21), whence, squaring and adding, and by means of (5), 1 dx 2 ds 2 1 } = ± √, {(a. dr)2 + ( d. dy)2 + (a. dz)' } * ; (23) ρ ds (20) and (23) being the same values of forms. ds though in different If s is equicrescent, (23) may be put in the form 378.] Let λ, μ, v be the direction-angles of the radius of on comparing which results with Art. 346, it appears that the radius of absolute curvature coincides in direction with the principal normal. 379.] To determine the angle of curvature. Since by equation (2), Art. 375, dr = ± ds ; therefore from (27) {(a.da)2 + (a.dy)2 + (a.dz)}"}"; ds ds ds Now equation (27) is remarkable and deserves attention; dx dy dz ds' d' ds are the direction-cosines of the tangent, and dr is the infinitesimal angle between it and the consecutive tangent. This result therefore may be generalized. Let cos a, cos B, cos y, cos a + d.cos a, cos ẞ+d.cos ß, cos y + d.cos y be the direction-cosines of two lines inclined to each other at an infinitesimal angle; then that infinitesimal angle is equal to {(d.cos a)2 + (d.cos B)2 + (d.cos y)2} *. (29) 380.] We may also, as follows, immediately obtain the value of p, and as the process exhibits the power of the infinitesimal method in its greatest simplicity, we recommend the reader to study it attentively. Let P, Q, R, fig. 128, be three consecutive points in the curve, all three lying in the plane of the paper which is the osculating plane. Then PQ = ds, QR = ds+d's; produce PQ to s, making Q8PQ; complete the parallelograms PQRV, VQSR, so that Qs = VR = PQ = ds, then the angle RQs = dr; and from q as a centre, and with qs as a radius, describe a small arc ST, which is therefore perpendicular to QR; hence QTQs ds; and therefore RT = d2s, and sT = ds x dr. Now the projections on the coordinate axes of PQ are dx, dy, dz; and of QR, or of pv, dx+d3x, dy + d2y, dz+d2z; therefore the projections of vo are d2x, d2y, d2z; and therefore but = (d2x)2 + (d2y)2 + (d2z)2 ; vQ2 = vQ2 = SR2, = ST2+TR2, = ds2 dr2 + (d2s)2; * Or thus: suppose cos a, &c. cos a + d. cos a, &c. to be the direction-cosines of two lines making an infinitesimal angle dr with each other; take along the lines two distances each equal to unity, and commencing from the point of intersection of the lines. Then the projections of these distances on the coordinate axes are respectively cosa...., cos a + d. cos a....; and therefore the projections of the line joining their extremities are d. cosa, d.cos ß, d.cosy, but this line measures dτ, since it subtends dr, at a distance unity; and therefore dr2 (d.cos a)2 + (d.cos B)2 + (d.cos y)2. = equating which values of vq2, we have ds2 dr2 = (d2x)2 + (d2y)2 + (d2z)2 — (d2s)2. (30) And therefore by (2), 1 (31) Hence also we have the following value of the angle of contingence, dr = 1 ds {(d2x)2 + (d2y)2 + (d2z) — (d2s)2} §. (32) 381.] Again consider fig. 128, and project the parallelogram PQRV on the plane of xy; the projections on the axes of ≈ and y of the sides PQ and QR are severally dx, dy, and dx+d2x, dy+day; so that, as was shewn in Art. 295, the area of the projected parallelogram is equal to dx d2y-dy d2x, that is, to the quantity represented by z in Art. 377. Similarly the projections of the parallelogram PQRV on the planes of yz and zx are x and y; also, as in Art. 295, area of PQRV = PQ × QR X sin RQS, = ds (ds+d2s) sin dτ, = ds2 x dτ, the same as equation (21). Hence also we have the following values of the direction-cosines of the binormal, which is perpendicular to the plane in which the parallelogram PQRS lies, viz. 382.] On torsion and the radius of torsion. Since in Art. 376 do is the angle between two consecutive binormals, and since the direction-cosines of the first binormal it follows from equation (29), Art. 379, that dw = Pdx {(a.¥)2 + (d. 1)2 + (a.Z)3}*; 2 2 d. (35) .. dw3 = (dx=xdr)2 + (dvydr) + (rdz-zdr), (36) p2 p2 (x2 + y2 + z2) (dx2 + dy2 + dz2) — (x dx + y dy+zdz)2 But differentiating the several terms of (6), we have dx = dy d3z-dz d3y, dy dz d3x-dx d3z, dz = dx d3y - dy d3x ; (37) (38) .. ydz―zdy = y (dx d3y — dy d3x) — z (dz d3x — dx d3z), = d (xd3x+xd3y+zd3z)_dr (xdx +xdy+zd): but xd+xdy+zd. =0; .. Y dz-zdy = dx (x d3x+Y d3y + z d3z); (39) and as similar values are true for the other terms of the nume Also substituting the second set of the values of the direction 383.] In reference to singular forms of the values of dr and do which have been determined by equations (27) and (42), it is to be observed that (a) If at any point of a curve dr = 0, or, which is the same thing, dx dy dz d. = 0, ds (44) then there is no change of direction of the tangent as we pass to a third point on the curve; that is, the curve becomes a straight line, or the curvature is suspended, and there is what is called a point of suspended curvature. (3) And if dr = 0, and changes its sign, then not only does the curve at the point run into and coincide with its tangent, but it intersects it; and we have what corresponds to a point of inflexion of plane curves, and is called a point of Inflected Curvature. (y) And if at all points of a curve in space dr = 0, then the line is straight, as is apparent from the three conditions (44). (8) If at any point dw= 0, or what is equivalent, the osculating plane does not change position as we pass from a third to a fourth consecutive point, and the torsion at that point is suspended, or the curve is plane; such a point is conveniently called a point of Suspended Torsion. (e) And if do = 0, and changes its sign, then the direction in which the torsion takes place is changed, and we have what is called a point of Inflected Torsion. (Š) And if at all points of a curve do 0, then the binormals are all parallel, and the curve is plane. This, it is to be observed, is the same condition as that before found in Art. 348. 384.] We proceed now to consider certain surfaces which are generated by planes and straight lines, the positions and directions of which depend on properties of curves of double curvature; and first let us consider the effects of the intersection of consecutive normal planes. Two consecutive normal planes of course generally intersect; for did they not, the elements of the curve to which they are perpendicular would be in the same straight line, which is true |