CHAP. III.] FIRST AND SECOND ARRANGEMENTS OF PLANES. 289

(3.) It may also be said, that if five successive sides (KL,.. ED) of one spherical hexagon be respectively and arcually equal to the five successive diagonals (AB, MI, BC, IF, CA) of another such hexagon (AMBICF), then the sixth side (DK) of the first is equal to the sixth diagonal (FM) of the second.

(4.) Or, if we adopt the conception mentioned in 180, (3.), of an arcual sum, and denote such a sum by inserting + between the symbols of the two summands, that of the added are being written to the left-hand, we may state the theorem, in connexion with the recent Fig. 59, by the formula:

III... ^DF+^ BA=0 EF + 0 BC, if DA=0 EC;

where B and F may denote any two points upon the sphere.

(5.) We may also express the same principle, although somewhat less simply, as follows (see again Fig. 59, and compare sub-art. (2.)):

IV... if ED + GH+ KL = 0, then DK + HE + LG = 0.

(6.) If, for a moment, we agree to write (comp. Art. 1),

V... AB = B-A,

we may then express the recent statement IV. a little more lucidly thus:

VI... if D-ETH-G+L-K= 0, then

K-DE-H+G-L 0.

(7) Or still more simply, if o, n', "be supposed to denote any three diplanar arcs, which are to be added according to the rule (180, (3.)) above referred to, the theorem may be said to be, that

or in words, that Addition of Ares on a Sphere is an Associative Operation.

(8.) Conversely, if any independent demonstration be given, of the truth of any one of the foregoing statements, considered as expressing a theorem of spherical geometry,† a new proof will thereby be furnished, of the associative property of multiplication of quaternions.

265. In the second arrangement (263) of the six planes, instead of representing the three given versors, and their partial or total products, by arcs, it is natural to represent them (174, II.) by angles on the sphere. Conceive then that the two versors, q and r, are represented, in Fig. 60, by the two spherical angles, EAB and ABE; and therefore (175) that their product, rq or s', is represented by the external vertical angle at E, of the triangle ABE. Let the

Some of these formulæ and figures, in connexion with the associative principle, are taken, though for the most part with modifications, from the author's Sixth Lecture on Quaternions, in which that whole subject is very fully treated. Comp. the Note to page 160.

† Such a demonstration, namely a deduction of the equation II. from the five equations I., by known properties of spherical conics, will be briefly given in the ensuing Section.

second versor r be also represented by the angle FBC, and the third versors by BCF; then the

A

/G

if D be a point determined

Fig. 60.

by the two conditions, that the angle ECD shall be equal to BCF, and DEC supplementary to BEA. On the other hand, if we conceive a point d' determined by the conditions that D'AF shall be equal to EAB, and AFD' supplementary to CFB, then the external angle at D', of the triangle AFD', will represent the second ternary product, q'q=sr.q, which (by the associative principle) must be equal to the first. Conceiving then that ED is prolonged to G, and FD' to H, the two spherical angles, GDC and AD'HI, must be equal in all respects; their vertices D and D' coinciding, and the rotations (174, 177) which they represent being not only equal in amount, but also similarly directed. Or, to express the same thing otherwise, we may enunciate (262) the Associative Principle by saying, that when the three angular equations, I... ABE = FBC, BCF = ECD,

are satisfied, then these three other equations,

II... DAF = EAB, FDA = CDE,

DECT-BEA,

AFD7CFB,

are satisfied also. For not only is this theorem of spherical geometry a consequence of the associative principle of multiplication of quaternions, but conversely any independent demonstration of the theorem is, at the same time, a proof of the principle.

266. The third arrangement (263) of the six planes may be illustrated by conceiving a gauche hexagon, AB'CA'BC', to be inscribed in a sphere, in such a manner that the intersection D of the three planes, c'AB', B'CA', A'BC', is on the surface; and therefore that the three small circles, denoted by these three last triliteral symbols, concur

Fig. 61.

Such as we shall sketch, in the following Section, with the help of the known properties of the spherical conics. Compare the Note to the foregoing Article.

CHAP. III.] THIRd arrangemenT, SPHERICAL HEXAGON. 291

in one point D; while the second intersection of the two other small circles, ABC, CA'B, may be denoted by the letter D', as in the annexed Fig. 61. Let it be also for simplicity at first supposed, that (as in the Figure) the five circular successions,

I... C'AB'D, AB'CD', B'CA'D, CA'BD', A'BC'D,

are all direct; or that the five inscribed these symbols I., are all uncrossed ones. allowed to introduce three versors, q, r, sions, as follows:

s,

quadrilaterals, denoted by Then (by 260, (9.)) it is each having two expres

although (by the cited sub-article) the last members of these three formulæ should receive the negative sign, if the first, third, and fourth of the successions I. were to become indirect, or if the corre

sponding quadrilaterals were crossed ones. We have thus (by 191) the derived expressions,

whereof, however, the two versors in the first formula would differ in their signs, if the fifth succession I. were indirect; and those in the second formula, if the second succession were such. Hence,

and since, by the associative principle, these two last versors are to be equal, it follows that, under the supposed conditions of construction, the four points, B, C', A, D', compose a circular and direct succession; or that the quadrilateral, BC'AD', is plane, inscriptible, and uncrossed.

*

267. It is easy, by suitable changes of sign, to adapt the recent reasoning to the case where some or all of the successions I. are indirect; and thus to infer, from the associative principle, this theorem of spherical geometry: If AB'CA'BC'

Of course, since the four points BC'AD' are known to be homospheric (comp. 260. (10.)), the inscriptibility of the quadrilateral in a circle would follow from its being plane, if the latter were otherwise proved: but it is here deduced from the equality of the two versors IV., on the plan of 260, (9.).

be a spherical hexagon, such that the three small circles c'AB', B'CA', A'BC' concur in one point D, then, Ist, the three other small circles, ABC, CA'B, BC's, concur in another point, D'; and IInd, of the six circular successions, 266, I., and BC'AD', the number of those which are indirect is always even (including zero). And conversely, any independent demonstration* of this geometrical theorem will be a new proof of the associative principle.

268. The same fertile principle of associative multiplication may be enunciated in other ways, without limiting the factors to be rersors, and without introducing the conception of a sphere. Thus we may say (comp. 264, (2.)), that if o. ABCDEF (comp. 35) be any pencil of six rays in space, and o. A'B'C' any pencil of three rays, and if the three angles AOB, COD, EOF of the first pencil be respectively equal to the angles B'oc', c'oA', A'OB' of the second, then another pencil of three rays, o. A"B"c", can be assigned, such that the three other angles BOC, DOE, FOA of the first pencil shall be equal to the angles B'Oc", c"OA", A"OB" of the third: equality of angles (with one vertex) being here understood (comp. 165) to include complanarity, and similarity of direction of rotations.

(1.) Again (comp. 264, (4.)), we may establish the following formula, in which the four vectors aßyd form a complanar proportion (226), but ɛ and are any two lines in space:

(2.) Another enunciation of the associative principle is the following:

for if we determine (120) six new vectors, not, and kλμ, so that

An elementary proof, by stereographic projection, will be proposed in the following Section.

(3.) Conversely, the assertion that this last equation or proportion VI. is true, whenever the twelve vectors a..μ are connected by the five proportions IV., is a form of enunciation of the associative principle; for it conducts (comp. IV. and V.) to the equation,

λιη λι η θ

=

εη

but, even with this last restriction, the three factor-quotients in VII. may represent any three quaternions.

SECTION 2.-On some Geometrical Proofs of the Associative Property of Multiplication of Quaternions, which are independent of the Distributive Principle.

269. We propose, in this Section, to furnish three geometrical Demonstrations of the Associative Principle, in connexion with the three Figures (59-61) which were employed in the last Section for its Enunciation; and with the three arrangements of six planes, which were described in Art. 263. The two first of these proofs will suppose the knowledge of a few properties of spherical conics (196, (11.)); but the third will only employ the doctrine of stereographic projection, and will therefore be of a more strictly elementary character. The Principle itself is, however, of such great importance in this Calculus, that its nature and its evidence can scarcely be put in too many different points of view.

270. The only properties of a spherical conic, which we shall in this Article assume as known,† are the three following: Ist, that through any three given points on a given sphere, which are not on a great circle, a conic can be described (consisting generally of two opposite ovals), which shall have a given great circle for one of its two cyclic arcs; IInd, that if a transversal arc cut both these arcs, and the conic, the intercepts (suitably measured) on this transversal are equal; and IIIrd, that if the vertex of a spherical angle move along the conic, while its legs pass always through two fixed points thereof, those legs

* Compare 224 and 262; and the Note to page 236.

The reader may consult the Translation (Dublin, 1841, pp. 46, 50, 55) by the present Dean Graves, of two Memoirs by M. Chasles, on Cones of the Second Degree, and Spherical Conics.