which gives again the angular relation XX.; the quotient XXIII. being thus a rector, as it ought by 393, XV. to be; and the trigonometric formula XXIV. being obtained from its expression, by observing that

XXV. . . Tp'r‍1 = PT:OT = sin POT=sin PB, and (V: S)pσ = Up'. cot PD, because op'o, but I p, p'o, or p'oo, but p, σ.

(8.) The rectangularity of the planes of т, к and т, p'a is also expressed by the equation,

XXVI... 0=S(VKT .Vp'σT) = SKTSρ'σT - T2Sp'σK;

in proving which we may employ the values,

XXVII... STK ̄11=1, Sp'σk1=(− r ̃3p'2Snp =) Sp'or1.

(9.) We may also interpret these equations XXVII., as expressing the system of the two relations,

XXVIII... --TILT, K-po;

from which it follows that 1, and therefore also that x, is a line in the plane so drawn through r, as to be perpendicular to the plane through and p'o, as before. (10.) And the two relations XXVIII. are both included in the following expression,

XXIX... K ̄l — T1 Vr'p'o: Spo.

(11.) We may also easily deduce, from the foregoing spherical construction, the following trigonometric expressions, for the arcual radius r=EP of the osculating small circle (5.), and for the angle a= PAE= EBP which it subtends at A or at B:

C

XXX... tanr= sin tan a;

XXXI... tan α = (cot A + cot в);

A and B here denoting, as in XI., the base angles of the triangle ABC with c for vertex, and c denoting as usual the buse AB, namely the portion of the arcual tangent (2.) to the conic, which is intercepted between the cyclic arcs.

(12.) The osculating plane and circle at P being thus fully and in various ways determined, we may next inquire (393) in what point q do they meet the conic again. In symbols, denoting by w the vector of this point, we have the three scalar equations,

XXXII... Skw = Skp, SλwSμw = SXpSμp, w2= p2, which are all evidently satisfied by the value = p, but can in general be satisfied also by one other vector value, which it is the object of the problem to assign. (13.) We satisfy the two first of these three equations XXXII., by assuming the expression,

XXXIII... w= § + } (x−1 7′ — xv′),

in which a is any scalar; in fact we have the relations,

XXXIV... SK = Sкp, Sλv=-2SXp, Sur' = 2Sμp,

0 = SXE=SμE = Sλr' = Sμv' = SK7' = Skv',

whence XXXIII. gives, XXXV... Sλw=xS\p, Sμw = x-1Sμp, &c.

CHAP. III.] INTERSECTION WITH OSCULATING CIRCLE.

And because

XXXVI. . . p = § + } (†' − v′),

545

we shall satisfy also the third equation XXXII., if we adopt for x any root of that new scalar equation, which is obtained by equating the square of the expression XXXIII. for w, to what that square becomes when x is changed to 1.

(14.) To facilitate the formation of this new equation, we may observe that the relations,

{=p-p", r' = p' + p", v' = p'-p", Spp' = 0, Spp" =-p'2,

which have all occurred before, give

the resulting equation is therefore, after a few slight reductions, the following biquadratic in x,

XXXVIII... 0 = (x − 1)3 (v22x − 7′2);

of which the cubic factor is to be rejected (comp. 393, (2.)), as answering only to the point P itself.

(15.) We have then the values,

XXXIX. . . x = 'v'-2, and

v '2

XL...oq=w= { +

comparing which last expression with the formulæ XVII., we see that the required point of intersection Q, of the sphero-conic with its osculating circle, can be constructed by the following rule. On the traces (4.), of the osculating plane on the two cyclic planes, determine two points T, and U1, by the conditions,

XLI... XT. XT1 = XU2,

XU.XU1 = XT2; then XLII... TQ = QUI,

or in words, the right line T11 is bisected by the sought point Q.

(16.) But a still more simple or more graphic construction may be obtained, by investigating (comp. 393, (4.)) the direction of the chord PQ. The vector value of this rectilinear chord is, by XXXVI. and XL.,

XLIII... PQ = w − p = } (v′2 − −′2) (v′-1 + 7′-1) = } (7′-2 — v'-2) r′ (7′ + v′) v'

p'2

=

'2 v's

r'p'-'v', because p′ = 1 (7′ + v′) ;

the chord PQ has therefore the direction (or its opposite) of the fourth proportional (226) to the three vectors, p', ', and – v', or PT, XT, and XU; if then we conceive this chord or its prolongations to meet the traces XT, XU in two new points T2, U2, we shall have (comp. 393, VIII.) the two inversely similar triangles (118),

XLIV... ▲ T2XU2 ∞ ́UXT.

(17.) To deduce hence a spherical construction for Q, we may conceive four planes, through the axis OKE, perpendicular respectively to the four following right lines in the osculating plane :

XLV. . . 7', - v', p', w—p, or XT, XU, PT, PQ;

which planes will cut the sphere in four great circles, whereof the four arcs,

XLVI... EF, EG, EP, EH,

are parts, if F, G, H (see again Fig. 80) be the feet of the three arcual perpendiculars from the pole E of the osculating circle on the two cyclic arcs CB, CA, and on the arcual chord PQ.

(18.) These four arcs XLVI. are therefore connected by the same angular relation as the four lines XLV.; and we have thus the very simple formula,

XLVII... GEH = PEF,

expressing an equality between two spherical angles at the pole E, which serves to determine the direction of the arc EH, and therefore also the positions of the points H and Q, by means of the relations,

(19.) If the arcual chord PQ, both ways prolonged, or any chord of the conic, cut the cyclic arcs CB and CA in the points R and s (Fig. 80), it is well known that there exists the equality of intercepts (comp. 270, (2.)),

XLIX... RP = 0 QS ;

and conversely this equation, combined with the formulæ (11.), or with the trigonometric expression,

L... tan PE = tanr=sin(cot A+ cot B),

for the tangent of the arcual radius of the osculating circle, enables us to determine what may be called perhaps the arcual chord of osculation PQ, by determining the spherical angle RPB, or simply P, from principles of spherical trigonometry alone, in a way which may serve as a verification of the results above deduced from quaternions.

(20.) Denoting by t the semitransversal RH = HS, and by s the semichord PH = HQ, the oblique-angled triangles RPB, SPA give the equations,

Equating then the values of cot 2s, deduced from LI. and LII., we eliminate s and t, and obtain a quadratic in tan P, of which one root is zero, when tan r has the value L.; such then might in this new way be inferred to be the tangent of the arcual radius of curvature of the conic, and the remaining root of the equation is then,

a formula which ought to determine the inclination P, or RPB, or QPA, of the chord Po to the tangent PA, but which does not appear at first sight to admit of any simple interpretation.*

We might however at once see from this formula, that PA - B at the plane limit; which agrees with the known construction 393, (4.), for the corresponding chord PQ in the case of the plane hyperbola.

CHAP. III.] HYPERBOLIC CYLIDER, ASYMPTOTIC PLANES. 547

(21.) On the other hand, the construction (17.) (18.), to which the quaternion analysis led us, gives

LIV... HEP GEP GEH GEP-PEF = FEB + GEA,

and therefore, by the four right-angled triangles, PHE, BFE, AGE, and BPE or EPA, conducts to this other formula,

in which a is the same auxiliary angle as in XXXI.; we ought therefore to find, as the proposed verification (19.), that this last equation LV. expresses virtually the same relation between A, B, C, and P, as the formula LIII., although there seems at first to be no connexion between them; and such agreement can accordingly be proved to exist, by a chain of ordinary trigonometric transformations, which it may be left to the reader to investigate.

(22.) A geometrical proof of the validity of the construction (17.) (18.) may be derived in the following way. The product of the sines of the arcual perpendiculars, from a point of a given sphero-conic on its two cyclic arcs, is well known to be constant; hence also the rectangle under the distances of the same variable point from the two cyclic planes is constant, and the curve is therefore the intersection of the sphere with an hyperbolic cylinder, to which those planes are asymptotic. It may then be considered to be thus geometrically evident, that the circle which osculates to the spherical curve, at any given point P, osculates also to the hyperbola, which is the section of that cylinder, made by the osculating plane at this point; and that the point q, of recent investigations, is the point in which this hyperbola is met again, by its own osculating circle at P. But the determination 393, (4.) of such a point of intersection, although above deduced (for practice) by quaternions, is a plane problem of which the solution was known; we may then be considered to have reduced, to this known and plane problem, the corresponding spherical problem (12.); and thus the inverse similarity of the two plane triangles XLIV., although found by the quaternion analysis, may be said to be geometrically explained, or accounted for: the traces XT and XU, or 7' and -v', of the osculating plane to the conic on the two cyclic planes (4.), being evidently the asymptotes of the hyperbola in question.

(23.) In quaternions, the constant product of sines, &c., is expressed by this form of the equation II. of the cone,

LVI... SUλp.SUμp=(g – S\μ): 2Tλμ = const. ;

and the scalar equation of the hyperbolic cylinder, obtained by eliminating p2 between I. and II., after the first substitution (1.), is

LVII... SλpSμp = §r2 (g – S\μ) = const. ;

while the expression XXXIII. for w may be considered as the vector equation of the hyperbola, of which the intersection Q with the circle, or with the sphere, is determined by combining that equation with the condition w2 = p2 (= − r2).

(24.) In the foregoing investigation, we have treated a sphero-conic in connexion with its cyclic ares (2.); but it would have been about equally easy to have treated the same curve, with reference to its focal points: or to the focal lines of the cone, of which it is the intersection with a concentric sphere. (Compare what has been called the bifocal transformation, in 360, (2.)).

(25.) We can however only state generally here the result of such an application of quaternions, as regards the construction of the osculating small circle to a spherical conic, considered relatively to its foci: which construction* can indeed be also geometrically deduced, as a certain polar reciprocal of the one given above. Two focal points (not mutually opposite) being called F and G, let PN be the normal are at P, which is thus equally inclined, by a well-known principle, to the two vector arcs, FP, GP; so that if the focus G be suitably distinguished from its own opposite, the spherical angle FPG is bisected by the arc PN, which is here supposed to terminate on the given arc FG. At N erect an arc QNR, perpendicular to PN, and terminating in Q and R on the two vector arcs. Perpendiculars, QE, RE, to these last arcs, will meet on the normal arc PN, in the sought pole (or spherical centre) E, of the sought small circle, which osculates to the conic at the given point P.

(26.) The two focal and arcual chords of curvature from P, which pass through F and G, and terminate on the osculating circle, are evidently bisected at Q and E, in virtue of the foregoing construction, which may therefore be thus enunciated :

:

The great circle QR, which is the common bisector of the two focal and arcual chords of curvature from a given point P, intersects the normal arc PN on the fixed arc FG, connecting the two foci; that is, on the arcual major axis of the conic.

(27.) The construction (5.) fails to determine the position of the auxiliary point D in Fig. 80, for the case when the given point P is on the minor axis of the conic; and in fact the expressions (4.) for p' and p′′ become infinite, when the denominator Όλμρ is zero. But it is easy to see that the auxiliary vector σ, which represents generally the trace of the normal plane to the curve on the plane of the two cyclic normals, becomes at the limit here considered the required axis of the osculating circle; and accordingly, if we assume simply (comp. (1.) and (2.)),

we have

=

LVIII... ' Vpo, and therefore p" Vp'o+Vpo',
LIX... σ = 0, and

Vp'p" o, when SApp = 0.

(28.) In general, if we determine three points L, M, s in the plane of Aμ, by the formula (comp. again (2.)),

then L and м will be the intersections of the cyclic normals A, μ with the tangent

*The reader can easily draw the Figure for himself. As regards the known rule, lately alluded to (in 393, (4.), and 394, (22.)), for determining the chord of intersection of a plane conic with its osculating circle, it will be found (for instance) in page 194 of Hamilton's Conic Sections (in Latin, London, 1758). The two spherical constructions, for the small circle osculating to a spherical conic, were early deduced and published by the present writer, as consequences of quaternion calculations. Compare the first Note to page 535.