III. . . q = (Sq + Vq =) w + ix + jy + kz;

in which the four scalars, wxyz, may be said to be the Four Constituents of the Quaternion. And it is evident (comp. 202, (5.), and 133), that if we write in like manner,

IV... q' = w' + ix' + jy + kz',

where ijk denote the same three given right versors (181) as before, then the equation

between these two quaternions, q and q', includes the four following scalar equations between the constituents:

VI... w' = w,

x' =

= x, y = y, 2' = 2;

which is a new justification (comp. 112, 116) of the propriety of naming, as we have done throughout the present Chapter, the General Quotient of two Vectors (101) a QUATernion. 222. When the Standard Quadrinomial Form (221) is adopted, we have then not only

I. Sq=w, and Vq = ix + jy + kz,

as before, but also, by 204, XI.,

II... Kq = (Sq - Vq =)w – ix − jy – kz.

And because the distributive property of multiplication of quaternions (212), combined with the laws of of the symbols ijk (182), or with the General and Fundamental Formula of this whole Calculus (183), namely with the formula,

¿2 = j2 = k2 = ijk = − 1,

gives the transformation,

III. . . (ix + jy + kz)2 = − (x2 + y2 + 22),

we have, by 204, &c., the following new expressions :

IV... NVq= (TVq)2 = − Vq2 = x2 + y2 + z2 ;

V... TVq = √(x2 + y2 + z2) ;

VI. UVq = (ix + jy + kz) : √ (x2 + y2 + z2) ;

.

VII... Nq = Tq2 = Sq2 – V q2 = w2 + x2 + y2 + z2 ;

VIII... Tq = √ (w2 + x2 + y2 + z2) ;

IX. . . Uq = (w + ix + jy + kz) : √ (w2 + x2 + y2 + z2) ;

(A)

X... SUq=w: √ (w2 + x2 + y2 + z2) ;
XI... VUq = (ix + jy + kz): √ (w2 + x2 + y2 + z2) ;

235

(1.) To prove the recent formula III., we may arrange as follows the steps of the multiplication (comp. again 182):

a result to which we have already alluded,* in connexion with the partial indeterminateness of signification, in the present calculus, of the symbol V-1, when considered as denoting a right radial (149), or a right versor (153), of which the plane or the axis is arbitrary.

(3.) If q'=q'q, then Nq" Ng'. Nq, by 191, (8.); but if q = w + &c., q' = w' + &c., q′′=w" + &c., then

and conversely these four scalar equations are jointly equivalent to, and may be summed up in, the quaternion formula,

XV...

w" + ix" +jy" + kz′′ = (w' + ix' + jy' + kz') (w + ix + jy + kz) ;

we ought therefore, under these conditions XIV., to have the equation,

XVI... "2
w ' + x′′2 + y''2 + z′′2 = (w'2 + x′2 + y'2 + z22) (w2 + x2 + y2 + z2) ;

which can in fact be verified by so easy an algebraical calculation, that its truth may be said to be obvious upon mere inspection, at least when the terms in the four quadrinomial expressions w"..z" are arranged† as above.

Compare the first Note to page 131; and that to page 162.

+ From having somewhat otherwise arranged those terms, the author had some little trouble at first, in verifying that the twenty-four double products, in the expansion of w'" + &c., destroy each other, leaving only the sixteen products of squares, or that XVI. follows from XIV., when he was led to anticipate that result through quaternions, in the year 1843. He believes, however, that the algebraic theorem XVI., as distinguished from the quaternion formula XV., with which it is here connected, had been discovered by the celebrated EULER.

223. The principal use which we shall here make of the standard quadrinomial form (221), is to prove by it the general associative property of multiplication of quaternions; which can now with great ease be done, the distributive* property (212) of such multiplication having been already proved. In fact, if we write, as in 222, (3.),

without now assuming that the relation q′′ = q'q, or any other relation, exists between the three quaternions q, q', q”, and inquire whether it be true that the associative formula,

holds good, we see, by the distributive principle, that we have only to try whether this last formula is valid when the three quaternion factors q, q, q" are replaced, in any one common order on both sides of the equation, and with or without repetition, by the three given right versors ijk; but this has already been proved, in Art. 183. We arrive then, thus, at the important conclusion, that the General Multiplication of Quaternions is an Associative Operation, as it had been previously seen (212) to be a Distributive one: although we had also found (168, 183, 191) that such Multiplication is not (in general) Commutative: or that the two products, q'q and qq', are generally unequal. We may therefore omit the point (as in 183), and may denote each member of the equation II. by the symbol q'q'q.

(1.) Let vVq, v' =Vq', v′′ =Vq"; so that v, v', v" are any three right quaternions, and therefore, by 191, (2.), and 196, 204,

Let this last right quaternion be called v,, and let Se'vs,, so that v'v=s, + v,; we shall then have the equations,

At a later stage, a sketch will be given of at least one proof of this Associative Principle of Multiplication, which will not presuppose the Distributive Principle.

and therefore generally, if v, v', v" be still right, as above,

III. . . V. v"Vo'v = vSv ́v′′ – vˆSv ̈v ;

a formula with which the student ought to make himself completely familiar, on account of its extensive utility.

+ up!

(2.) With the recent notations,

V.v"Sv'v=Vv's, = v′′s,= v′′Svv′;

we have therefore this other very useful formula,

IV. . . V. v'v'v = vSv′v′′ – v′Sv'v+v′′Svv',

where the point in the first member may often for simplicity be dispensed with; and in which it is still supposed that

hence this last vector, which is evidently complanar with the two indices Iv and Iv', is at the same time (by 208) perpendicular to the third index Iv", and therefore (by (1.)) complanar with the third quaternion q'.

(4.) With the recent notations, the vector,

VI... Iv, IVv'v = IV (Vq'.Vq),

is (by 208, XXII.) a line perpendicular to both Iv and Iv'; or common to the planes of q and q'; being also such that the rotation round it from Iv' to Iv is positive: while its length,

TIv, or Tv, or TV. vv, or TV (Vq'. Vq),

bears to the unit of length the same ratio, as that which the parallelogram under the indices, Iv and Iv', bears to the unit of area.

(5.) To interpret (comp. IV.) the scalar expression,

VII... Sv'v'v = Sv′′v, = S. v′′Vv'v,

(because Sv's, = 0), we may employ the formula 208, V.; which gives the the transformation,

VIII... Sv'v'v Tv". Tv,. cos (π − x) ;

where To" denotes the length of the line Iv', and Tv, represents by (4.) the area (positively taken) of the parallelogram under Iv' and Iv; while x is (by 208), the angle between the two indices Iv", Iv,. This angle will be obtuse, and therefore the cosine of its supplement will be positive, and equal to the sine of the inclination of the line Iv" to the plane of Iv and Iv', if the rotation round Iv" from Iv' to Iv be negative, that is, if the rotation round Iv from Iv' to Iv" be positive; but that cosine will be equal the negative of this sine, if the direction of this rotation be reversed. We have therefore the important interpretation:

IX... Sv"v'v= + volume of parallelepiped under Iv, Iv', Iv";

the upper or the lower sign being taken, according as the rotation round Ir, from Iv' to Iv", is positively or negatively directed.

(6.) For example, we saw that the ternary products ijk and kji have scalar values, namely,

and accordingly the parallelepiped of indices becomes, in this case, an unit-cube; while the rotation round the index of i, from that of j to that of k, is positive (181). (7.) In general, for any three right quaternions vv'v", we have the formula, X... Svv'v' = - Sv"v'v ;

and when the three indices are complanar, so that the volume mentioned in IX. vanishes, then each of these two last scalars becomes zero; so that we may write, as a new Formula of Complanarity;

XI... Sv'v'v = 0, if Iv" ||| Iv', Iv (123):

while, on the other hand, this scalar cannot vanish in any other case, if the quaternions (or their indices) be still supposed to be actual (1, 144); because it then represents an actual volume.

(8.) Hence also we may establish the following Formula of Collinearity, for any three quaternions :

XII... S(Vq".Vq'.Vq)=0, if IVq" ||| IVq', IVq;

that is, by 209, if the planes of q, q', q′′ have any common line.

(9.) In general, if we employ the standard trinomial form 221, II., namely, v' = ix' + &c., v" ix" + &c.,

v= =Vq=ix+jy + k2,

the laws (182, 183) of the symbols i, j, k give the transformation,

XIII... Sv"v'v = x′′ (z'y — y'z) + y′′ (x'z – z′x) + z′′ (y'x − x'y) ;

and accordingly this is the known expression for the volume (with a suitable sign) of the parallelepiped, which has the three lines op, op, op" for three co-initial edges, if the rectangular co-ordinates* of the four corners, O, P, P', p" be 000, xyz, x'y'z', x"y"z".

(10.) Again, as another important consequence of the general associative property of multiplication, it may be here observed, that although products of more than two quaternions have not generally equal scalars, for all possible permutations of the factors, since we have just seen a case X. in which such a change of arrangement produces a change of sign in the result, yet cyclical permutation is permitted, under the sign S; or in symbols, that for any three quaternions (and the result is easily extended to any greater number of such factors) the following formula holds good: XIV... Sq"qq=Sqq'q'.

In fact, to prove this equality, we have only to write it thus,

XIV'... S(q'q'. q) = S (q · q′′q′),

and to remember that the scalar of the product of any two quaternions remains unaltered (198, I.), when the order of those two factors is changed.

* This result may serve as an example of the manner in which quaternions, although not based on any usual doctrine of co-ordinates, may yet be employed to deduce, or to recover, and often with great ease, important co-ordinate expressions.