which was there seen to represent the same locus, considered as a spheric surface, with o for centre, and aa for one of its radii, and write it as Nq = a2, we can then by calculation return to the form

or finally,

N (q-a2)=a3N (q-1), or T (q-a2)=aT (q-1),

T(p-a2a)=aT (pa), as in 186, (5.);

this first form of that sub-article being thus deduced from the second, namely from

(5.) It is far from being the intention of the foregoing remarks, to discourage attention to the geometrical interpretation of the various forms of expression, and general rules of transformation, which thus offer themselves in working with quaternions; on the contrary, one main object of the present Chapter has been to establish a firm geometrical basis, for all such forms and rules. But when such a foundation has once been laid, it is, as we see, not necessary that we should continually recur to the examination of it, in building up the superstructure. That each of the two forms, in 186, (5.), involves the other, may be proved, as above, by calculation ; but it is interesting to inquire what is the meaning of this result: and in seeking to interpret it, we should be led anew to the theorem of the Apollonian Locus.

(6.) The result (4.) of calculation, that

N (q-a2)=a2N (q − 1), if Nq = a2,

may be expressed under the form of an identity, as follows:

IX... N(g-Ng) Ng. N (q-1);

in which q may be any quaternion.

=

from the latter; because each may be put under the form (comp. 196, XII.),

196, (6.)), each of these two equations expresses that the locus of P is the sphere which passes through o, and has its centre at A; or which has OB 2a for a dia

meter.

(10.) By changing q to q 1 in (8), we find that

if Tq= 1, then S

1

0, and reciprocally.

(12.) Each of these two equations (11.) expresses that the locus of P is the sphere through A, which has its centre at o; and their proved agreement is a recognition, by quaternions, of the elementary geometrical theorem, that the angle in a semicircle is a right angle.

SECTION 13.-On the Right Part (or Vector Part) of a Quaternion; and on the Distributive Property of the Multiplication of Quaternions.

B

B

201. A given vector OB can always be decomposed, in one but in only one way, into two component vectors, of which it is the sum (6); and of which one, as OB' in Fig. 50, is parallel (15) to another given vector OA, while the other, as OB" in the same Figure, is perpendicular to that given line oA; namely, by letting fall the perpendicular BB' on OA, and drawing OB" = B′B, 80 that оB'BB" shall be a rectangle. In other words, if a and ẞ be any two given, actual, and co-initial vectors, it is always possible to deduce from them, in one definite way, two other co-initial vectors, B' and 3", which need not however both be actual (1); and which shall satisfy (comp. 6, 15, 129) the conditions,

A

Fig. 50.

B' vanishing, when ẞa; and ẞ" being null, when ẞ || a; but both being (what we may call) determinate vector-functions of a and B. And of these two functions, it is evident that B' is the orthographic projection of ẞ on the line a; and that ẞ" is the corresponding projection of B on the plane through o, which is perpendicular to a.

202. Hence it is easy to infer, that there is always one, but only one way, of decomposing a given quaternion,

into two parts or summands (195), of which one shall be, as in

RIGHT PART OF A QUATERNION.

191

CHAP. I.] 196, a scalar, while the other shall be a right quotient (132). Of these two parts, the former has been already called (196) the scalar part, or simply the Scalar of the Quaternion, and has been denoted by the symbol Sq; so that, with reference to the recent Figure 50, we have

I... Sq=S(OB: OA) = OB': OA; or, S (B: a) = B': a. And we now propose to call the latter part the RIGHT PART* of the same quaternion, and to denote it by the new symbol Vq;

writing thus, in connexion with the same Figure,

1=

II... Vq= V (OB: OA) = OB": OA; or, V(B: a) = ẞ": a. The System of Notations, peculiar to the present Calculus, will thus have been completed; and we shall have the following general Formula of Decomposition of a Quaternion into two Summands (comp. 188), of the Scalar and Right kinds : III... q = Sq + Vq = Vq + Sq,

or, briefly and symbolically,

IV... 1 = S+ V = V + S.

(1.) In connexion with the same Fig. 50, we may write also,

V (OB: 0A) = B'B: OA,

because, by construction, B'B = OB".

(2.) In like manner, for Fig. 36, we have the equation,

(4.) In general, it is evident that

V... q=0, if Sq=0, and Vq=0; and reciprocally.

* This Right Part, Vg, will come to be also called the Vector Part, or simply the VECTOR, of the Quaternion; because it will be found possible and useful to identify such part with its own Index-Vector (133). Compare the Notes to pages 119, 136, 174.

203. We had (196, XIX.) a formula which may now be written thus,

to express the projection of Oв on os, or of the vector 3 on a; and we have evidently, by the definition of the new symbol Vq, the analogous formula,

to express the projection of ẞ on the plane (through o), which is drawn so as to be perpendicular to a; and which has been considered in several former sub-articles (comp. 186, (6.), and 196, (1.)). It follows (by 186, &c.) that

III. . . TB" = TV2. Ta = perpendicular distance of в from oa;

a

this perpendicular being here considered with reference to its length alone, as the characteristic T of the tensor implies. It is to be observed that because the factor, V, in the recent

formula II. for the projection ẞ", is not a scalar, we must write that factor as a multiplier, and not as a multiplicand; although we were at liberty, in consequence of a general convention (15), respecting the multiplication of vectors and scalars, to denote the other projection B' under the form,

CHAP. I.]

GEOMETRICAL EXAMPLES.

193

expresses that the locus of P is the indefinite right line BB", in Fig. 50, which is drawn through the point в, parallel to the line oa.

has been seen to express that the locus of P is the plane through B, perpendicular to the line oa; if then we combine it with the recent equation (2.), we shall express that the point P is situated at the intersection of the two last mentioned loci; or that it coincides with the point B.

(4.) Accordingly, whether we take the two first or the two last of these recent forms (2.), (3.), namely,

we can infer this position of the point P: in the first case by inferring, through 202,

=

= 0, whence p − ẞ= 0, by 142; and in the second case by inferring,

through 202, VI., that =

a a

; so that we have in each case (comp. 104), or as a

consequence from each system, the equality p = ẞ, or OP=OB; or finally (comp. 20) the coincidence, P= B.

expresses that the locus of the point P is the cylindric surface of revolution, which passes through the point B, and has the line OA for its axis; for it expresses, by III., that the perpendicular distances of P and B, from this latter line, are equal. (6.) The system of the two equations,

expresses that the locus of P is the (generally) elliptic section of the cylinder (5.), made by the plane through o, which is perpendicular to the line oc.

(7.) If we employ an analogous decomposition of p, by supposing that

p=p' +p", p'la, p" + α,

the three rectilinear or plane loci, (1.), (2.), (3.), may have their equations thus briefly written:

p" =0; p" = ẞ";

p' = ß' :

while the combination of the two last of these gives p = ẞ, as in (4.).

(8.) The equation of the cylindric locus, (5.), takes at the same time the form, Tp" =TB";

which last equation expresses that the projection p" of the point P, on the plane through o perpendicular to OA, falls somewhere on the circumference of a circle, with o for centre, and OB" for radius: and this circle may accordingly be considered as the base of the right cylinder, in the sub-article last cited.

204. From the mere circumstance that Vq is always a right quotient (132), whence UVq is a right versor (153), of