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which shows, in a new way, how to resolve a linear equation in quaternions, when put under what we may call (comp. 347, (1.)) the Standard Quadrinomial Form, XXXIV.

(11.) Accordingly, if we operate on the formula XXXVIII. with ƒ, attending to the equations XXXIII., and dividing by (prst), we get this new equation, XXXIX... (p'r's't') fq=p' (q'r's't') -r' (q's't'p')+s' (q't'p'r') -t' (q'p'r's'); fq=q', by XXV.


(12.) It has been remarked (9.), that p, r, s, t, in recent formulæ, may be any four quaternions, which do not satisfy the equation XXXII.; we may therefore assume,

XL. p=1, r=i, 8=j, t=k,

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with the laws of 182, &c., for the symbols i, j, k, because those laws give here,

XLI... (lijk) = −2;

and then it will be found that the equations XXXIII. give simply,

XLII...p' =ƒ1, r'=-fi, s'=-fj, t'=-fk;

so that the standard quadrinomial form XXXIV. becomes, with this selection of prst,

XLIII... fq=f1.Sq-fi.Siq-fj.Sjq-fk. Skq,

and admits of an immediate verification, because any quaternion, q, may be expressed (comp. 221) by the quadrinomial,

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the letters abcde being here used to denote five known quaternions, while wryz are four sought scalars, the problem of quaternion inversion comes to be that of the separate determination (comp. 312) of these four scalars, so as to satisfy the one equation XLVII.; and it is resolved (comp. XXV.) by the system of the four following formulæ :


[w(abcd)=(ebcd); x(abcd)=(aecd);
Ly (abcd) = (abed); z(abcd)=(abce);

the notations (6.) being retained.

(14.) Finally it may be shown, as follows, that the biquadratic equation I., for linear functions of quaternions, includes* the cubic I'., or 350, I., for vectors. Sup

* In like manner it may be said, that the cubic equation includes a quadratic one, when we confine ourselves to the consideration of vectors in one plane; for which case m = 0, and also p=0, if p be a line in the given plane: for we have then x=m=m', or

p3 — m”4+ m2 = 0,




pose, for this purpose, that the linear and quaternion function, fq, reduces itself to the last term of the general expression 364, XII., or becomes,

XLIX... fq=&Vq, so that L...e= =0, ɛ=ɛ'=0, ƒ1=ƒ'1=0;

the coefficients n, n', n”, n”” take then, by XIII., the values,

LI...n= =0, 'n'=m, n"=m', n"=m";

and the biquadratic I. becomes,

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But fq is now a vector, by XLIX., and it may be any vector, p; also the operation f is now equivalent to that denoted by , when the subject of the operation is a vector; we may therefore, in the case here considered, write this last equation LII. under the form,

LIII... 0=(-m + m'p — m"p2 + Ø3)p,

which agrees with 351, I., and reproduces the symbolical cubic, when the symbol of the operand (p) is suppressed.



SECTION 1.-Remarks Introductory to this Concluding


366. WHEN the Third Book of the present Elements was begun, it was hoped (277) that this Book might be made a much shorter one, than either of the two preceding. That purpose it was found impossible to accomplish, without injustice to the subject; but at least an intention was expressed (317), at the commencement of the Second Chapter, of rendering that Chapter the last: while some new Examples of Geo

with this understanding as to the operand. In fact, the cubic gives here (because m = 0),

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if o be already the result of an operation with o, on any vector p: that is if it be, as above supposed, a line in the given plane.

metrical Applications, and some few Specimens of Physical ones, were promised.

367. The promise, thus referred to, has been perhaps already in part redeemed; for instance, by the investigations (315) respecting certain tangents, normals, areas, volumes, and pressures, which have served to illustrate certain portions of the theory of differentials and integrals of quaternions. But it may be admitted, that the six preceding Sections have treated chiefly of that Theory of Quaternion Differentials, including of course its Principles and Rules; and of the connected and scarcely less important theory of Linear or Distributive Functions, of Vectors and Quaternions: Examples and Applications having thus played hitherto a merely subordinate or illustrative part, in the progress of the present Volume.

368. Such was, indeed, designed from the outset to be, upon the whole, the result of the present undertaking: which was rather to teach, than to apply, the Calculus of Quaternions. Yet it still appears to be possible, without quite exceeding suitable limits, and accordingly we shall now endeavour, to condense into a short Third Chapter some Additional Examples, geometrical and physical, of the application of the principles and rules of that Calculus, supposed to be already known, and even to have become by this time familiar to the reader. And then, with a few general remarks, the work may be brought to its close.

SECTION 2.-On Tangents and Normal Planes to Curves in Space.

369. It was shown (100) towards the close of the First Book, that if the equation of a curve in space, whether plane or of double curvature, be given under the form,

I. . . p = p(t) = &t,

where t is a scalar variable, and is a functional sign, then the derived vector,

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* Accordingly, even references to former Articles will now be supplied more sparingly than before.


represents a line which is, or is parallel to, the tangent to the curve, drawn at the extremity of the variable vector p. If then we suppose that T is a point situated upon the tangent thus drawn to a curve PQ at P and that u is a point in the corresponding normal plane, so that the angle TPU is right, and if we denote the vectors op, ot, ou by p, T, v, the equations of the tangent line and normal plane at P may now be thus expressed:

III... V(T-P) p'=0;

IV... S(v-p) p′ =0;

the vector being treated as the only variable in III., and in like manner as the only variable in IV., when once the curve PQ is given, and the point P is selected.

(1.) It is permitted, however, to express these last equations under other forms; for example, we may replace p' by dp, and thus write, for the same tangent line and normal plane,

V... V(rp) dp = 0 ;

VI... S(v-p) dp = 0 ;

where the vector differential dp may represent any line, parallel to the tangent to the curve at P, and is not necessarily small (compare again 100).

(2.) We may also write, as the equation of the tangent,

VII. . . r = p + xp', where x is a scalar variable;

and as the equation of the normal plane,

VIII... d2T(v − p)=0, or VIII'... dT (v− p)=0, if dv=0; because this partial differential of T(v − p), or of PU, is (by 334, XII., &c.), IX... dT(v - p) = S(U(v− p).dp).

(3.) For the circular locus 314, (1.), or 337, (1.), of which the equation is, X... patß, with Ta= 1, and Saß=0,


the equation of the tangent is, by VII., and by the value 337, VI. of p', XI... r=p+yap, where y is a new scalar variable;

the perpendicularity of the tangent to the radius being thus put in evidence. (4.) For the plane but elliptic locus, 314, (2.), or 337, (2), for which, XII... p= V.a1ß, with Ta = 1, but not Saß = 0,

the value 337, VIII. of p' shows that the tangent, at the extremity of any one semidiameter f, is parallel to the conjugate semidiameter of the curve; that is, to the one obtained by altering the excentric anomaly (314, (2.)), by a quadrant: or to the value of p which results, when we change t to t +1.


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the tangent line (p') to the helix is therefore inclined to the axis (a) of the cylinder whereon that curve is traced, at a constant angle (a), whereof the trigonometrical tangent (tan a) is given by this formula XVII.; and accordingly, the numerator Tẞ of that formula represents the semicircumference of the cylindric base; while the denominator 2c is an expression for half the interval between two successive spires, measured in a direction parallel to the axis. We may then write,

XVIII... TT3=2c tan a = 2c cot b,

if a thus denote the constant inclination of the helix to the axis, while b denotes the constant and complementary inclination of that curve to the base, or to the circles which it crosses on the cylinder.

(6.) In general, the parallels p' to the tangents to a curve of double curvature, which are drawn from a fixed origin o, have a certain cone for their locus; and for the case of the helix, the equation of this cone is given by the formula XVII., or by any legitimate transformation thereof, such as the following,

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it is therefore, in this case, a cone of revolution, with its semiangle = a.

(7.) As an example of the determination of a normal plane to a curve of double curvature, we may observe that the equation XIII. of the helix gives,

XX... p2= ẞ2 — c2t2, and therefore XXI... Spp' = − c2t ;

the equation IV. becomes therefore, for the case of this curve,

XXII... 0= Sp'v+ ct, with the value XIV. of p'.

(8.) If then it be required to assign the point u in which the normal plane to the helix meets the axis of the cylinder, we have only to combine this equation XXII. with the condition v || a, and we find, by XIII. and XIV.,

XXIII. . . OU = v = cta: Sap' = cta, XXIV... Sa (v − p)=0 ;

the line PU is therefore perpendicular to the axis, being in fact a normal to the cylinder.

370. Another view of tangents and normal planes may be proposed, which shall connect them in calculation with Taylor's Series adapted to quaternions (342), as follows.

(1.) Writing I. . . pt = po + u¿tp'o, or briefly, I'. . . pt = p + utp', the coffiecient ut or u will generally be a quaternion, but its limiting value will be positive unity, when t tends to zero as its limit; or in symbols,

II... u
..uolim. u = 1.


(2.) Admitting this, which follows either from Taylor's Series, or (in so simple a case) from the mere definition of the derived vector p', we may conceive that vector p' to be constructed by some given line PT, without yet supposing it to be known that this line is tangential at P to the curve PQ, of which the variable vector is oQ=P1, while OP = pop, so that the line PQ = utp' is a vector chord from P, which diminishes indefinitely with the scalar variable, t, and is small, if t be small.

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