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(16.) Whether the mean semiaxis (b) be real or imaginary, the surface III. (supposed to be itself real) is always, by the form XLIV. of its equation, the locus of a system of real ellipses (comp. 404, (1.)), in planes parallel to the director plane XLVII., which have their centres on the focal line a, and are orthogonally projected into circles on a plane perpendicular to that line.

(17.) The same surface is also the locus of a second system of such ellipses, related similarly to the second focal line a', and to the second director plane XLVII'.; and it appears that these two systems of elliptic sections of a surface of the second order, which from some points of view are nearly as interesting as the circular sections, may conveniently be called its Centro-Focal Ellipses.

(18.) For example, when the first quaternion form (204, (14.), or 404, I.) of the equation of the ellipsoid is employed, one system of such ellipses coincides with the system (204, (13.)) of which, in the first generation* of the surface, the ellipsoid

Besides that first generation (I) of the Ellipsoid, which was a double one, in the sense that a second system (17.) of generating ellipses might be employed, and which served to connect the surface with a concentric sphere, by certain relations of homology (274); and the second double generation or construction (II), by means of either of two diacentric spheres (217, (4.), (6.), (7.), and 220, (3.)), which was illustrated by Fig. 53 (page 226): several other generations of the same important surface were deduced from quaternions in the Lectures, to which it is only possible here to refer. A reader, then, who happens to have a copy of that earlier work, may consult page 499 for a generation (III) of a system of two reciprocal ellipsoids, with a common mean axis (2b), by means of a moving sphere, of which the radius (=b) is given, but of which the centre has the original ellipsoid for its locus; while the corresponding point on the reciprocal surface, and also the normals at the two points, are easily deduced from the construction. In page 502, he will find another and perhaps a simpler generation (IV), of the same pair of reciprocal ellipsoids, by means of quadrilaterals inscribed in a fixed sphere (the common mean sphere, comp. 216, (10.)); the directions of the four sides of such a quadrilateral being given, and one pair of opposite sides intersecting in a point of one surface, while the other pair have for their intersection the corresponding point of the other (or reciprocal) ellipsoid. In the page last cited, and in the following page, there is given a new double generation (V) of any one ellipsoid; its circular sections (of either system) being constructed as intersections of two equal spheres (or spheric surfaces), of which the line of centres retains a fixed direction, while the spheres slide within two equal and right cylinders, whose axes intersect each other (in the centre of the generated surface), and of which the common radius is the mean semiaxis (b). Finally, in page 699 of the same volume, there will be found a new generation (VI) of the original ellipsoid (abc), analogous to the generation (IV) by the fixed (mean) sphere, but with new directions of the sides of the quadrilaterals, which are also (in this last generation) inscribed in the circles of a certain mean ellipsoid (or prolate spheroid) of revolution, which has the mean axis (2b) for its major axis, and has two medial foci on that axis, whose common distance from the centre is represented by the expression,

V(a2-b2) V (b2 - c2)
V (a2 - b2 + c2)


was treated as the locus; and an analogous generation of the two hyperboloids, by geometrical deformation of two corresponding surfaces of revolution, with certain resulting homologies (comp. sub-arts. to 274), through substitution of (centro-focal) ellipses for circles, conducts to equations of those hyperboloids of the same unifocal form; namely, if a and ẞ have significations analogous to those in the cited equation of the ellipsoid (so that ẞ and not a is here a focal line),

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the upper or the lower sign being taken, according as the surface consists of one sheet or of two.

(19.) It may also be remarked that as, by changing ẞ to a in the corresponding equation of the ellipsoid, we could return (comp. 404, (1.)) to a form (403, XI.) of the equation of the sphere, so the same change in XLVIII. conducts to equations of the equilateral hyperboloids of revolution, of one sheet and of two, under the very simple forms (comp. 210. XI.),

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in which it seems unnecessary to insert points after the signs S, and of which the geometrical interpretations become obvious when then they are written thus (comp. 199, V.),

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(20.) The real cyclic forms of the equation of the surface III. might be deduced from the unifocal form XLIV., by the general method of the subarticles to 359; but since we have ready the rectangular form X., it is simpler to obtain them from that form, with the help of the identity,

LIII... - p2 = (SpU (a +a'))2 + (SpUVaa')a+ (SpU (a − a'))2,

by eliminating the first of these three terms for the case of a single-sheeted hyperbo

the common tangent planes, to this mean (or medial) ellipsoid, and to the given (or generated) ellipsoid (abc), which are parallel to their common axis (2b), being parallel also to the two umbilicar diameters of the latter surface.

* The same forms, but with σ for p, and ẞ for a, may be deduced from XLVIII. on the plan of 274, (2.), (4.), by assuming an auxiliary vector σ such that






and V


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V ; the homologies, above alluded to, between the general

hyperboloid of either species, and the equilateral hyperboloid of revolution of the same species, admitting also thus of being easily exhibited.



loid (for which b-2>a-2>0>c-2); the second for an ellipsoid (c-2> b-2>a-2 > 0); and the third for a double-sheeted hyperboloid (a-2>0>c-2>b-2).

(21.) Whatever the species of the surface III. may be, we can always derive from the unifocal form XLIV. of its equation what may be called an Exponential Transformation; namely the vector expression,

LIV... p = xa + yVaß, with LV... x2ƒa + y2ƒUVaa' = 1;

the scalar exponent, t, remaining arbitrary, but the two scalar coefficients, x and y, being connected by this last equation of the second degree: provided that the new constant vector ẞ be derived from a, a', and e, by the formula,

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which gives after a few reductions (comp. the expression 315, III. for a', when Ta = 1),

LVII... Vaẞ= UVaa', S(a' - ea) ẞ=0, Saa'ß = 0;

LVIII... Vatß=ẞS. at + UVaa'. S. at-1; LIX... V. aVa'ß= a1UVaa' = T ̄11; LX... S(a' — ea)p=x(e + Saa'), Vap=ya1UVaa' ;


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LXI... fa=a2bc, and LXII. . . ƒß=ƒUVaa' = b2.


(22.) If we treat the exponent, t, as the only variable in the expression LIV. for P1 then (comp. 314, (2.)) that exponential expression represents what we have called (17.) a centro-focal ellipse; the distance of its centre (or of its plane) from the centre of the surface, measured along the focal line a, being represented by the coefficient x; and the radius of the right cylinder, of which the ellipse is a section, or the radius of the circle (16.) into which that ellipse is projected, on a plane being represented by the other coefficient, y: while tπ is the excentric anomaly. (23.) If, on the contrary, we treat the exponent t as given, but the coefficients x and y as varying together, so as to satisfy the equation LV. of the second degree, the expression LIV. then represents a different section of the surface III., made by a plane through the line a, which makes with the focal plane (of a, a') an angle ἐπ



; this latter section (like the former) being always real, if the surface itself be such but being an ellipse for an ellipsoid, and an hyperbola for either hyperboloid, because

LXIII... fa.ƒUVaa' = a-2c-2 by LXI. and LXII.

(24.) And it is scarcely necessary to remark, that by interchanging a and a' we obtain a Second Exponential Transformation, connected with the second system (17.) of centro-focal ellipses, as the first exponential transformation LIV. is connected with the first system (16.).

(25.) The asymptotic cone fp = 0 has likewise its two systems of centro-focal ellipses, and its equation admits in like manner of two exponential transformations, of the form LIV.; the only difference being, that the equation LV. is replaced by the following,

LXIV... x2fa + y2ĆVaa' = 0,

in which, for a real cone, the coefficients of z2 and y2 have opposite signs by (23.). (26.) Finally, as regards the confocal relation of the surfaces III., which may represent any confocal system of surfaces of the second order, it may be perceived

from (4.) that an essential character of such a relation is expressed by the equation,

LXV. . . . Vv ̧¢v,=Vv¢ ̧v ;

which may perhaps be called, on that account, the Equation of Confocals.

(27.) It is understood that the two confocal surfaces here considered, are reprepresented by the two scalar equations,

LXVI.... Spøp =1, Spp,p=1, or LXVI... fp=1, ƒ„p=1;

and that the two linear and vector functions, v and v, of an arbitrary vector p, which represent normals to the two concentric and similar and similarly posited surfaces,

LXVII... fp = const., fp = const.,

passing through any proposed point P, are expressed as follows,

LXVIII... v=qp, v1 = $,p.

(28.) It is understood also, that the two surfaces LXVI. or LXVI. are not only concentric, as their equations show, but also coaxal, so far as the directions of their axes are concerned: or that the two vector quadratics (comp. 354),

LXIX... Vpop = 0, and LXX... Vpo,p=0,

are satisfied by one common system of three rectangular unit lines. And with these understandings, it will be found that the equation LXV., which has been called above the Equation of Confocals, is not only necessary but sufficient, for the establishment of the relation required.

(29.) It is worth while however to observe, before closing the present series of subarticles, that the equations XII., and those formed from them by introducing e2 and 2, give the following among other relations :

LXXI. . . ƒUv1 = (b2 — b12) ̄1 = − ƒ¡Uv; ƒ¡Uvq=(b1a — by2)·1 = −ƒ1⁄2Uv¡ ; &c. ; LXXII. . .ƒ(v1, v2)=f1(V2, V)=ƒ1⁄2(v, v1) = 0 ;


and therefore,

LXXIII. . . ƒı { (bg2 — b12)4Uv2 + (b18 — b2)1Uv } = 0 ;

whence it is easy to see that the two vectors under the functional sign fi in this last expression have the directions of the generating lines of the single-sheeted hyperboloid (e1) through P, if we suppose that b22 >b12>0>b2, so that the confocal (e2) is here an ellipsoid, and (e) a double-sheeted hyperboloid.

(30.) But if o be taken to denote the variable vector of the auxiliary surface XXIV., the equation of that surface may by (7.) and (8.) be brought to the following rectangular form, with the meaning XXI. of w,

LXXIV... 1 = Sowo = (Spo)2 - 21aSaoSa'o = b2 (SoUv)2

+b12(SσUv1)2+by2 (SoU¥2)3;

hence, with the inequalities (29.), its cyclic normals, or those of its asymptotic cone Sowo = 0, or the focal lines of the reciprocal cone Sow lo= 0, that is of the cone XXXVI., or finally the focal lines of the focal* cone (12.), which rests on the focal hyperbola, have the directions of the lines LXXIII.; those focal lines are therefore

* A more general known theorem, including this, will soon be proved by quaternions.

CHAP. III.] CONdition of contact with right line. 653

(by what has just been seen) the generating lines of the hyperboloid (e1), which passes through the given point P.

(31.) And for an arbitrary σ we have the transformation,

LXXV... 1-2(Spo)2 – Saoa'o = e (So Uv)2 + e1 (SoUv1)3 + €2(SoUv2)3.


408. The general equation of conjugation,

I... (fp, p') = 1,

405, III.

connecting the vectors p, p' of any two points P, P' which are conjugate with respect to the central but non-conical surface fp = 1, may be called for that reason the Equation of Conjugate Points; while the analogous equation,

II... f(p, p')=0,

which replaces the former for the case of the asymptotic cone fp = 0, may be called by contrast the Equation of Conjugate Directions: in fact, it is satisfied by any two conjugate semidiameters, as may be at once inferred from the differential equation f(p, dp) = 0 of the surface fp=const. (comp. 362). Each of these two formulæ admits of numerous applications, among which we shall here consider the deduction, and some of the transformations, of the Equation of a Circumscribed Cone,


III... (f(p, p')-1)=(fp-1) (fp' − 1);

may also be considered as the Condition of Contact, of the right line PP' with the surface fp = 1.

(1.) In this last view, the equation III. may be at once deduced, as the condition of equal roots in the scalar and quadratic equation (comp. 216, (2.), and 316, (30.)),


IV... 0 =ƒ(xp+x′'p') − (x + x')2,

V... 0 = x2(fp - 1) + 2xx′ (ƒ(p, p′) − 1) + x22 (ƒp′ − 1);

which gives in general the two vectors of intersection, as the two values of the ex

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(2.) If we treat the point r' as given, and denote the two secants drawn from it in any given direction 7 by t1 ́1r and t1⁄2‍1, then t1 and t2 are the roots of this other quadratic, f(p+t17) = 1, or

VI. . . 0 = f(tp' + r) — t2 = t2 ( ƒp' − 1) + 2tf(p', r) + fr;

denoting then by for the harmonic mean of these two secants, so that 2toti + 12, and writing p = p' + to'r, we have

VII. . . to(1 − ƒp')=f(p', t), _ƒ(p, p′) = 1 ;

* For the notation used, Art. 362 may be again referred to.

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