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is integrable, or represents a system of surfaces, without the expression
Evlp being an exact differential, as it was in 410, (b). In this case,
there exists some scalar factor, n, such that Snydp is the exact diffe-
rential of a scalar function of p, without the assumption that this vec-
top is itself a function of a scalar variable, t; and then if we write
(pp. 701, 702, comp. p. xxx),

dv=pdo, d. nv = ddp,
tiis new vector function & will be self-conjugate, although the function
is not such now, as it was in the equation (U1).

(e). In this manner it is found (p. 702), that the Condition* of In-
tegrability of the equation (Y2) is expressed by the very simple for-

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in which y is a vector function of p, not generally linear, and deduced
from on the plan of the Section III. ii. 6 (p. 442), by the relation,
pdp-p'dp = 2Vydp;

being the conjugate of p, but not here equal to it.


(d). Connexions (pp. 702, 703) of the Mixed Transformations in
the last cited Section, with the known Modular and Umbilicar Gene-
rations of a surface of the second order.

(). The equation (p. 704),

T-V.ẞVya)=T(a - V. & Vẞp),


in which a, B, y are any three vector constants, represents a central
quadric, and appears to offer a new mode of generationt of such a sur-
face, on which there is not room to enter, at this late stage of the


The vector of the centre of the quadric, represented by the

equation fp - 28ep = const., with fp = Sppp, is generally x=-1ε

= (p. 704);

case of paraboloids, and of cylinders.

(9). The equation (p. 705),

Sqpq'pq"p+ Spop + Syp+ C = 0,


represents the general surface of the third degree, or briefly the General
Cubic Surface; C being a constant scalar, y a constant vector, and 7,
, three constant quaternions, while op is here again a linear,
Vector, and self-conjugate function of p.

(4). The General Cubic Cone, with its vertex at the origin, is thus
represented in quaternions by the monomial equation (same page),


*It is shown, in a Note to p. 702, that this monomial equation (Y”2) be-
comes, when expanded, the known equation of six terms, which expresses the con-
dition of integrability of the differential equation pdx+qdy + rdz = 0.

In a Note to p. 649 (already mentioned in p. xxviii), the reader will find
references to the Lectures, for several different generations of the ellipsoid, derived
from quaternion forms of its equation.

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(c). When the condition (C3) is satisfied, the forces compound

themselves generally into one couple, of which the axis = £Vaß, what-
ever may be the position of the assumed origin o of vectors.

(d). When

Vaẞ=0, (D3), with or without (C3),

the forces have no tendency to turn the body round that point o; and
when the equation (A3) holds good, as in (a), for an arbitrary vector
y, the forces do not tend to produce a rotation round any point c,
so that they completely balance each other, as before, and both the
conditions (C3) and (D3) are satisfied.

(e). In the general case, when neither (C3) nor (D3) is satisfied, if

be an auxiliary quaternion, such that

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with T2ẞ>0, then SQ = c = central moment divided by total force;

*It is easy to prove that the moment of the force ß, acting at the end of the
vector a from o, and estimated with respect to any unit-line from the same ori-
gin, or the energy with which the force so acting tends to cause the body to turn
round that line, regarded as a fixed axis, is represented by the scalar, – Staß, or
Saß; so that when the condition (D3) is satisfied, the applied forces have no
tendency to produce rotation round any axis through the origin: which origin
becomes an arbitrary point c, when the equation of equilibrium (A3) holds good.


and VQ is the vector y of a point c upon the central axis which does not vary with the origin o, and which there are reasons for considering as the Central Point of the system, or as the general centre of applied forces in fact, for the case of parallelism, this point c coincides with what is usually called the centre of parallel forces.

(9). Conceptions of the Total Moment Laß, regarded as being generally a quaternion; and of the Total Tension, – Σaß, considered as a scalar to which that quaternion with its sign changed reduces itself for the case of equilibrium (a), and of which the value is in that case independent of the origin of vectors.

(h). Principle of Virtual Velocities,

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the vector representing the accelerating force, or me the moving force, acting on a particle m of which the vector at the time t is a; and da being any infinitesimal variation of this last vector, geometrically compatible with the connexions between the parts of the system, which need not here be a rigid one.

(b). For the case of a free system, we may change each da to ε + Via, ε and being any two infinitesimal vectors, which do not change in passing from one particle m to another; and thus the general equation (H3) furnishes two general vector equations, namely,

Em (Da-)=0, (13), and ΣmVa (Di3α-)=0; (J3) which contain respectively the law of the motion of the centre of grarity, and the law of description of areas.

(e). If a body be supposed to be rigid, and to have a fixed point o, then only the equation (J3) need be retained; and we may write, Dia = Via,

(K3) being here a finite vector, namely the Vector Axis of Instantaneous Rotation: its versor U denoting the direction of that axis, and its tensor Ti representing the angular velocity of the body about it, at the time t.

(d). When the forces vanish, or balance each other, or compound themselves into a single force acting at the fixed point, as for the case of a heavy body turning freely about its centre of gravity, then 2mVag=0, (L3); and if we write, p=2maVai,


so that again denotes a linear, vector, and self-conjugate function, we shall have the equations,

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the vector y being what we may call the Constant of Areas, and the scalar 2 being the Constant of Living Force.

(e). One of Poinsot's representations of the motion of a body, under the circumstances last supposed, is thus reproduced under the form, that the Ellipsoid of Living Force (P3), with its centre at the fixed point o, rolls without gliding on the fixed plane (Q3), which is parallel to the Plane of Areas (Sty = 0); the variable semidiameter of contact, , being the vector-axis (c) of instantaneous rotation of the body.

(f). The Moment of Inertia, with respect to any axis through o, is equal to the living force (h2) divided by the square (Ti2) of the semidiameter of the ellipsoid (P3), which has the direction of that axis; and hence may be derived, with the help of the first general construction of an ellipsoid, suggested by quaternions, a simple geometrical representation (p. 711) of the square-root of the moment of inertia of a body, with respect to any axis AD passing through a given point A, as a certain right line BD, if CD = CA, with the help of two other points B and C, which are likewise fixed in the body, but may be chosen in more ways than one.

(g). A cone of the second degree,

Siv = 0, (S3), with v=y2pt - h2p2,


is fixed in the body, but rolls in space on that other cone, which is the locus of the instantaneous axis ; and thus a second representation, proposed by Poinsot, is found for the motion of the body, as the rolling of one cone on another.

(h). Some of Mac Cullagh's results, respecting the motion here considered, are obtained with equal ease by the same quaternion analysis; for example, the line y, although fixed in space, describes in the body an easily assigned cone of the second degree (p. 712), which cuts the reciprocal ellipsoid,

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in a certain sphero-conic: and the cone of normals to the last mentioned cone (or the locus of the line + hey ́1) rolls on the plane of areas (StY = 0).

(i). The Three (Principal) Axes of Inertia of the body, for the given point o, have the directions (p. 712) of the three rectangular and vector roots (comp. (P), p. xii., and the paragraph 415, (a), p. xlii.) of the equation

Vipi=0, (V3), because, for each, Du=0;


and if A, B, C denote the three Principal Moments of inertia corresponding, then the Symbolical Cubic in p (comp. the formula (N) in page xii.) may be thus written,

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(j). Passage (p. 713), from moments referred to axes passing through a given point o, to those which correspond to respectively parallel axes, through any other point 2 of the body.



ARTICLE 418.-On the motions of a System of Bodies, considered
as free particles m, m', . . which attract each other according to the
law of the Inverse Square ..

(a). Equation of motion of the system,

EmSD ada + ¿P= 0, (X3), if P= Σmm'T (a− a')1; (Y3)

a is the vector, at the time t, of the mass or particle m; P is the po-
tential (or force-function); and the infinitesimal variations da are ar-

(b). Extension of the notation of derivatives,

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(c). The differential equations of motion of the separate masses
m,.. become thus,

mD2a + DaP= 0, . . ;


and the laws of the centre of gravity, of areas, and of living force,
are obtained under the forms,



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B, y being two vector constants, and H a scalar constant.

(d). Writing,


(P+T) dt, (Es), and V- =f [ 2 Tαt = F+ tH, (F.)

F= ['" (P+T) α,

F may be called the Principal Function, and V the Characteristic
Function, of the motion of the system; each depending on the final
vectors of position, a, a', . . and on the initial vectors, ao, a'o,.. ; but
F depending also (explicitly) on the time, t, while V (= the Action)
depends instead on the constant H of living force, in addition to those
final and initial vectors: the masses m, m', . . being supposed to be
known, or constant.

(e). We are led thus to equations of the forms,

mDia + DaF=0,.. (Gs); mDoa + DaF=0,.. (H1);

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whereof the system (G) contains what may be called the Interme-
diate Integrals, while the system (H) contains the Final Integrals,
of the differential Equations of Motion (A4).

(ƒ). In like manner we find equations of the forms,

DaV=-mDia,.. (J1); DaV=mD。a,.. (K1); DμV=t; (Ls)
the intermediate integrals (e) being here the result of the elimination

* References are given to two Essays by the present writer, "On a General
Method in Dynamics," in the Philosophical Transactions for 1834 and 1835, in
which the Action (V), and a certain other function (S), which is here denoted by F,
were called, as above, the Characteristic and Principal Functions. But the ana-
lysis here used, as being founded on the Calculus of Quaternions, is altogether
unlike the analysis which was employed in those former Essays.

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