XVIII... S (dUq: Uq)=0; or XVIII'... dUq: Uq= S-10;

but this vector character of the quotient dUq: Uq can easily be confirmed, as follows. Taking the conjugate of that quotient, we have, by VII. (comp. 192, II.; 158; and 324, XI.),

XIX... K (dUq. Ug ̃ ̄1) = KUq ̄1, dKUq = Ug.d(Ug1) = − dUq. Uq ́1 ;

whence

XX... (1+K) (dUq. Ug1) = 0;

which agrees (by 196, II.) with XVIII.

(10.) The scalar character of the tensor, Tq, enables us always to write, as in the ordinary calculus,

XXI... dlTq=dTq: Tq;

but ITq=Slq, by 316, V.; the recent formula XIII. may therefore by VIII. be thus written,

XXII... Sdlq = dSlq = dTq : dq = S(dq: q); or XXII'... dlq — q ̄1 dq = S~1 0. (11.) When dg |||q, this last difference vanishes, by 333, II.; and the equation XV. takes the form,

XXIII. . . dlUq = Vdlq = dVlq.

And in fact we have generally, IUq = Vlq, by 316, XX., although the differentials of these two equal expressions do not separately coincide with the members of the recent formula XV., when 9 and dg are diplanar. We may however write generally (comp. XXII.),

XXIV... dlUq-dUq: Ug = V (dlq-dq: q)= dlq-dq: q.

335. We have now differentiated the six simple functions 334, I., which are formed by the operation of the six characteristics,

K, S, V, N, T, U;

and as regards the differentiation of the compound functions 334, II., which are formed by combinations of those former operations, it is easy on the same principles to determine them, as may be seen in the few following examples.

(1.) The axis Ax. q of a quaternion has been seen (291) to admit of being represented by the combination UVq; the differential of this axis may therefore, by 334, IX. and XIV., be thus expressed :

The differential of the axis is therefore, generally, a line perpendicular to that axis, or situated in the plane of the quaternion; but it vanishes, when the plane (and therefore the axis) of that quaternion is constant; or when the quaternion and its differential are complanar.

(2.) Hence,

III. . . dUVq = 0, if IV... dg ||| 9;

and conversely this complanarity IV. may be expressed by the equation III.

(4.) But in general, for any two quaternions, q and g', we have (comp. 223, (5.)) the transformations,

VII... S(Vq.q) = S (Vg'. Vg)=S. q'Vq;

and when we thus suppress the characteristic V before dq: q, and insert it before Uq, under the sign S in the last expression VI., we may replace the new factor VUq by TVUq. UVUq (188), or by TVUq. UVq (274, XIII.), or by - TVUq: UVq (204, V.), where the scalar factor TVUq may be taken outside (by 196, VIII.); also for q1: UVq we may substitute 1: (UVq.q), or 1: qUVq, because UVq ||| 9; the formula VI. may therefore be thus written,

(5.) Now it may be remembered, that among the earliest connexions of quaterternions with trigonometry, the following formulæ occurred (196, XVI., and 204, XIX.),

IX... SUq=cos 4 q, TVUq=sin Lq;

we had also, in 316, these expressions for the angle of a quaternion,

X... 4 q= TVlq = TlUq;

we may therefore establish the following expression for the differential of the angle of a quaternion,

(6.) The following is another way of arriving at the same result, through the differentiation of the sine instead of the cosine of the angle, or through the calculation of dTVUq, instead of dSUq. For this purpose, it is only necessary to remark that we have, by 334, XII. XIV., and by some easy transformations of the kind lately employed in (4.), the formula,

dividing which by SUq, and attending to IX. and X., we arrive again at the expression XI., for the differential of the angle of a quaternion.

(7.) Eliminating S (dq: qUVq) between VIII, and XII., we obtain the differential equation,

XIII... SUq. dSUq+TVUq. dTVUq = 0 ;

of which, on account of the scalar character of the differentiated variables, the integral is evidently of the form,

XIV. . . (SUq)2 + (TVU9)2 = const. ;

and accordingly we saw, in 204, XX., that the sum in the first member of this equation is constantly equal to positive unity.

(8.) The formula XI. may also be thus written,

XV... dq= S (V(dq: q): UVg);

with the verification, that when we suppose dg ||| 7, as in IV., and therefore dUVq = 0 by III., the expression under the sign S becomes the differential of the quotient, Vlq: UVq, and therefore, by 316, VI., of the angle q itself.

336. An important application of the foregoing principles and rules consists in the differentiation of scalar functions of vectors, when those functions are defined and expressed according to the laws and notations of quaternions. It will be found, in fact, that such differentiations play a very extensive part, in the applications of quaternions to geometry; but, for the moment, we shall treat them here, as merely exercises of calculation. The following are a few examples.

(1.) Let p denote, in these sub-articles, a variable vector; and let the following equation be proposed,

so that r is a (generally variable) scalar. Differentiating, and observing that, by 279, III., pp' + p'p = 2Spp', if p' be any second vector, such as we suppose dp to be, we have, by 322, VIII., and 324, VII., the equation,

II... rdr + Spdp = 0;

or III...dr=-r-1Spdf=rSp ̈1dp.

In fact, if r be supposed positive, it is here, by 282, II., the tensor of p; so that this last expression III. for dr is included in the general formula, 334, XIII.

(2.) If this tensor, r, be constant, the differential equation II. becomes simply, IV... Spdp = 0, if p2=const., or if dTp=0.

(3.) Again, let the proposed equation be (comp. 282, XIX.),

V. . . r2 = T (ip + pk), with dɩ = 0, dî = 0,

so that and are here two constant vectors. Then, squaring and differentiating, we have (by 334, XI., because Kip = pi, &c.),

or more briefly,

VI... 2r3 dr=dN (10+ pk) = $(pi + xp) (idp + dpk)

= (12+k2) Spdp + 25kpedp; SEND = SANSSLBYS!

VII... 2r-1 dr = Svdp,

if v be an auxiliary vector, determined by the equation,

VIII. . . r^y = (12 + k2) p + 2Vxpi;

CHAP. II.]

SCALAR FUNCTIONS OF VECTORS.

417

(5.) The equation V. gives (comp. 190, V.), when squared without differentiation,

XI... r = N(ip + pk) = (ip + px) (p + xp)

= (12 + k2) p2 + iрkp + pkpi

= (12 + k2) p2 + 2Siрxp

= (i −k)2 p2 + 4Sip Skp = &c.,

by transformations of the same kind as before; we have therefore, by the recent expressions for ry, the following remarkably simple relation between the two variable vectors, p and v,

(6.) When the scalar, r, is constant, we have, by VII., the differential equation,

XIII... Svdρ = 0; whence also XIV... Spdv = 0, by XII.;

a relation of reciprocity thus existing, between the two vectors p and v, of which the geometrical signification will soon be seen.

(7.) Meanwhile, supposing r again to vary, we see that the last expression VI. for 2r3dr may be otherwise obtained, by taking half the differential of either of the two last expanded expressions XI. for r; it being remembered, in all these little calculations, that cyclical permutation of factors, under the sign S, is permitted (223, (10.)), even if those factors be quaternions, and whatever their number may be: and that if they be vectors, and if their number be odd, it is then permitted, under the sign V, to invert their order (295, (9.)), and so to write, for instance, Vipk instead of Vept, in the formula VIII.

(8.) As another example of a scalar function of a vector, let p denote the proximity (or nearness) of a variable point P to the origin o; so that

Then,

XV... p = (-p2) ̄1 = Tp ́1, or XV'... p2 + p2 = 0.

XVI... dp = Svdp, if XVII... v=p3p = p2Up;

v being here a new auxiliary vector, distinct from the one lately considered (VIII.), and having (as we see) the same versor (or the same direction) as the vector p itself, but having its tensor equal to the square of the proximity of P to o; or equal to the inverse square of the distance, of one of those two points from the other.

337. On the other hand, we have often occasion, in the applications, to consider vectors as functions of scalars, as in 99, but now with forms arising out of operations on quaternions, and therefore such as had not been considered in the First Book. And whenever we have thus an expression such as either of the two following, I...p(t), or II... p= (s, t), p=4(s,

for the variable vector of a curve, or of a surface (comp. again 99), s and t being two variable scalars, and (t) and (s, t) denoting any functions of vector form, whereof the latter is here supposed to be entirely independent of the former, we may then employ (comp. 100,

* We are therefore not employing here the temporary notation of some recent Articles, according to which we should have had, døq = ¢ (q, dq).

(4.) and (9.), and the more recent sub-articles, 327, (5.), (6.), and 329, (5.)) the notation of derivatives, total or partial; and so may write, as the differentiated equations, resulting from the forms I. and II. respectively, the following:

III... dp = p't. dt = p'dt = D1p.dt;

IV... dp = dp + d1p = D1p.ds + D1p.dt;

of which the geometrical significations have been already partially seen, in the sub-articles to 100, and will soon be more fully developed.

(1.) Thus, for the circular locus, 314, (1.), for which

V... paß, Ta= 1, Saß = 0,

we have, by 333, VIII., the following derived vector,

VI. . . p' = Dip == atẞ = Tap.

αρ.

(2.) And for the elliptic locus, 314, (2.), for which

VII. . . p = V. a'ß, Ta=1, but not Saß = 0,

we have, in like manner, this other derived vector,

(3.) As an example of a vector-function of more scalars than one, let us resume the expression (308, XVIII.),

IX... p = rk' ja kj-sk-t ;

in which we shall now suppose that the tensor r is given, so that p is the variable vector of a point upon a given spheric surface, of which the radius is r, and the centre is at the origin; while s and t are two independent scalar variables, with respect to which the two partial derivatives of the vector p are to be determined.

(4.) The derivation relatively to t is easy; for, since ijk are vector-units (295), and since we have generally, by 333, VIII.,

π

π

X... d.ax= alde, and therefore

XI... Dt. a* =

Dx,

if Ta= 1, and if x be any scalar function of t, we may write, at once, by 279, IV.,

π

XIL... Dep = 1⁄2 (kp — pk) =πVkp ;

and we see that

XIII...

SpDtp = 0,

a result which was to be expected, on account of the equation,

XIV... p2 + r2 = 0,

which follows, by 308, XXIV., from the recent expression IX. for p.

(5.) To form an expression of about the same degree of simplicity, for the other partial derivative of p, we may observe that js+1 kj is equal to its own vector part (its scalar vanishing); hence