296. There are a few particular but remarkable cases, of ternary and other products of vectors, which it may be well to mention here, and of which some may be worth a student's while to remember: especially as regards the products of successive sides of closed polygons, inscribed in circles, or in spheres.

(1.) If A, B, C, D be any four concircular points, we know, by the sub-articles to 260, that their anharmonic function (ABCD), as defined in 259, (9.), is scalar; being also positive or negative, according to a law of arrangement of those four points, which has been already stated.

(2.) But, by that definition, and by the scalar (though negative) character of the square of a vector (282), we have generally, for any plane or gauche quadrilateral ABCD, the formula:

I... e2 (ABCD) = AB. BC. CD. DA = the continued product of the four sides; in which the coefficient e2 is a positive scalar, namely the product of two negative or of two positive squares, as follows:

II. . . e2 = BC2. DA2 = BC2. DA2>0.

(3.) If then ABCD be a plane and inscribed quadrilateral, we have, by 260, (8.), the formula,

III... AB. BC. CD. DA = a positive or negative scalar,

according as this quadrilateral in a circle is a crossed or an uncrossed one.

(4.) The product aẞy of any three complanar vectors is a vector, because its scalar part Saẞy vanishes, by 294, (3.) and (4.); and if the factors be three successive sides AB, BC, CD of a quadrilateral thus inscribed in a circle, their product has either the direction of the fourth successive side, DA, or else the opposite direction, or in symbols,

IV... AB. BC.CD: DA> or < 0,

according as the quadrilateral ABCD is an uncrossed or a crossed one.

(5.) By conceiving the fourth point D to approach, continuously and indefinitely, to the first point A, we find that the product of the three successive sides of any plane triangle, ABC, is given by an equation of the form:

V... AB. BC.CA = AT;

A

P

AT being a line (comp. Fig. 63) which touches the circumscribed circle, or (more fully) which touches the segment ABC of that circle, at the point A; or represents the initial direction of motion, along the cir- T cumference, from A through B to C: while the length of this tangential product-line, AT, is equal to, or represents, with the usual reference to an unit of length, the product of the lengths of the three sides, of the same inscribed triangle ABC.

Fig. 63.

(6.) Conversely, if this theorem respecting the product of the sides of an inscribed triangle be supposed to have been otherwise proved, and if it be remembered, then since it will give in like manner the equation,

CHAP. I.] PRODUCTS OF SIDES OF INSCRIBED POLYGONS. 325

VI... AC.CD.DA=AU,

D

if D be any fourth point, concircular with A, B, C, while AU is, as in the annexed Figures 63, a tangent to the new segment ACD, we can recover easily the theorem (3.), respecting the product of the sides of an inscribed quadrilateral; and thence can return to the corresponding theorem (260, (8.)), respecting the anharmonic function of any such figure B ABCD: for we shall thus have, by V. and VI., the equation,

VII... AB. BC.CD. DA=(AT. AU): (ca.ac),

T

A

U

Fig. 63, bis.

in which the divisor CA. AC or N. AC, or AC2, is always positive (282, (1.)), but the dividend AT. AU is negative (281, (9.)) for the case of an uncrossed quadrilateral (Fig. 63), being on the contrary positive for the other case of a crossed one (Fig. 63, bis).

(7.) If P be any point on the circle through a given point a, which touches at a given origin o a given line or=7, as represented in Fig. 64, we shall then have by (5.) an equation of the form,

form of the equation of the circle, determined by the given conditions.

(8.) Geometrically, the last formula IX." expresses, that the line p ̈1-a-1, or Rp-Ra, or A'r' (see again Fig. 64), if os' a ̈1 = Ra = R. OA, and op' = p-1 = R. OP, is parallel to the given tangent r at 0; which agrees with Fig. 58, and with Art.

260.

(9.) If B be the point opposite to o upon the circle, then the diameter OB, or ß, as being +7, so that 7B-1 is a vector, is given by the formula,

X... Tẞ-1 = Vτa1; or X'... B =-τ:ται;

in which the tangent r admits, as it ought to do, of being multiplied by any scalar, without the value of 3 being changed.

(10.) As another verification, the last formula gives,

XI... OBTẞ= Ta: TVUra ̄1 = QA : sin AOT.

(11.) If a quadrilateral OABC be not inscriptible in a circle, then, whether it be plane or gauche, we can always circumscribe (as in Fig. 65) two circles, OAB and OBC, about the two triangles, formed by drawing the diagonal OB; and then, on the plan of (6.), we can draw two tangents OT, OU, to the two segments OAB, OBC, so as to represent the two ternary products,

after which we shall have the quaternary product,

XII... OA. AB. BC.CO = OT. OU: OB2;

where the divisor, OB2, or BO. OB, or N. OB, is a positive scalar, but the dividend or. Ou, and therefore also the quotient in the second member, or the T product in the first member, is a quaternion.

(12.) The axis of this quaternion is perpendicular to the plane TOU of the two tangents; and therefore to the plane itself of the quadrilateral OABC, if that be a plane figure; but if it be gauche, then the axis is normal to the circumscribed sphere at the point o: being also in all cases such, that the rotation round it, from or to ou, is positive.

Fig. 65.

(13.) The angle of the same quaternion is the supplement of the angle Tou between the two tangents above mentioned; it is therefore equal to the angle u’or, if ou' touch the new segment OCB, or proceed in a new and opposite direction from o (see again Fig. 65); it may therefore be said to be the angle between the two arcs, OAB and OCB, along which a point should move, in order to go from o, on the two circumferences, to the opposite corner B of the quadrilateral OABC, through the two other corners, A and c, respectively: or the angle between the arcs OCB, OAB.

(14.) These results, respecting the axis and angle of the product of the four successive sides, of any quadrilateral OABC, or ABCD, apply without any modification to the anharmonic quaternion (259, (9.)) of the same quadrilateral; and although, for the case of a quadrilateral in a circle, the axis becomes indeterminate, because the quaternary product and the anharmonic function degenerate together into scalars, or because the figure may then be conceived to be inscribed in indefinitely many spheres, yet the angle may still be determined by the same rule as in the general case this angle being =π, for the inscribed and uncrossed quadrilateral (Fig. 63); but = 0, for the inscribed and crossed one (Fig. 63, bis).

(15.) For the gauche quadrilateral OABC, which may always be conceived to be inscribed in a determined sphere, we may say, by (13.), that the angle of the quaternion product, ≤ (OA. AB. BC.CO), is equal to the angle of the lunule, bounded (generally) by the two arcs of small circles OAB, OCB; with the same construction for the equal angle of the anharmonic,

L (OABC), or < (OA: AB. BC: co).

(16.) It is evident that the general principle 223, (10.), of the permissibility of cyclical permutation of quaternion factors under the sign S, must hold good for the case when those quaternions degenerate (294) into vectors; and it is still more obvious, that every permutation of factors is allowed, under the sign T: whence cyclical permutation is again allowed, under this other sign SU; and consequently also (comp. 196, XVI.) under the sign 4.

(17.) Hence generally, for any four vectors, we have the three equations,

XIII... Saßyd = Sẞyda; XIV... SUαßyò = SUßyda;

XV... Laßyd = L Byda;

CHAP. I.]

PENTAGON IN A SPHERE.

327

and in particular, for the successive sides of any plane or gauche quadrilateral ABCD, we have the four equal angles,

XVI... 4 (AB. BC. CD. DA) = ≤ (BC. CD. DA. AB)= &c. ; with the corresponding equality of the angles of the four anharmonics, XVII... 4 (ABCD) = ≤ (BCDA) = ≤ (CDAB) = ≤ (DABC);

or of those of the four reciprocal anharmonics (259, XVII.),

XVII'. . . < (ADCB) = ≤ (BADC) = ≤ (CBAD) = L (DCBA). *(18.) Interpreting now, by (13.) and (15.), these last equations, we derive from them the following theorem, for the plane, or for space :—

Let ABCD be any four points, connected by four circles, each passing through three of the points: then, not only is the angle at A, between the arcs ABC, ADC, equal to the angle at C, between CDA and CBA, but also it is equal (comp. Fig. 66) to the angle at B, between the two other arcs BCD and BAD, and to the angle at D, between the arcs DAB, DCB.

(19.) Again, let ABCDE be any pentagon, inscribed in a sphere; and conceive that the two diagonals AC, AD are drawn. We shall then have three equations, of the forms,

XVIII... AB. BC.CA = AT; AC. CD.DA = AU;

AD. DE. EA = AV;

Fig. 66.

where AT, AU, AV are three tangents to the sphere at A, so that their product is a fourth tangent at that point. But the equations XVIII. give

XIX... AB.BC.CD.DE. EA = (AT.AU.AV): (AC2. AD2)

=AW = a new vector, which touches the sphere at A.

We have therefore this Theorem, which includes several others'under it :

"The product of the five successive sides, of any (generally gauche) pentagon inscribed in a sphere, is equal to a tangential vector, drawn from the point at which the pentagon begins and ends."

(20.) Let then P be a point on the sphere which passes through o, and through three given points A, B, C; we shall have the equation,

XX... 0=S(OA.AB.BC.CP.PO) = Sa (ẞ − a) ( y − B) (p − y) (− p)

= a2Sẞyp + ẞ2Syap + y2Saßp - p2Saßy.

(21.) Comparing with 294, XIV., we see that the condition for the four co-initial vectors a, ẞ, y, p thus terminating on one spheric surface, which passes through their common origin o, may be thus expressed:

o' on

XXI... if p = xa+yẞ+zy, then p2 = xa2+ yẞ2 + zy2. (22.) If then we project (comp. 62) the variable point P into points A, B, the three given chords OA, OB, Oc, by three planes through that point P, respectively parallel to the planes BOC, COA, AOB, we shall have the equation:

XXII. . . Op2 = OA. OA' + OB. OB'+oc.oc'.

(23.) That the equation XX. does in fact represent a spheric locus for the point P, is evident from its mere form (comp. 282, (10.)); and that this sphere passes

through the four given points, O, A, B, C, may be proved by observing that the equation is satisfied, when we change p to any one of the four vectors, 0, a, ß, y. (24.) Introducing an auxiliary vector, OD or d, determined by the equation, XXIII... Saẞy = a2Vẞy + ẞ2Vya+yoVaß,

or by the system of the three scalar equations (comp. 294, (25.)),

or

XXIV... a2 = Sda, B2= Sdß, y2=Sdy,
XXIV'... Sda 1=S&B-1= Sdy-1= 1,

the equation XX. of the sphere becomes simply,

XXV... p2 = Sop, or XXV'. . . Sồp-1 = 1;

so that D is the point of the sphere opposite to o, and ô is a diameter (comp. 282, IX'.; and 196, (6.)).

(25.) The formula XXIII., which determines this diameter, may be written in this other way:

or

XXVI... Saẞy = Va (B− a) (y −ẞ)y;

XXVI... 6.0ABC. OD=-V (OA.AB.BC.CO);

where the symbol OABC, considered as a coefficient, is interpreted as in 294, XLIV.; namely, as denoting the volume of the pyramid OABC, which is here an inscribed

one.

(26.) This result of calculation, so far as it regards the direction of the axis of the quaternion OA.AB.BC.CO, agrees with, and may be used to confirm, the theorem (12.), respecting the product of the successive sides of a gauche quadrilateral, OABC ; including the rule of rotation, which distinguishes that axis from its opposite. (27.) The formula XXIII. for the diameter & may also be thus written :

and the equation XX. of the sphere may be transformed to the following:

which expresses (by 294, (34.), comp. 260, (10.)), that the four reciprocal vectors,

XXIX... OA' = a' = a ̄1, OB' = ẞ′ = B-1, oc' = y' = y1, or' = p2 = p1,

are termino-complanar (64); the plane A'B'C'r', in which they all terminate, being parallel to the tangent plane to the sphere at o: because the perpendicular let fall on this plane from o is

XXX... d'= 8-1,

as appears from the three scalar equations,

then

XXXI... Sa'd = Sß'd = Sy'd=1.

(28.) In general, if D be the foot of the perpendicular from o, on the plane ABC,

because this expression satisfies, and may be deduced from, the three equations, XXXIII... Sad-1 Sẞd-1 Syd-1 = 1.

=

-1

As a verification, the formula shows that the length To, of this perpendicular, or altitude, OD, is equal to the sextuple volume of the pyramid OABC, divided by the double area of the triangular base Abc. (Compare 281, (4.), and 294, (3.), (33.).)