CHAP. III.] HOMOLOGIES OF ELLIPSOID AND SPHERE.

and this point is on the plane just mentioned (comp. 216, XI.), because

(4.) Quite similar results would have followed, if we had assumed

299

the other cyclic plane, with y instead of d for its normal, might therefore have been taken (as asserted in 216, (10.)), as another plane of homology of ellipsoid and sphere, with the same centre of homology as before: namely, the point at infinity on the line ẞ, or on the axis (204, (15.)) of one of the two circumscribed cylinders of revolution (comp. 220, (4.)).

(5.) The same ellipsoid is, in two other ways, homologous to the same mean sphere, with the same two cyclic planes as planes of homology, but with a new centre of homology, which is the infinitely distant point on the axis of the second circumscribed cylinder (or on the line AB' of the sub-article last cited).

(6.) Although not specially connected with the ellipsoid, the following general transformations may be noted here (comp. 199, XII., and 204, XXXIV'.):

XL.. TVVq=V{}(Tq − Sq)}; XII. . . tan ≤ q = (TV: S) √ q =,
(7.) The equations 204, XVI. and XXXV., give easily,
XIII. . . UVq = UVUq ; XIV. . . UIVq = Ax. q ;

or the more symbolical forms,

XIII'... UVU = UV ;

Tq - Sq Tq + Sq

XV... TIVq= TVq;

and the identity 200, IX. becomes more evident, when we observe that

(8.) We have also generally (comp. 200, (10.) and 218, (10.)),

XVIII. . . U (rg + Kqr) = U (Sr. Sq + Vr.Vq) = r ̄1 (r2q2)§ q ̄1,

in which q and r may be any two quaternions, is not perhaps of any great importance in itself, but will be found to furnish a student with several useful exercises in transformation.

(10.) When it was said, in 257, (1.), that zero had only itself for a square-root, the meaning was (comp. 225), that no binomial expression of the form x + iy (228) could satisfy the equation,

XIX. . . 0 = q2 = (x + iy)2 = (x2 − y2)+2ixy,

• This formula was given, but in like manner without proof, in page 587 of the author's Lectures on Quaternions.

for any real or imaginary values of the two scalar coefficients x and y, different from zero ;* for if biquaternions (214, (8.)) be admitted, and if h again denote, as in 256, (2.), the imaginary of algebra, then (comp. 257, (6.) and (7.)) we may write, generally, besides the real value 01 = 0, the imaginary expression,

XX. . . 0} = v + hv', if Sv = Sv' = Svv' = Nv' - Nv = 0 ;

v and v' being thus any two real right quaternions, with equal norms (or with equal tensors), in planes perperpendicular to each other.

(11.) For example, by 256, (2.) and by the laws (183) of ijk, we have the transformations,

XXI. . . (i + hj)2 = ¿2 − j2 + h (ij + ji) = 0 + h0 = 0;

so that the bi-quaternion i + hj is one of the imaginary values of the symbol 0.

(12.) In general, when bi-quaternions are admitted into calculation, not only the square of one, but the product of two such factors may vanish, without either of them separately vanishing: a circumstance which may throw some light on the existence of those imaginary (or symbolical) roots of equations, which were treated of in 257. (13.) For example, although the equation

XXII. . . q2 - 1 = (g − 1) (g + 1) = 0

has no real roots except + 1, and therefore cannot be verified by the substitution of any other real scalar, or real quaternion, for q, yet if we substitute for q the bi-quaternion† v + hv', with the conditions 257, XIII., this equation XXII. is verified.

(14.) It will be found, however, that when two imaginary but non-evanescent factors give thus a null product, the norm of each is zero; provided that we agree to extend to bi-quaternions the formula Nq = Sq2-Vq2 (204, XXII.); or to define that the Norm of a Biquaternion (like that of an ordinary or real quaternion) is equal to the Square of the Scalar Part, minus the Square of the Right Part: each of these two parts being generally imaginary, and the former being what we have

called a Bi-scalar.

(15.) With this definition, if q and q' be any two real quaternions, and if h be, as above, the ordinary imaginary of algebra, we may establish the formula:

XXIII... N(q+hq') = (Sq +hSq')2 − (Vq + hVq')*;

or (comp. 200, VII., and 210, XX.),

XXIV... N (q+hq') = Ng - Ng' + 2hS. qKq'.

(16.) As regards the norm of the sum of any two real quaternions, or real vectors (273), the following transformations are occasionally useful (comp. 220, (2.) ): XXV... N(q'+ q) = N (Tq'. Ug + Tq. Uq');

XXVI... N(B+ a) = N(Tẞ. Ua + Ta. Uẞ);

in each of which it is permitted to change the norms to the tensors of which they are the squares, or to write T for N.

* Compare the Note to page 276.

This includes the expression + hi, of 257, (1.), for a symbolical square-root of positive unity. Other such roots are + hj, and + hk.

BOOK III.

ON QUATERNIONS, CONSIDERED AS PRODUCTS OR POWERS OF VECTORS; AND ON SOME APPLICATIONS OF QUATERNIONS.

CHAPTER I.

ON THE INTERPRETATION OF A PRODUCT OF VECTORS, OR POWER OF A vector, AS A QUATERNION.

SECTION 1.-On a First Method of interpreting a Product of Two Vectors as a Quaternion.

ART. 275. In the First Book of these Elements we interpreted, Ist, the difference of any two directed right lines in space (4); IInd, the sum of two or more such lines (5-9); IIIrd, the product of one such line, multiplied by or into a positive or negative number (15); IVth, the quotient of such a line, divided by such a number (16), or by what we have called generally a SCALAR (17); and Vth, the sum of a system of such lines, each affected (97) with a scalar coefficient (99), as being in each case itself (generally) a Directed Line* in Space, or what we have called a VECTOR (1).

276. In the Second Book, the fundamental principle or pervading conception has been, that the Quotient of two such Vectors is, generally, a QUATERNION (112, 116). It is however to be remembered, that we have included under this general conception, which usually relates to what may be called an Oblique Quotient, or the quotient of two lines in space making either an acute or an obtuse angle with each other

The Fourth Proportional to any three complanar lines has also been since interpreted (226), as being another line in the same plane.

(130), the three following particular cases: Ist, the limiting case, when the angle becomes null, or when the two lines are similarly directed, in which case the quotient degenerates (131) into a positive scalar; IInd, the other limiting case, when the angle is equal to two right angles, or when the lines are oppositely directed, and when in consequence the quotient again degenerates, but now into a negative scalar; and IIIrd, the intermediate case, when the angle is right, or when the two lines are perpendicular (132), instead of being parallel (15), and when therefore their quotient becomes what we have called (132) a Right Quotient, or a RIGHT QUATERNION: which has been seen to be a case not less important than the two former ones.

277. But no Interpretation has been assigned, in either of the two foregoing Books, for a PRODUCT of two or more Vectors; or for the SQUARE, or other POWER of a Vector: so that the Symbols,

I... ẞa, yẞa,.. and II. . . a2, a3, II... a3,.. a ̈1, in which a, ẞ, y .. denote vectors, but t denotes a scalar, remain as yet entirely uninterpreted; and we are therefore free to assign, at this stage, any meanings to these new symbols, or new combinations of symbols, which shall not contradict each other, and shall appear to be consistent with convenience and analogy. And to do so will be the chief object of this First Chapter of the Third (and last) Book of these Elements: which is designed to be a much shorter one than either of the foregoing.

278. As a commencement of such Interpretation we shall here define, that a vector a is multiplied by another vector ẞ, or that the latter vector is multiplied into the former, or that the product ßa is obtained, when the multiplier-line ß is divided by the reciprocal Ra (258) of the multiplicand-line a; as we had proved (136) that one quaternion is multiplied into another, when it is divided by the reciprocal thereof. In symbols, we shall therefore write, as a first definition, the formula:

Compare the Notes to pages 146, 159.

CHAP. I.] INTERPRETATION OF A PRODUCT OF TWO VECTORS. 303

. .

I. Ba=B: Ra; where II... Ra = - Ua: Ta (258, VII.). And we proceed to consider, in the following Section, some of the general consequences of this definition, or interpretation, of a Product of two Vectors, as being equal to a certain Quotient, or Quaternion.

SECTION 2.- On some Consequences of the foregoing Interpretation.

279. The definition (278) gives the formula:

it gives therefore, by 259, VIII., the general relation, II. . . ẞa = Kaß; or II'. . . aß = Kẞa.

The Products of two Vectors, taken in two opposite orders, are therefore Conjugate Quaternions; and the Multiplication of Vectors, like that of Quaternions (168), is (generally) a NonCommutative Operation.

(1.) It follows from II. (by 196, comp. 223, (1.)), that

III... Sẞa + Saß = † (ẞa + aß).

(2.) It follows also (by 204, comp. again 223, (1.)), that IV... Vẞa = Vaẞ= (ẞaaß).

280. Again, by the same general formula 259, VIII., we have the transformations,

it follows, then, from the definition (278), that

II... ẞ(a + a) = ẞa + Ba';

whence also, by taking conjugates (279), we have this other general equation,

III. . . (a + a) ẞ = aß + a'ß.

Multiplication of Vectors is, therefore, like that of Quaternions (212), a Doubly Distributive Operation.

281. As we have not yet assigned any signification for a ternary product of vectors, such as yẞa, we are not yet pre