and in this form we can easily see the meaning of the cubic. For, let P1, P2, P3 be three vectors such that

(-91)P1 = 0, (92) P2 = 0, (-93)P3 = 0.

Then any vector p may be expressed by the equation

PS.P1P2P3 = P1S.P2P3P + P¿S.P3P1P + P3S.P1P2P (§ 91),

and we see that when the complex operation, denoted by the lefthand member of the above symbolic equation, is performed on p, the first of the three factors makes the term in P1 vanish, the second and third those in p2 and p respectively. In other words, by the successive performance upon a vector of the operations -91, -921 -93, it is deprived successively of its resolved parts in the directions of P1, P2, P3 respectively; and is thus necessarily reduced to zero, since P1, P2, P3 are (because we have supposed 91, 92, 93 to be distinct) distinct and non-coplanar vectors.

166.] If we take P1, P2, P3 as rectangular unit-vectors, we have -p= P1S+ P2SP2P + P3SP3P,

whence

op=-911801-J2P2S P2P —J3P3SP3P;

or, still more simply, putting i, j, k for P1, P2, P3, we find that any self-conjugate function may be thus expressed

Op-giSip-g2jSjp-g3kSkp,

provided, of course, i, j, k be taken as roots of the equation Τρφρ = 0.

167.] A very important transformation of the self-conjugate linear and vector function is easily derived from this form.

We have seen that it involves three scalar constants only, viz. 91, 92, 93. Let us enquire, then, whether it can be reduced to the following form Op=fp+hV. (i+ek)p(i—ek),

which also involves but three scalar constants f, h, e. Here, again, i, j, k are the roots of Урфр = 0.

Substituting for p the equivalent

p =―iSip—jSjp-kSkp,

expanding, and equating coefficients of i, j, k in the two expressions for op, we find

These give at once

-91=-f+h (2—1+ e2),

-92-f-h(1-e2),
-93=-f-h (2e2 + 1 —e2).

-(91-92) = 2h,

-(92-93)=2e2h.

Hence, as we suppose the transformation to be real, and therefore e2 to be positive, it is evident that 91-92 and 92-93 have the same sign; so that we must choose as auxiliary vectors in the last term of op those two of the rectangular directions i, j, k for which the coefficients g have the greatest and least values.

168.] We may, therefore, always determine definitely the vectors A, μ, and the scalar f, in the equation

φρ = fρ + Γ.λρμ

when is self-conjugate, and the corresponding cubic has not equal roots, subject to the single restriction that

Τ.λμ

is known, but not the separate tensors of A and με This result is important in the theory of surfaces of the second order, and will be considered in Chapter VII.

169.] Another important transformation of when self-conjugate is the following,

Op aaVap+bẞSßp,

=

where a and b are scalars, and a and ẞ unit-vectors. This, of course, involves six scalar constants, and belongs to the most general form pp =—g11Sp1p—J2P2SP2P-J3P3SP3P,

where P1, P2, P3 are the rectangular unit-vectors for which p and op are parallel. We merely mention this form in passing, as it belongs to the focal transformation of the equation of surfaces of the second order, which will not be farther alluded to in this work. It will be a good exercise for the student to determine a, ß, a and b, in terms of 91, 92, 93, and P1, P2, P3.

170.] We cannot afford space for a detailed account of the singular properties of these vector functions, and will therefore content ourselves with the enuntiation and proof of one or two of the most important.

Change to +9, and therefore p' to '+g, and m to my, we have

1

m„V(p'′+g)−1λ(p'′ + g) ̄1μ = ($+g) Vλμ;

a formula which will be found to be of considerable use.

m

h

Similarly S.p(4+)−1p = "/ Sp$¬1p + SpXp + hp2.

h

mSpo-1p

9

That is, the functions

h

mg S.p(+9)-1p, and S.p(p+h)−1p

h

gh

are identical, i. e. when equated to constants represent the same series of surfaces, not merely when

g=h,

but also, whatever be g and h, if they be scalar functions of p which satisfy the equation mS.pp-1p = ghp2.

This is a generalization, due to Hamilton, of a singular result obtained by the author *.

172.] The equations

S.p($+g)1p = 0,

S.p (4+h)-1p = 0, }

are equivalent to mSpp-1p+g8pxp+g2p2 = 0,

Hence

mSpp-1p+hSpxp+ h2 p2 = 0.

m (1−x)Spp ̄1p+(g− hx) Spxp+(g2 — h2x) p2 = 0,

whatever scalar be represented by x.

(1)

That is, the two equations (1) represent the same surface if this identity be satisfied. As particular cases let

*Note on the Cartesian equation of the Wave-Surface. Quarterly Math. Journal, Oct. 1859.

173.] In various investigations we meet with the quaternion

1=афа+ ВФВ+уфу,

where a, ß, y are three unit-vectors at right angles to each other. It admits of being put in a very simple form, which is occasionally of considerable importance..

We have, obviously, by the properties of a rectangular unitsystem 4 = βγφα + γαφβ + αβφγ.

Vq=a(Sẞpy—Sy¢ß) + ß(Sypa — Sapy)+y (Sapß— Sßøa)

= aSß(p—p')y+ßSy (p—p′) a +ySa(p—¢ ́′)ß

=aS.Bey+BS.yea+yS.aeß

− (aSae + BSße + ySye) = e.

[We may note in passing that this quaternion admits of being expressed in the remarkable form

We will recur to this towards the end of the work.]

Many similar singular properties of 4 in connection with a rectangular system might easily be given; for instance,

Γ (αφβφγ + βφγφα + γ φαφβ)

= mV (ap′-1a + ßp'−1ß + yp′−1y) = mV.V¢′−1p = $e ; which the reader may easily verify by a process similar to that just given, or (more directly) by the help of § 145 (4). A few others will be found among the Examples appended to this Chapter.

174.] To conclude, we may remark that as in many of the immediately preceding investigations we have supposed to be self-conjugate, a very simple step enables us to pass from this to the non-conjugate form.

For, if ' be conjugate to 4, we have

Adding, we have

Sp(4+$′)σ = So($+¢ ́′)p;

so that the function (+) is self-conjugate.

where, if o be not self-conjugate, e is some real vector, and therefore

&p= {($+4′)p + (p−p')p

= {($+$')p + {Vep.

Thus every non-conjugate linear and vector function differs from a conjugate function solely by a term of the form

Vep.

The geometric signification of this will be found in the Chapter on Kinematics.

175.] We have shewn, at some length, how a linear and vector equation containing an unknown vector is to be solved in the most general case; and this, by § 138, shews how to find an unknown quaternion from any sufficiently general linear equation containing it. That such an equation may be sufficiently general it must have both scalar and vector parts: the first gives one, and the second three, scalar equations; and these are required to determine completely the four scalar elements of the unknown quaternion.

where is any vector whatever. In each of these cases, only one scalar condition being given, the solution contains three scalar indeterminates. A similar remark applies to the following:

in each of which is any scalar, and any unit vector. 177.] Again, the reader may easily prove that

V.aVq = B,