kind considered in the Section III. ii. 6, whether self-conjugate or not; but it requires a little more attention to perceive, that an equation of this other form,

LIII... T(p-V. ẞVya)=T(a - V. YVẞp),

represents such a surface, whatever the three vector constants a, ß, y may be. The discussion of this last form would present some circumstances of interest, and might be considered to supply a new mode of generation, on which however we cannot enter here.

(22.) The surfaces of the second order, considered hitherto in the present Section, have all had the origin for centre. But if, retaining the significations of ø, ƒ, and F, we compare the two equations,

LIV. . . ƒ (p − k) = C, and LV. . . fp – 2Sɛp = C',

we shall see (by 362, &c.) that the constants are connected by the two relations,

(23.) If then we meet an equation of the form LV., in which (as has been usual) we have still fp= Spop = a scalar and homogeneous function of p, of the second dimension, we shall know that it represents generally a surface of that order, with the expression (comp. 347, IX., &c.),

LVIII... = -1e = m ̄1e = Vector of Centre.

(24.) It may happen, however, that the two relations,

LIX... m= =0, Tε>0,

exist together; and then the centre may be said to be at an infinite distance, but in a definite direction: and the surface becomes a Paraboloid, elliptic or hyperbolic, according to conditions which are easy consequences from what has been already shown. (25.) On the other hand it may happen that the two equations,

LX... m = 0, ε=0,

are satisfied together; and then the vector of the centre acquires, by LVIII., an indeterminate value, and the surface becomes a Cylinder, as has been already suthiciently exemplified.

(26.) It would be tedious to dwell here on such details; but it may be worth

general rule, to avoid writing v=pp, except for central quadrics; and to confine ourselves to the notation dy = ødp, as in some recent and several earlier sub-articles, when we wish, for the sake of association with other investigations and results, to treat the function p as linear (or distributive); because we shall thus be at liberty to treat the surface as general, notwithstanding this property of 4. As regards the methods of generating a quadric, it may be worth while to look back at the Note to page 649, respecting the Six Generations of the Ellipsoid, which were given by the writer in the Lectures, with suggestions of a few others, as interpretations of quaternion equations.

CHAP. III.] CUBIC CONE, SCREW SURFACE, SKEW CENTRE.

705

while to observe, that the general equation of a Surface of the Third Degree may be thus written:

LXI... Sqpq'pq"p + Sppp + Syp + C = 0;

C and y being any scalar and vector constants; pp any linear, vector, and self-conjugate function; and q, q', q′′ any three constant quaternions: while p is, as usual, the variable vector of the surface.

(27.) In fact, besides the one scalar constant, C, three are included in the vector y, and six others in the function (comp. 358); and of the ten which remain to be introduced, for the expression of a scalar and homogeneous function of p, of the third degree, the three versors Uq, Uq', Uq" supply nine (comp. 312), and the tensor T.qq'q" is the tenth.

(28.) And for the same reason the monomial equation,

LXII... Sqpq'pq′′p = 0,

with the same significations of q, q', q′′, represents the general Cone of the Third Degree, or Cubic Cone, which has its vertex at the origin of vectors.

(29.) If then we combine this last equation with that of a secant plane, such as Sɛp+1=0, we shall get a quaternion expression for a Plane Cubic, or plane curve of the third degree: and if we combine it with the equation p2 + 1 = 0 of the unitsphere, we shall obtain a corresponding expression for a Spherical Cubic,* or for a curve upon a spheric surface, which is cut by an arbitrary great circle in three pairs of opposite points, real or imaginary.

(30.) Finally, as an example of sections of surfaces, represented by transcendental equations, let us consider the Screw Surface, or Helicoid, † of which the vector equation may be thus written (comp. the sub-arts. to 314):

LXIII.. p=c(x + a) a + ya*ɣ, with Ta = 1, y=Vaß, and y>0;

a being the unit axis, while ẞ, y are two other constant vectors, a, c two scalar constants, and x, y two variable scalars.

(31.) Cutting this surface by the plane of By, or supposing that

LXIV... 0= Syßp = ß2 Sap - SaẞSßp, and writing LXV... c = bSaß, we easily find that the scalar and vector equations of what we may call the Screw Section may be thus written :

LXVI... b (x + a) = yS. a2 ̄1;

LXVII... p=y(yS. a* – BS. a*-1).

(32.) Derivating these with respect to a, and eliminating ẞ and y', we arrive at the equation,

LXVIII. . . p = (x + a) p′+ zy, if _LXIX. . . 2bz = ñy2;

Compare the Note to page 43; see also the theorem in that page, which contains perhaps a new mode of generation of cubic curves in a given plane: or, by an easy modification, of the corresponding curves upon a sphere.

Already mentioned in pages 383, 502, 514, 557. The condition y>0 an swers to the supposition that, in the generation of the surface, the perpendiculars from a given helix on the axis of the cylinder are not prolonged beyond that axis.

but zy in LXVIII. is the vector of the point, say G, in which the tangent to the section at the point (x, y), or P, intersects the given line y, namely the line in the plane of that section which is perpendicular to the axis a: we see then, by LXIX., that this point of intersection depends only on the constant, b, and on the variable, y, being independent of the constant, a, and of the variable, x.

(33.) To interpret this result of calculation, which might have been otherwise found with the help of the expression 372, XII. (with ẞ changed to y) for the normal v to a screw-surface, we may observe, first, that the equation LXVII., which may be written as follows,

LXX...p=yV.ar+1ẞ, and gives LXXI... TVap=yTy,

would represent an ellipse, if the coefficient y were treated as constant; namely, the section of the right cylinder LXXI. by the plane LXIV.; the vector semiaxes (major and minor) of this ellipse being yẞ and yy (comp. 314, (2.)).

(34.) By assigning a new value to the constant a, we pass to a new screw surface (30.), which differs only in position from the former, and may be conceived to be formed from it by sliding along the axis a; while the value of x, corresponding to a given y, will vary by LXVI., and thus we shall have a new screw section (31.), which will cross the ellipse (33.) in a new point q: but the tangent to the section at this point will intersect by (32.) the minor axis of the ellipse in the same point G as before.

S

G

с

P

Fig. 85.

(35.) We shall thus have a Figure* such as the following (Fig. 85); in which if F be a focus of the ellipse BC, and G (as above) the point of convergence of the tangents to the screw sections at the points P, Q, &c., of that ellipse, it is easy to prove, by pursuing the same analysis a little farther, Ist, that the angle (g), subtended at this focus F by the minor semiaxis oc, which is also a radius (r) of the cylinder LXXI., is equal to the inclination of the axis (a) of that cylinder to the plane of the ellipse, as may indeed be inferred from elementary principles; and IInd, what is less obvious, that the other angle (h), subtended at the same focus (F) by the interval OG, or by what may be called (with reference to the present construction, in which it is supposed that b< 0, or that the angles made by Dip and ẞ with a are either both acute, or both obtuse) the Depression (s) of the Skew Centre (G), is equal to the inclination of the same axis (a) to the helix on the same cylinder, which is obtained (comp. 314, (10.)) by treating y as constant, in the equation LXIII. of the Screw Surface.

*Those who are acquainted, even slightly, with the theory of Oblique Arches (ot skew bridges), will at once see that this Figure 85 may be taken as representing rudely such an arch and it will be found that the construction above deduced agrees with the celebrated Rule of the Focal Excentricity, discovered practically by the late Mr. Buck. This application of Quaternions was alluded to, in page 620 of the Lectures.

CHAP. III.]

STATICS OF A RIGID BODY.

707

SECTION 8.-On a few Specimens of Physical Application of

Quaternions, with some Concluding Remarks.

416. It remains to give, according to promise (368), before concluding this work, some examples of physical applications of the present Calculus: and as a first specimen, we shall take the Statics of a Rigid Body.

(1.) Let a,.. an ben Vectors of Application, and let B1, . . ẞn be n corresponding Vectors of Force, in the sense that n forces are applied at the points A1, . . An of a free but rigid system, and are represented as usual by so many right lines from those points, to which lines the vectors OB1, . . O, are equal, though drawn from a common origin; and let y(=oc) be the vector of an arbitrary point c of space. Then the Equation† of Equilibrium of the system or body, under the action of these n applied forces, may be thus written :

(2.) The supposed arbitrariness (1.) of y enables us to break up the formula I. or I', into the two vector equations :

of each of which it

easy to assign, as follows, the physical signification.

(3.) The equation II. expresses that if the forces, which are applied at the points A1.. of the body, were all transported to the origin o, their statical resultant, or vector sum, would be zero.

(4.) The equation III. expresses that the resultant of all the couples, produced in the usual way by such a transference of the applied forces to the assumed origin, is null.

(5.) And the equation I., which as above includes both II. and III., expresses that if all the given forces be transported to any common point C, the couples hence arising will balance each other: which is a sufficient condition of equilibrium of the system.

(6.) When we have only the relation,

IV. . . S (Σβ. Σαβ) = 0,

without ẞ vanishing, the applied forces have then an Unique_Resultant = Σβ, acting along the line of which I. or I'. is the equation, with y for its variable vec

tor.

* The reader may compare the remarks on hydrostatic pressure, in pages 434, 435.

We say here, "equation:" because the single quaternion formula, I. or I'., contains virtually the six usual scalar equations, or conditions, of the equilibrium at present considered.

(7.) And the physical interpretation of this condition IV. is, that when the forces are transported to o, as in (3.) and (4.) the resultant force is in the plane of the resultant couple.

(8.) When the equation II., but not III., is satisfied, the applied forces compound themselves into One Couple, of which the Axis = 2Vaß, whatever may be the posi tion of the origin.

(9.) When neither II. nor III. is satisfied, we may still propose so to place the auxiliary point C, that when the given forces are transferred to it, as in (5.), the resultant force Σß may have the direction of the axis ΣV(a− y)ẞ of the resultant couple, or else the opposite of that direction; so that, in each case, the condition,*

shall be satisfied by a suitable limitation of the auxiliary vector y.

(10.) This last equation V. represents therefore the Central Axis of the given system of applied forces, with y for the variable vector of that right line; or the aris of the screw-motion which those forces tend to produce, when they are not in balance, as in (1.), and neither tend to produce translation alone, as in (6.), nor rotation alone, as in (8.).

(11.) In general, if q be an auxiliary quaternion, such that

VI. . . 9Σβ = Σαβ,

its vector part, Vq, is equal by (V.) to the Vector-Perpendicular, let fall from the origin on the central axis; while its scalar part, Sq, is easily proved to be the quotient, of what may be called the Central Moment, divided by the Total Force: so that Vq=0 when the central axis passes through the origin, and Sq=0 when there exists an unique resultant.

(12.) When the total force 2ẞ does not vanish, let Q be a new auxiliary quaternion, such that

with

VIII... c=SQ= Sq, and IX. . . y = oc = VQ,

for its scalar and vector parts; then c28 represents, both in quantity and in direction, the Axis of the Central Couple (9.), and y is the vector of a point c which is on the central axis (10.), considered as a right line having situation in space: while the position of this point on this line depends only on the given system of applied forces, and does not vary with the assumed origin o.

(13.) Under the same conditions, we have the transformations,

Χ... Σαβ = (c + γ) Σβ;

XII... Vaß = c£ß+Vy£ß;

ΧΙ. . . ΤΣαβ = (c - γε) ΤΣβ ;

XIII... (EVaß)2 = c2 (£ß)2 + (VyΣB)2;

*The equation V. may also be obtained from the condition,

V'... TEV (a - y)ß = a minimum,

when y is treated as the only variable vector; which answers to a known property

of the Central Moment.