we are then led in this way to the formula I., as the Equation of the Polar Plane of the point r', if that plane be here supposed to be defined by its well-known harmonic property (comp. 215, (16.), and 316, (31.), (32.)). (3.) At the same time we obtain this other form of the condition of contact III., as that of equal roots in VI., VIII. . . ƒ(p', r)2 = fr. (ƒp' − 1), the first member being an abridgment of (ƒ(p', r))2: and because this last equation VIII. is homogeneous with respect to 7, it represents a cone, namely the Cone of Tasgents (r) to the given surface fp = 1, from the given point P'. Accordingly it is easy to prove that the equation III. may be thus written, under which last form it is seen to be homogeneous with respect to p-p'. (4.) Without expressly introducing 7, the transformation IX. shows that the equation III. represents some cone, with the given point p' for its vertex; and because the intersection of this cone with the given surface is expressed by the square of the equation I. of the polar plane of that point, the cone must be (as above stated) circumscribed to the surface fp = 1, touching it along the curve (real or imaginary) in which that surface is cut by that plane I. (5.) Another important transformation, or set of transformations, of the equation III. may be obtained as follows. In general, for any two vectors p and p', if the scalar constant m, the vector function, and the scalar function F, be derived from the linear and vector function 4, which is here self-conjugate (405), by the method of the Section III. ii. 6, we have successively, X. . . ƒ(p, p')2 - ƒp. fp' = Sppp'. Sp'¢p – Sppp. Sp'op' = S(Vpp'.V¢pøp') =S.pp'Vpp' = mS. pp'p-1Vpp' =mFVpp' ; and thus the equation III. of the circumscribed cone becomes, XI... mFVpp' +ƒ(p-p') = 0, or XII... mFVrp' + fr = 0, if r = p-p' be a tangent from P'. Or because om, and m = − c1c2c3 = − a ̄b & ̃ ̈2, by 406, XXIV., we may write (with 7p-p') either or XIII... 0 ST1r + Sup1v, if v = Vrp' Vpp', as the condition of contact of the line PP' with the surface fp = 1. (6.) A geometrical interpretation, of this last form XIV. of that condition, can easily be assigned as follows. Supposing at first for simplicity that the surface is an ellipsoid, let P be the point of contact, so that fp = 1, ƒ(p, r) = 0; and let the tangent PP' be taken equal to the parallel semidiameter or, so that fr=f(p-p') = 1. Then, with the signification XIII. of v, the equation XIV. becomes, XV... √Fu Tv. V FUv=abc; in which the factor Tu represents the area of the parallelogram under the conjugate semidiameters OP, OT of the given surface fp = 1; while the other factor V FUv represents the reciprocal of the semidiameter of the reciprocal surface Fv = 1, which is perpendicular to their plane POT; or the perpendicular distance between that plane, and a parallel plane which touches the given ellipsoid: so that their product V Fu is equal, by elementary principles, to the product of the three semiaxes, as stated in the formula XV. And the result may easily be extended by squaring, to other central surfaces. CHAP. III.] AXES OF CIRCUMSCRIBED CONE. 655 (7.) It may be remarked in passing, that if p, σ, 7 be any three conjugate semidiameters of any central surface fp = 1, so that XVI. . . fp = fo=fr=1, and XVII. . . ƒ(p, σ) = f(o, 7)=ƒ(T, p) = 0, and if xp+ya+zr be any other semidiameter of the same surface, we have then the scalar equation, XVIII. . . ƒ(xp + yo + zr) = x2 + y2 + z2 = 1; a relation between the coefficients, x, y, z, which has been already noticed for the ellipsoid in 99, (2.), and in 402, I., and is indeed deducible for that surface, from principles of real scalars and real vectors alone: but in extending which to the hyperboloids, one at least of those three coefficients becomes imaginary, as well as one at least of the three vectors p, σ, T. (8.) Under the same conditions XVI. XVII., we have also, XIX... Vpo = +abcør=+(− m) ̈3ør ; XX... r = ± (− m)}p ̄1Vpo=7(− m)1Vpppo ; XXI. Spor=abc=± (− m)1 ; together with this very simple relation, ΧΧΙΙ. . . ρστ. φρφσφτ= - 1. (9.) Under the same conditions, if xp + yo+zr and x'p + y'o + z'r have only conjugate directions, that is, if they have the directions of any two conjugate semidiameters, the six scalar coefficients must satisfy (comp. II.) the equation, XXIII. . . xx' + yy' + zz' = 0. (10.) The equation VIII., with p for p', may be written under the form, XXIV... 0 SOT = STWT, if XXV. . . σ = WT = OpSpPT + PT(1 −ƒp), = = a new linear and vector function, which represents a normal to the cone of tangents from P, to the surface fp = 1. Inverting this last function, we find the equation in ☛ of the reciprocal cone, or of the cone of normals to the circumscribed cone from P, is therefore, XXVII... Sow-1σ = 0, or XXVIII. . . Fo= (Spo)2, or finally a remarkably simple form, which admits also of a simple interpretation. In fact, the line σ: Spa is the reciprocal of the perpendicular, from the centre o, on a tangent plane to the cone, which is also a tangent plane to the surface; it is therefore one of the values of the vector v (comp. (6.), and 373, (21.)), and consequently it is a semidiameter of the reciprocal surface Fv = 1. (11.) As an application of the equation XXVIII., let the surface be the confocal (e), represented by the equation 407, III. or X., of which the reciprocal is represented by 407, XVII. or XVIII. Substituting for Fo its value thus deduced, the equation of the reciprocal cone (10.), with σ for a side, becomes,* - XXIX. . . 2/2SaoSa ́o – (Spr)2 = b22, or XXIX'. . . Saoa'o – 12(Spo)2 = eo2; if then the vertex P be fixed, but the confocal vary, by a change of e, or of b2 which It may be observed that, when b= 0, this equation XXIX. represents the asymptotic cone to the auxiliary surface 407, XXIV.; and at the same time the reciprocal of that focal cone, 407, XXXVI., which rests on the focal hyperbola. varies with it, the cone XXIX. will also vary, but will belong to a biconcyclic system; whence it follows that the (direct or) circumscribed cones from a given point are all biconfocal: and also, by 407, (30.), that their common focal lines are the generating lines of the confocal hyperboloid* of one sheet, which passes through their common vertex. (12.) Changing e to e, in XXIX., and using the transformation 407, LXXV., with the identity (comp. 407, LIII.), σ we find that if ☛ be a normal to the cone of tangents from P to (e,), it satisfies the equation, XXX... 0 = (e- e,) (So Uv)2 + (e1 − e,) (SoUv1)2 + (e2 − e,) (SoUv2)* ; and therefore that if r be a tangent from the same point P, to the same confocal (e,), it satisfies this other condition, XXXI. . . 0 = (e − e,)`1 (SrUv)2 + (e1 − e,) ̄1 (SrUv1)a + (e2 − e,)-1 (SrUv1⁄2)2, which thus is a form of the equation of the circumscribed cone to (e), with its vertex at a given point P: the confocal character (11.) of all such cones being hereby exhibited anew. (13.) It follows also from XXXI., that the ares of every cone thus circumscribed have the directions of the normals v, v1, v2 to the three confocals through P; and this known theorem† may be otherwise deduced, from the Equation of Confocals (407, LXV.), by our general method, as follows. That equation gives v, – v || 4,v (because pv,=,v), and therefore, XXXII. . . (v, – v) Svv ̧ = q,v(ƒ„p − 1), VvvSvv ̧+ Vvp,v(1 −ƒ,p) = 0 ; changing then V to S, and v to T, we see that v, v1, v2, as being the roots (354) of this last vector quadratic XXXII., have the directions of the axes of the cone, with T for side, XXXIII. ..ƒ,(p, r)2 +ƒ‚r. (1 −ƒ.p) = 0 ; that is, by VIII., the directions of the axes of the cone of tangents, from P to (e). (14.) As an application of the formula XIV., with the abridged symbols – and v of (5.) for p-p' and Vpp', the condition of contact of the line PP' with the confocal (e) becomes, by the expressions 407, III., XVIII., and VII. for the functions f, F, and the squares a2, b2, c2, the following quadratic in e: XXXIV... (Sar) – 2eSarSa'r + (Sa'r)2 + (1 − e2) 72 = 1−2 (Sava'v — ev3) ; there are therefore in general (as is known) two confocals, say (e) and (e,), of a given system, which touch a given right line; and their parameters,‡ e and è̟, are the two roots of the last equation: for instance, their sum is given by the formula, XXXV... (e + e,)72 = 1-2v2 – 2SarSa'r. * This theorem (which includes that of 407, (30.)) is cited from Jacobi, and is proved, in page 143 of Dr. Salmon's Treatise, referred to in several former Notes. + Compare the second Note to page 648. e= This name of parameter is here given, as in 407, to the arbitrary constant a2 + c2 of which the value distinguishes one confocal (e) of a system from another. CHAP. III.] CIRCUMSCRibed right CONES. 657 (15.) Conceive then that p is a given semidiameter of a given confocal (e), and that do is a tangent, given in direction, at its extremity; the equation XXXIV. will then of course be satisfied,* if we change r to do, and v to Vpdp, retaining the given value of e; but it will also be satisfied, for the same p and dp (or for the same and v), when we change e to this new parameter, XXXVI... e,=-e+2SaUdp. Sa Udp – l-2 (VpUdp)2; that is to say, the new confocal (e), with a parameter determined by this last formula, will touch the given tangent to the given confocal (e). (16.) If we at once make 120 in the equation 407, III. of a Confocal System of Central Surfaces, leaving the parameter e finite, we fall back on the system 406, XXXV. of Biconfocal Cones; but if we conceive that 12 only tends to zero, and that e at the same time tends to positive infinity, in such a manner that their product tends to a finite limit, r2, or that XXXVII... lim. 70, lim.e=o, lim. el2 = r2, then the equation of the surface (e) tends to this limiting form, a system of biconfocal cones is therefore to be combined with a system of concentric spheres, in order to make up a complete confocal system. (17.) Accordingly, any given right line PP' is in general touched by only one cone of the system just referred to, namely by that particular cone (e), for which (comp. XXXIV.) we have the value, XXXIX... e= = Sava'v1, or XXXIX... e + Saa' = 2SavSa'v-1, with v=Vpp', as before, so that v is perpendicular to the given plane OPP', which contains the vertex and the line; in fact, the reciprocals of the biconfocal cones 406, XXXV., when a, a' are treated as given unit lines, but e as a variable parameter, compose the biconcyclic† system (comp. 407, XVIII.), XL... Sava'v = ev2. But, besides the tangent cone thus found, there is a tangent sphere with the same centre o; of which, by passing to the limits XXXVII., the radius r may be found from the same formula XXXIV. to be, and such is in fact an expression (comp. 316, L.) for the length of the perpendicular from the origin on the given line PP'. (18.) In general, the equation XXXIV. is a form of the equation of the cone, with p for its variable vector, which has a given vertex P', and is circumscribed to a given confocal (e). Accordingly, by making e=- -Saa' in that formula, we are * In fact it follows easily from the transformations (5.), that fp.fap-a-2b2c2F'Vpdp=ƒ(p, dp)2. + The bifocal form of the equation of this reciprocal system of cones XL. was given in 406, XXV., but with other constants (λ, μ, g), connected with the cyclic form (406, I.) of the equation of the given system. led (after a few reductions, comp. 407, XXVII.) to an equation which may be thus written, XLII... 0=12 (Saa'r)2 + 2Sap'rSa'p'T, T= with the variable side r = p- p', as before; and which differs only by the substitution of p' and r for p and v, from the equation 407, XXXVI. for that focal cone, which rests on the focal hyperbola. The other (real) focal cone which has the same arbitrary vertex P', but rests on the focal ellipse, has for equation, as is found by changing e to 1 in the same formula XXXIV. (19.) It is however simpler, or at least it gives more symmetric results, to change e, in XXXI. to - Saa' for the focal hyperbola, and to +1 for the focal ellipse, in order to obtain the two real focal cones with P for vertex, which rest on those two curves; while that third and wholly imaginary focal cone, which has the same vertex, but rests on the known imaginary focal curve, in the plane of b and c, is found by changing e, to 1. This imaginary focal cone, and the two real ones which rest as above on the hyperbola and ellipse respectively, may thus be represented by the three equations, XLIV... 0 = a 2(SrUv)2 + a1 ̃2(S+Uv1)2 + a2 ̃2(STUv1⁄2)2; XLV... 0 = b-2(StUv)2 + b1-2(StUv1)2 + b2 ̃2(STU1)2; 7 being in each case a side of the cone, and v, vi, v1⁄2 having the same significations as before. (20.) On the other hand, if we place the vertex of a circumscribed cone at a point P of a focal curve, real or imaginary, the enveloped surface being the confocal (e), we find first, by XXX., for the reciprocal cones, or cones of normals σ, with the same order of succession as in (19.), the three equations, XLVII... a2 (SUvo)2 = a,2 ; and next, for the circumscribed cones themselves, or cones of tangents r, the connected equations: L... a2 (VUvr)2 + a‚2 = 0 ; LI... b2(VUvr)2 + b,2 = 0 ; all which have the forms of equations of cones of revolution, but on the geometrical meanings of the three last of which it may be worth while to say a few words. (21.) The cone L. has an imaginary vertex, and is always itself imaginary; but the two other cones, LI. and LII., have each a real vertex P, with 62 >0 for the first, and c2 <0 for the second; b being the mean semiaxis of the ellipsoid, which passes through a given point of the focal hyperbola, and c2 being the negative and algebraically least square of a scalar semiaxis of the double-sheeted hyperboloid, which passes through a given point of the focal ellipse: while, in each case, v has the direction of the normal to the surface, which is also the tangent to the curve at that point, and is at the same time the axis of revolution of the cone. (22.) The semiangles of the two last cones, LI. and LII., have for their respective sines the two quotients, |