2dp)= 0; 629 which agrees with the equation 392, VI., although deduced in a quite different manner, and conducts anew to the expression II. for x-p, under the form, (7.) Again, if OD = 8 be the diameter from the origin, of any sphere through that point o, which passes also through any three other given points A, B, C, with OA = a, &c., we have by 296, XXVI. the formula, XX... 8Saßy = Va(B-a) (y-ẞ)y; writing then (comp. XVII.), and XXI... a= dp, B-a=dp+d2p, y-ẞ= dp + 2d2p + d3p, XXII... d=2ps = 2(σ - p), where σ is (as in 395, &c.) the vector os (from an arbitrary origin o) of the centre s of the osculating sphere to a curve of double curvature at P, we have by infinitesimals, suppressing terms which are of the seventh and higher orders, because the first member is only of the sixth order, and reducing* by the rules of quaternions, XXIII... (o-p)Sdpd2pd3p = 4Vdp (dp + d2p) (dp +2d2p + d3p) (3dp + 3d3p + d3p) = 3Vdpd2pSdpd2p + dp2Vd3pdp ; which agrees precisely with the formula 395, XIII., although obtained by a process so different. (8.) Finally as regards the osculating plane, and the second curvature, of a curve in space, infinitesimals give at once for that plane the equation, XXIV. . . S (w − p) dpd2p = 0, agreeing with 376, V.; and if three consecutive elements of the curve be represented (comp. XXI.) by the differential expressions, XXV... PQ = dp, QR = dp + d2p, RS = dp + 2d2p + d3p, the second curvature r1, defined as in 396, is easily seen to be connected as follows with the angle of a certain auxiliary quaternion q, which differs infinitely little from unity: XXVI... r-ids = 4%, if XXVII. V(QR RS) = 1+ Vdpd3p * Of the eighteen terms which would follow the sign of operation V, if the second member of XXIII. were fully developed, one is of the fourth order, but is a scalar; three are of the fifth order, but have a scalar sum; nine are of orders higher than the sixth; and two terms of the sixth order are scalars, so that there remain only three terms of that order to be considered. In this manner it is found that the second member in question reduces itself to the sum of the two vector parts, and V. (dpd2p)2 = 3Vdpd2p. Sdpd2p, dp2V(dpd3p+3d3pdp)= dp2Vd3pdp; and thus the third member of XXIII. is obtained. which agrees with the formula 397, XXVII., and has been illustrated, in the subarticles to 397 and 398, by numerous geometrical applications. (9.) On the whole, then, it appears that although the logic of derived vectors, and of differentials of vectors considered as finite lines, proportional to such dericatives, is perhaps a little clearer than that of infinitesimals, because it shows more evidently (especially when combined with Taylor's Series adapted to Quaternions, 342, 375) that nothing is neglected, yet it is perfectly possible to combine* quaternions, in practice, with methods founded on the more usual notion of Differentials, as infinitely small differences: and that when this combination is judiciously made, abridgments of calculation arise, without any ultimate error. SECTION 7.-On Surfaces of the Second Order; and on Curvatures of Surfaces. 402. As early as in the First Book of these Elements, some specimens were given of the treatment or expression of Surfaces of the Second Order by Vectors; or by Anharmonic Equations which were derived from the theory of vectors, without any introduction, at that stage, of Quaternions properly so called. Thus it was shown, in the sub-articles to 98, that a very simple anharmonic equation (xz = yw) might represent either a ruled paraboloid, or a ruled hyperboloid, according as a certain condition (ac = bd) was or was not satisfied, by the constants of the surface. Again, in the sub-articles to 99, two examples were given, of vector expressions for cones of the second order (and one such expression for a cone of the third order, with a conjugate ray (99, (5.)); while an expression of the same sort, namely, I. p = xa + yẞ + 27, with x2+y+ z2 = 1, was assigned (99, (2.)) as representing generally an ellipsoid,† with a, ß, y, or oa, OB, OC, for three conjugate semidiameters. And finally, *Compare the first Note to page 623. It will however be of course necessary, in any future applications of quaternions, to specify in which of these two senses, as a finite differential, or as an infinitesimal, such a symbol as dp is employed. In like manner the expression, II. . . p = xa+yß+2y, with x2 + y2 − z2 = 1, or = - - 1, represents a general hyperboloid, of one sheet, or of two, with aẞy for conjugate semidiameters: while, with the scalar equation x2 + y2-22 0, the same vector expression represents their common asymptotic cone (not generally of revolution). = CHAP. III.] QUATERNION EQUATIONS OF SPHere. 631 in the sub-articles (11.) and (12.) to Art. 100, an instance was furnished of the determination of a tangential plane to a cone, by means of partial derived vectors. 403. In the Second Book, a much greater range of expression was attained, in consequence of the introduction of the peculiar symbols, or characteristics of operation, which belong to the present Calculus; but still with that limitation which was caused, by the conception and notation of a Quaternion being confined, in that Book, to Quotients of Vectors (112, 116, comp. 307, (5.)), without yet admitting Products or Powers of Directed Lines in Space: although versors, tensors, and even norms of such vectors were already introduced (156, 185, 273). (1.) The Sphere,† for instance, which has its centre at the origin, and has the vector OA, or a, with a length Ta = a, for one of its radii, admitted of being represented, not only (comp. 402, I.) by the vector expression, but also by any one of the following equations, in which it is permitted to change a representing a system of circles, with the spheric surface for their locus. 204, (4.) *The notation Na, for (Ta)2, although not formally introduced before Art. 273, had been used by anticipation in 200, (3.), page 188. That is to say, the spheric surface through A, with o for centre. Compare the Note to page 197. (2.) Other forms of equation, for the same spheric surface, may on the same principles be assigned; for example we may write, under which last form, the sphere may be considered to be generated by the revolution of the circle, which has been already spoken of as the Apollonian* Locus. (3.) And from any one to any other, of all these various forms, it is possible, and easy to pass, by general Rules of Transformation,† which were established in the Second Book: while each of them is capable of receiving, on the principles of the same Book, a Geometrical Interpretation. (4.) But we could not, on the principles of the Second Book alone, advance to such subsequent equations of the same sphere, as XXIII... p2 = a2, or XXIV... p2 + a2 = 0, 282, VII. XIII. whereof the latter includes (282, (9.)) the important equation p2 + 1 = 0, or p2 = − 1, of what we have called the Unit-Sphere (128); nor to such an exponential expression for the variable vector p of the same spheric surface, as in which j and k belong to the fundamental system ijk of three rectangular unitlines (295), connected by the fundamental Formula A of Art. 183, namely, ia = j2 = k2 = ijk = − 1, (A) while s and t are two arbitrary and scalar variables, with simple geometrical‡ significations: because we were not then prepared to introduce any symbol, such as p2, or k', which should represent a square or other power of a vector.§ And similar re *Compare the first Note to page 128. This richness of transformation, of quaternion expressions or equations, has been noticed, by some friendly critics, as a characteristic of the present Calculus. In the preceding parts of this work, the reader may compare pages 128, 140, 183, 573, 574, 575; in the two last of which, the variety of the expressions for the second curvature (r-1) of a curve in space may be considered worthy of remark. On the other hand, it may be thought remarkable that, in this Calculus, a single expression, such as that given by the first formula (389, IV.) of page 532, adapts itself with equal ease to the determination of the vector (*) of the centre of the osculating circle, to a plane curve, and to a curve of double curvature, as has been sufficiently exemplified in the foregoing Section. Compare the second Note to page 365. § It is true that the formula A was established in the course of the Second Book (page 160); but it is to be remembered that the symbols ijk were there treated as denoting a system of three right versors, in three mutually rectangular planes (181): CHAP. III.] EQUATION OF ELLIPSOID RESUMED. 633 marks apply to the representation, by quaternions, of other surfaces of the second order. 404. A brief review, or recapitulation, of some of the chief expressions connected with the Ellipsoid, for example, which have been already established in these Elements, with references to a few others, may not be useless here. (1.) Besides the vector expression p=xa+yß+zy, with the scalar relation x2 + y2+ 22 = 1, and with arbitrary vector values of the constants a, ß, y, which was lately cited (402) from the First Book, or the equations 403, I., without the conditions 403, I., II'. which are peculiar to the sphere, there were given in the Second Book (204, (13.), (14.)) equations which differed from those lately numbered as 403, XI. XII. XIII. XIV. XV., only by the substitution of V for V for instance, there was the equation, α 204, (14.) analogous to 403, XI., and representing generally* an ellipsoid, regarded as the locus of a certain system of ellipses, which were thus substituted for the circlest (403, XV.) of the sphere, by a species of geometrical deformation, which led to the establishment of certain homologies (developed in the sub-articles to 274). although it has since been found possible and useful, in this Third Book, to identify those right versors with their own indices or axes (295), and so to treat them as a system of three rectangular lines, as above. L * In the case of parallelism of the two vector constants (3 || a), the equation I. represents generally a Spheroid of revolution, with its axis in the direction of a; while in the contrary case of perpendicularity (ß ± a), the same equation I. represents an elliptic Cylinder, with its generating lines in the direction of ß. Compare 204, (10.), (11.), and the Note to page 224. + The equation I. might also have been thus written, on the principles of the Second Book, )( 1)+()=1; a whence it would have followed at once (comp. 216, (7.)), that the ellipsoid I. is cut in two circles, with a common radius = Tẞ, by the two diametral planes, In fact, this equation I'. is what was called in 359 a cyclic form, while I. itself is what was there called a focal form, of the equation of the surface; the lines a1±ß-1 being, by the Third Book, the two (real) cyclic normals, while ẞ is one of the two (real) focal lines of the (imaginary) asymptotic cone. Compare the Note to page 474. |