pendent of both the variables, s and t, while their differentials are arbitrary, and are independent of each other, we shall thus have separately (comp. V., and 337, XIII., XVII.),

XV... SpDsp = 0, SpDtp = 0.

The radius p of the sphere XI. is therefore in this way seen to have the direction of the normal at its own extremity, because it is perpendicular to two distinct tangents, Dsp and Dip, at that point; which are indeed, in the present case, perpendicular to each other also (337, (8.)).

(3.) Instead of treating the two scalar variables, x and y, or s and t, &c., as both entirely arbitrary and independent, we may conceive that one is an arbitrary (but scalar) function of the other; and then the vector v, determined by the equation III., will be seen anew to be the normal at the extremity P of p, because it is perpendicular to the tangent at P to an arbitrary curve upon the surface, which passes through that point: or (otherwise stated) because it is a line in an arbitrary normal plane at P, if a normal plane to a curve on a surface be called (as usual) a normal plane to that surface also.

(4.) For example, if we conceive that s in VIII. is thus an arbitrary function of t, the last expression XIV. will take the form,

XVI... 0 = ¿dfp = S. p (s'Dsp + Dtp) dt, if ds = s′dt ; whence, dt being still arbitrary, we have the one scalar equation,

XVII... S.p(s'Dsp + Dtp) = 0,

or XVIII...p↓ s ́D2p + D‡p,

and although, on account of the arbitrary coefficient s', this one equation XVII. is equivalent to the system of the two equations XV., yet it immediately signifies, as in XVIII., that the directed radius p, of the sphere XI., is perpendicular to the arbitrary tangent, s'Dsp + Dep; or to the tangent to an arbitrary spherical curve through P, the centre o and tensor Tp (or undirected radius, r) remaining as before.

(5.) As regards the logic of the subject, it may be worth while to read again the proof (331), of the validity of the rule for differentiating a function of a function; because this rule is virtually employed, when after thus reducing, or conceiving as reduced, the scalar function fp of a vector p, to another scalar function such as Ft of a scalar t, by treating p as equal to some vector function pt of this last scalar, we infer that

XIX... dFt = dføt = 28. vdpt, if dfp = 2Svdp, as before.

(6.) And as regards the applications of the formulæ VI. and VII., or of the equations given by them for the normal and tangent plane to a surface generally, the difficulty is only to select, out of a multitude of examples which might be given: yet it may not be useless to add a few such here, the case of the sphere having of course been only taken to illustrate the theory, because the normal property of its radii was manifest, independently of any calculation.

(7.) Taking then the equation of the ellipsoid, under the form,

of which the first differential may (see the sub-articles to 336) be thus written,

XXI... 0=S. {(t − x)p + 2 (iSkp + KSip)} dp = Svdp,

and introducing an auxiliary vector, on or , such that

XXII... ON = { = − 2 (t − k ) 2 (iSkp + кSip),

CHAP. III.] NORMALS TO SURFACES OF THE SECOND ORDER. 505

we have vp, and may write, as the equation of the normal at the extremity P of p, the following,

XXIII. . . V. (§ − p) (w− p) = 0, or XXIV... w = p + œ (§ − p),

-

in which is a scalar variable (comp. 369, VII.); making then x=1, we see that is the vector of the point N in which the normal intersects the plane of the two fixed lines, k, supposed to be drawn from the origin, which is here the centre of the ellipsoid.

(8.) If we look back on the sub-articles to 216 and 217, we shall see that these lines ↳ have the directions of the two real cyclic normals, or of the normals to the two (real) cyclic planes; which planes are now represented by the two equations, Sкp=0.

XXV... Sip = 0,

Accordingly the equation XX. of the ellipsoid may be put (comp. 336, 357, 359) under the cyclic forms,

XXVI... Spøp = (12 + x2) p2 + 2Siρkp

= (1-k)2 p2 + 4SipSkp = (x2 – 12)2 = const. ;

hence each of the two diametral planes XXV. cuts the surface in a circle, the common radius of these two circular sections being

where b denotes, as in 219, (1.), the length of the mean semiaxis of the ellipsoid; and in fact, this value of Tp can be at once obtained from the equation XX., by making either p = − pɩ, or pк=Kp, in virtue of XXV.

==

(9.) By the sub-article last cited, the greatest and least semiares have for their lengths,

XXVIII. . . a = Ti + Tk, c = Ti-TK;

and the construction in 219, (2.) shows (by Fig. 53, annexed to 217, (4.)) that these three semiaxes a, b, c have the respective directions of the lines,

XXIX... ¿TK-KTI, Vik, iTx+kTi;

all which agrees with the rectangular transformation,

in deducing which (comp. 359, (1.)) from 357, VIII., by means of the formulæ 357, XX. and XXI., we employ the values (comp. XXVI.),

XXXI. . . g = i2 + k2, λ = 21,

(10.) The fixed plane (7.), of the cyclic normals and (8.), is therefore also the plane of the extreme semiaxes, a and c (9.), or that which may be called perhaps the principal plane* of the ellipsoid: namely, the plane of the generating tri

*This plane may also be said to be the plane of the principal elliptic section (219, (9.)); or it may be distinguished (comp. the Note to page 231) as the plane of the focal hyperbola, of which important curve we shall soon assign the equation in quaternions.

angle (218), (1.)), in that construction of the surface (217, (6.) or (7.)) which is illustrated by Fig. 53, and was deduced as an interpretation of the quaternion equation XX., or of the somewhat less simple form 217, XVI., with the value Ti2- T2 of t2.

(11.) Let n denote the length of that portion of the normal, which is intercepted between the surface and the principal plane (10.), so that, by (7.),

XXXII. . . n = NP = T(p), n2 = - (p − E)2,

= 1,

with the value XXII. of E. Let σ = os be the vector of a point s on the surface of a new or auxiliary sphere, described about the point N as centre, with a radius and therefore tangential to the ellipsoid at P; and let us inquire in what curve or curves, real or imaginary, does this sphere cut the ellipsoid.

(12.) The equations (comp. 371, (5.)) of the sought intersection are the two following,

XXXIII. . . (♂ – {)2 + n2 = 0, and XXXIV.......T(10 + ok) = k2 − 12 ;

whereof the first expresses that s is a point of the sphere, and the second that it is a point of the ellipsoid; while p or op enters virtually into XXXIII., through and n, but is here treated as a constant, the point P being now supposed to be a given one. (13.) We shall remove (18) the origin to this point P of the ellipsoid, if we write,

XXXV...σ=p+ o', or XXXV'... σ' = σ − p = PS ;

and thus we obtain the new or transformed equations,

XXXVI. . . 0 = o22 + 2S (p − €) o',

XXXVII. . . 0 = N(10′ + o'k) + 2Svo';

in which (as in (7.), comp. also 210, XX.),

and

XXXVIII... v=
= (1 − k)2p + 2 (iSkp + kSip) = (i − k)2 (p − E),
XXXIX. . . N (10′ + o'k) = (1 − x)2 o'2 + 4 Sio'Sko'.

(14.) Eliminating then '2, we obtain from the two equations XXXVI. and XXXVII. this other,

XL... Sto'. Sko' = 0;

which like them is of the second degree in o', but breaks up, as we see, into two linear and scalar factors, representing two distinct planes, parallel by XXV. to the two diametral and cyclic planes of the ellipsoid. The sought intersection consists then of a pair of (real) circles, upon that given surface; namely, two circular (but not diametral) sections, which pass through the given point P.

(15.) Conversely, because the equations XXXVII. XXXVIII. XXXIX. XL. give XXXVI. and XXXIII., with the foregoing values of 1⁄2 and n, it follows that these two plane sections of the ellipsoid at P are on one common sphere, namely that which has N for centre, and n for radius, as above; and thus we might have found, without differentials, that the line PN is the normal at P; or that this normal crosses the principal plane (10.), in the point determined by the formula XXII.

(16.) In general, the cyclic form of the equation of any central surface of the second order, namely the form (comp. 357, II.),

XLI... Spop=g'p2 + 2SXpSμp = C=const.,

shows that the two circles (real or imaginary) in which that surface is cut by any two planes,

XLII... Sλp=1, Sμp=m,

CHAP. III.]

RECIPROCAL SURFACES.

507

drawn parallel respectively to the two real cyclic planes, which are jointly represented (comp. XL., and 216, (7.)) by the one equation,

XLIII... SApSμp = 0,

are homospherical, being both on that one sphere of which the equation is,

(17.) But the centre (say N) of this new sphere, has for its vector (say ),

XLV... ON = { = − g'-'(lμ + mλ);

it is therefore situated in the plane of the two real cyclic normals, λ and μ; and if 7 and m in XLV. receive the values XLII., then this new is the vector of intersection of that plane, with the normal to the surface at P: because it is (comp. 15.)) the vector of the centre of a sphere which touches (though also cutting, in the two circular sections) the surface at that point.

(18.) We can therefore thus infer (comp. again (15.)), without the differential calculus, that the line,

XLVI... g'(p − ) = g′p + λSμp + μSλp = $p,

as having the direction of NP, is the normal at P to the surface XLI.; which agrees with, and may be considered as confirming (if confirmation were required), the conclusion otherwise obtained through the differential expression (361),

XLVII... dSpop = 2Svdp=2Sppdp;

the linear function pp being here supposed (comp. 361, (3.)) to be self-conjugate. (19.) Hence, with the notation 362, I., the equation of the tangent plane to a central surface of the second order, at the same point P, may by VII. be thus written,

XLVIII... f(w, p)= C, if Sppp = C = const.;

in which it is to be remembered, that

XLIX... f(w, p) = f(p, w) = Swop = Spow.

(20.) And if we choose to interpret this equation XLVIII., which is only of the first degree (362) with respect to each separately of the two vectors, p and w, or op and OR, and involves them symmetrically, without requiring that P shall be a point on the surface, we may then say (comp. 215, (13.), and 316, (31.)), that the formula in question is an equation of conjugation, which expresses that each of the two points P and R, is situated in the polar plane of the other.

(21.) In general, if we suppose that the length and direction of a line v are so adjusted as to satisfy the two equations (comp. 336, XII. XIII. XIV.),

L... Svp = 1, Svdp = 0, and therefore also LI... Spdv=0; then, because the equation VII. of the tangent plane to any curved surface may now be thus written,

LII... Sv (v - v ̄1) = 0,

it follows that represents, in length and direction, the perpendicular from o on that tangent plane at P; so that v itself represents the reciprocal of that perpendicular, or what may be called (comp. 336, (8.)) the vector of proximity, of the tangent plane to the origin. And we see, by LI., that the two vectors, p and v, if drawn from a common origin, terminate on two surfaces which are, in a known and

important sense (comp. the sub-arts. to 361), reciprocals* of one another: the line p-1, for instance, being the perpendicular from o on the tangent plane to the second surface, at the extremity of the vector v.

374. In the two preceding Articles, we have treated the symbol dp as representing (rigorously) a tangent to a curve on a given surface, and therefore also to that surface itself; and thus the formula Svdp=0 has been considered as expressing that has the direction of the normal to that surface, because it is perpendicular to two tangents (372), and therefore generally to every tangent (373), which can be drawn at a given point P. But without at present introducing any other signification for this symbol dp, we may interpret in another way, and with a reference to chords rather than to curves, the differential equation,

I... dfp=2Svdp,

supposed still to be a rigorous one (in virtue of our definitions of differentials, which do not require that dp should be small); and may still deduce from it the normal property of the vector v, but now with the help of Taylor's Series adapted to quaternions (comp. 342, 370). In fact, that series gives here a differenced equation, of the form,

II... Afp=2SvAp+ R;

where R is a scalar remainder (comp. again 342), having the property that

whence

III... lim. (R: TAp)=0, if lim. TAp=0;
IV... lim. (Afp: TAP) = 2 lim. SUAP,

whatever the ultimate direction of Ap may be. If then we conceive that

* Compare the Note to page 484.

It is permitted, for example, by general principles above explained, to treat the differential dp as denoting a chordal vector, or to substitute it for Ap, and so to represent the differenced equation of the surface under the form (comp. 342),

0 = Afp = (ed-1) fp = dƒp + }d2ƒp + &c. ;

but with this meaning of the symbol dp, the equation dfp = 0, or Svdp = 0, is no longer rigorous, and must (for rigour) be replaced by such an equation as the following,

0 = 2Svdp + Sdvdp + R, if dfp = 2Svdp, as before;

the remainder R vanishing, when the surface is only of the second order (comp. 362, (3.)). Accordingly this last form is useful in some investigations, especially in those which relate to the curvatures of normal sections: but for the present it seems to be clearer to adhere to the recent signification of dp, and therefore to treat it as still denoting a tangent, which may or may not be small.