intercept a constant interval, upon each cyclic arc, separately taken. Admitting these three properties, we see that if, in Fig. 59, we conceive a spherical conic to be described, so as to pass through the three points B, F, H, and to have the great circle DAEC for one cyclic arc, the second and third equations I. of 264 will prove that the arc GLIM is the other cyclic arc for this conic; the first equation I. proves next that the conic passes through к; and if the arcual chord FK be drawn and prolonged, the two remaining equations prove that it meets the cyclic arcs in D and м; after which, the equation II. of the same Art. 264 immediately results, at least with the arrangement* adopted in the Figure.

(1.) The Ist property is easily seen to correspond to the possibility of circumscribing a circle about a given plane triangle, namely that of which the corners are the intersections of a plane parallel to the plane of the given cyclic arc, with the three radii drawn to the three given points upon the sphere: but it may be worth while, as an exercise, to prove here the IInd property by quaternions.

(2.) Take then the equation of a cyclic cone, 196, (8.), which may (by 196, XII.) be written thus:

and p' being thus two rays (or sides) of the cone, which may also be considered to be the vectors of two points P and P' of a spherical conic, by supposing that their lengths are each unity. Let r and 7' be the vectors of the two points T and r' on the two cyclic arcs, in which the arcual chord PP' of the conic cuts them; so that

The theorem may then be stated thus: that

V. . . if p = xT+x'r', then VI. . . p' = x'r + xr′;

or that this expression VI. satisfies II., if the equations I. III. IV. V. be satisfied. Now, by III. V. VI., we have

whence it follows that the first members of I. and II. are equal, and it only remains to prove that their second members are equal also, or that Tp'=Tp, if Tr'=T7. Accordingly we have, by V. and VI.,

*Modifications of that arrangement may be conceived, to which however it would

be easy to adapt the reasoning.

CHAP. III.] PROOF BY STEREOGRAPHIC PROJECTION.

295

271. To prove the associative principle, with the help of Fig. 60, three other properties of a spherical conic shall be supposed known:* Ist, that for every such curve two focal points exist, possessing several important relations to it, one of which is, that if these two foci and one tangent arc be given, the conic can be constructed; IInd, that if, from any point upon the sphere, two tangents be drawn to the conic, and also two arcs to the foci, then one focal arc makes with one tangent the same angle as the other focal arc with the other tangent; and IIIrd, that if a spherical quadrilateral be circumscribed to such a conic (supposed here for simplicity to be a spherical ellipse, or the opposite ellipse being neglected), opposite sides subtend supplementary angles, at either of the two (interior) foci. Admitting these known properties, and supposing the arrangement to be as in Fig. 60, we may conceive a conic described, which shall have E and F for its two focal points, and shall touch the arc BC; and then the two first of the equations I., in 265, will prove that it touches also the arcs AB and CD, while the third of those equations proves that it touches AD, so that ABCD is a circumscribed† quadrilateral: after which the three equations II., of the same article, are consequences of the same properties of the curve.

272. Finally, to prove the same important Principle in a more completely elementary way, by means of the arrangement represented in Fig. 61, or to prove the theorem of spherical geometry enunciated in Art. 267, we may assume the point D as the pole of a stereographic projection, in which the three small circles through that point shall be represented by right lines, but the three others by circles, all being in one common plane. And then (interchanging accents) the theorem comes to be thus stated:

If a', B', c' be any three points (comp. Fig. 62) on the sides BC, CA, AB of any plane triangle, or on those sides prolonged, then, Ist, the three circles,

B

Fig. 62.

The reader may again consult pages 46 and 50 of the Translation lately cited. In strictness, there are of course four foci, opposite two by two.

† The writer has elsewhere proposed the notation, Er (..) ABCD, to denote the relation of the focal points E, F to this circumscribed quadrilateral.

I... C'AB', A'BC', B'CA',

will meet in one point D; and IInd, an even number (if any) of the six (linear or circular) successions,

II. . . AB ̊C, BC ́A, CA'B, and II'... C'AB'D, A'bc'd, b'ca'd, will be direct; an even number therefore also (if any) being indirect. But, under this form,* the theorem can be proved by very elementary considerations, and still without any employment of the distributive principle (224, 262).

(1.) The first part of the theorem, as thus stated, is evident from the Third Book of Euclid; but to prove both parts together, it may be useful to proceed as follows, admitting the conception (235) of amplitudes, or of angles as representing rotations, which may have any values, positive or negative, and are to be added with attention to their signs.

(2.) We may thus write the three equations,

III... AB'C = nπ, вC'A=n'π,

to express the three collineations, AB'C, &c. of Fig. 62; the integer, n, being odd or even, according as the point B' is on the finite line AC, or on a prolongation of that line; or in other words, according as the first succession II. is direct or indirect : and similarly for the two other coefficients, n' and n".

(3.) Again, if OPQR be any four points in one plane, we may establish the formula,

IV... POQ+ QOR = POR + 2mπ,

with the same conception of addition of amplitudes; if then D be any point in the plane of the triangle ABC, we may write,

V... AB'D + DB'C=NT, вC'D + DC'A=n'π, CA'D + DA'B = n′′π; and therefore,

VI. . . (AB'D + DC'A) + (BC′D + DA′B) + (CA'D + DB'C) = (n + n' + n′′) «. (4.) Again, if any four points OPQR be not merely complanar but concircular, we have the general formula,

VII... OPQ+QRо=Pπ,

the integer p being odd or even, according as the succession OPQR is direct or indi

* The Associative Principle of Multiplication was stated nearly under this form, and was illustrated by the same simple diagram, in paragraph XXII. of a communication by the present author, which was entitled Letters on Quaternions, and has been printed in the First and Second Editions of the late Dr. Nichol's Cyclopædia of the Physical Sciences (London and Glasgow, 1857 and 1860). The same communication contained other illustrations and consequences of the same principle, which it has not been thought necessary here to reproduce (compare however Note C); and others may be found in the Sixth of the author's already cited Lectures on Quaternions (Dublin, 1853), from which (as already observed) some of the formula and figures of this Chapter have been taken.

CHAP. III.] ADDITIONAL FORMULÆ, NORM OF a vector. 297

rect; if then we denote by D the second intersection of the first and second circles I., whereof c' is a first intersection, we shall have

VIII... AB'D + DC'A=Pπ,

p and p' being odd, when the two first successions trary case.

(5.) Hence, by VI., we have,

BC'D + DA'B = p'ñ,

II'. are direct, but even in the con

p+p' + p" = n + n' + n" ;

IX... CA'D + DB'c=p"π, where X. the third succession II'. is therefore always circular, or the third circle I. passes through the intersection D of the two first; and it is direct or indirect, that is to say, p" is odd or even, according as the number of even coefficients, among the five previously considered, is itself even or odd; or in other words, according as the number of indirect successions, among the five previously considered, is even (including zero), or odd.

(6.) In every case, therefore, the total number of successions of each kind is even, and both parts of the theorem are proved: the importance of the second part of it (respecting the even partition, if any, of the six successions II. II'.) arising from the necessity of proving that we have always, as in algebra,

XI.. sr.q=+8. rq, and never

if q, r, s be any three actual quaternions.

XII... sr. q = s.rq,

(7.) The associative principle of multiplication may also be proved, without the distributive principle, by certain considerations of rotations of a system, on which we cannot enter here.

SECTION 3.-On some Additional Formulæ.

273. Before concluding the Second Book, a few additional remarks may be made, as regards some of the notations and transformations which have already occurred, or others analogous to them. And first as to notation, although we have reserved for the Third Book the interpretation of such expressions as ẞa, or a2, yet we have agreed, in 210, (9.), to abridge the frequently occurring symbol (Ta)2 to Ta2; and we now propose to abridge it still further to Na, and to call this square of the tensor (or of the length) of a vector, a, the Norm of that Vector: as we had (in 190, &c.), the equation Tq2= Nq, and called Nq the norm of the quaternion q (in 145, (11.)). We shall therefore now write generally, for any vector a, the formula,

I. . . (Ta)2 = Ta2 = Na.

(1.) The equations (comp. 186, (1.) (2.) (3.) (4.) ),

II... Np = 1;

III... Np = Na; IV... N(pa) = Na;
V... N(pa)= N(ß − a),

represent, respectively, the unit-sphere; the sphere through A, with o for centre ; the sphere through o, with a for centre; and the sphere through B, with the same

centre A.

(2.) The equations (comp. 186, (6.) (7.) ),

VI... N(p+a)= N(p − a) ;

VII... N (p − ẞ) = N (p − a),

-

represent, respectively, the plane through o, perpendicular to the line oa; and the plane which perpendicularly bisects the line AB.

274. As regards transformations, the few following may here be added, which relate partly to the quaternion forms (204, 216, &c.) of the Equation of the Ellipsoid.

(1.) Changing K(K: p) to Rp: RK, by 259, VIII., in the equation 217, XVI. of the ellipsoid, and observing that the three vectors p, Rp, and Rê are complanar, while 1: Tp TRp by 258, that equation becomes, when divided by TRp, and when the value 217, (5.) for t2 is taken, and the notation 273 is employed:

=

of which the first member will soon be seen to admit of being written† as T(p + pk), and the second member as k2 - 12.

(2.) If, in connexion with the earlier forms (204, 216) of the equation of the same surface, we introduce a new auxiliary vector, σ or os, such that (comp. 216, VIII.)

SL + v 6) B = p +238 C,

the equation may, by 204, (14.), be reduced to the following extremely simple form: III... To Tẞ;

which expresses that the locus of the new auxiliary point s is what we have called the mean sphere, 216, XIV.; while the line rs, or σp, which connects any two corresponding points, P and s, on the ellipsoid and sphere, is seen to be parallel to the fixed line ẞ; which is one element of the homology, mentioned in 216, (10.). (3.) It is easy to prove that

IV...
S

σ

=S s!

a

and therefore V... S 818-889

if p' and ' be the vectors of two new but corresponding points, P' and s', on the ellipsoid and sphere; whence it is easy to infer this other element of the homology, that any two corresponding chords, PP' and ss', of the two surfaces, intersect each other on the cyclic plane which has d for its cyclic normal (comp. 216, (7.)): in fact, they intersect in the point T of which the vector is,

In the verification 216, (2.) of the equation 216, (1.), considered as repre