, the inclination, 7, the angular distance of X from node, and the formula

The foregoing very convenient algorithm, viz., the employment of

131. But previously to the foregoing investigations, viz., in the memoir "Du Mouvement de Rotation &c.," Mém. de Berlin for 1758 (pr. 1765), Euler had obtained incidentally a very elegant solution of the problem of the transformation of coordinates; this is in fact identical with the next mentioned one, the letters l, m, n; X, μ, v being used in the place of 5, 5', 5'' ; n, n', n''.

132. In the memoir "Formulæ generales pro translatione &c." (1775), Euler gives the following formula for the transformation of coordinates, viz., if the position of the set of axes XYZ in reference to the set ayz is determined by

with the following equations connecting the six angles, viz., if —A2=cos (n'—n") cos (n"—n) cos (n−n'),

Y

y

then

133. It is right to notice that these values of 4, 4, 4" give the twelve equations a2+B+y=1, &c., but they do not give definitely a=B'y"-ẞ"y', &c., but only a=+(B'y"-"y'); that is, in the formulæ in question the two sets of axes are not of necessity displacements the one of the other. In the same memoir Euler considers two sets of rectangular axes, and assuming that they are displacements the one of the other (this assumption is not made as explicitly as it should have been), he remarks that the one set may be made to coincide with the other set by means of a finite rotation about a certain axis (which may conveniently be termed the Resultant Axis). This consideration leads him to an equation which ought to be satisfied by the coefficients of transformation, but which he is not able to verify by means of the foregoing expressions in terms of 4, 5', L'', n, n', n''.

134. I remark that Euler's equation in fact is

a',

=0,

β ,7 -(B'y"—ẞ"y')—(y′′a—ya")—(aß'—a'ß)+a+ß'+y”—1=0,

B' Y

a", B", Y"

"

in which form it is an immediate consequence of the equations

which are true for the proper, but not for the improper transformation. 135. In the undated addition to the memoir, Euler states the theorem of the resultant axis as follows:-"Theorema. Quomodocunque sphæra circa centrum suum convertatur, semper assignari potest diameter cujus directio in situ translato conveniat cum situ originali ;" and he again endeavours to obtain a verification of the foregoing analytical theorem.

136. The theory of the Resultant Axis was further developed by Euler in the memoir "Nova Methodus Motum &c." (1775), and by Lexell in the me

moir "Nonnulla theoremata generalia &c." (1775): the geometrical investigations are given more completely and in greater detail in Lexell's memoir. The result is contained in the following system of formula for the transformation of coordinates, viz., if a, ß, y are the inclinations of the resultant axis to the original set, and if 4 is the rotation about the resultant axis, or say the resultant rotation, then we have

zcosycosa(1-coso)+cosẞsino cosycosß(1—coso) — cosasino cos2y+sin2ycosp

Euler attempts, but not very successfully, to apply the formula to the dynamical problem of the rotation of a solid body: he does not introduce them into the differential equations, but only into the integral ones, and his results are complicated and inelegant. The further simplification effected by Rodrigues was in fact required.

137. Jacobi's paper, "Euleri formulæ &c." (1827), merely cites the last

mentioned result.

138. I find it stated in Lacroix's 'Differential Calculus,' t. i. p. 533, that the following system for the transformation of coordinates was obtained by Monge (no reference is given in Lacroix), viz., the system being as above,

and the quantities a, ẞ', y" being arbitrary, then putting

26=√NP+√MQ, 2y' = √PQ+ √ MN, 2a′′=√QN+√MP,

2a'=√NP-√MQ, 23′′=√PQ_√MN, 2y =√QN−√MP.

These are formulæ very closely connected with those of Rodrigues.

139. The theory was perfected by Rodrigues in the valuable memoir "Des lois géométriques &c." (1840). Using for greater convenience λ, μ, v in the place of his m, n, p, he in effect writes

and this being so, the coefficients of transformation are

1+λ2—μ2 — v2, 2(λμ+v) , 2(xv-μ)

1—λ2+μ2—v2, 2(μv+1)

all divided by the common denominator 1+1+2+v. Conversely, if the coefficients of transformation are as usual represented by

a β

a', B', Y',

a", ẞ", y",

then λ2, μ3, v3, λ, μ, v are respectively equal to

1+a-ẞ'-y", 1-a+ß-y", 1-a-ẞ'+y",

each of them divided by 1+a+ß'+y".

B-a'

The memoir contains very elegant formula for the composition of finite rotations, and it will be again referred to in speaking of the kinematics of a solid body.

140. Sir W. R. Hamilton's first papers on the theory of quaternions were published in the years 1843 and 1844: the fundamental idea consists in the employment of the imaginaries i, j, k, which are such that

¿2=j2=k2=—1, jk=—kj=i, ki=—ik=j, ij=—ji=k,

whence also

W2+X2+Y2+Z2=(w2+x2+ y2+z2) (w'2 +x12+y'2+z'2).

It is hardly necessary to remark that Sir W. R. Hamilton in his various publications on the subject, and in the Lectures on Quaternions,' Dublin, 1853, has developed the theory in detail, and has made the most interesting applications of it to geometrical and dynamical questions; and although the first explicit application of it to the present question may have been made in my own paper next referred to, it seems clear that the whole theory was in its original conception intimately connected with the notion of rotation.

141. Cayley, "On certain Results relating to Quaternions" (1845).—It is shown that Rodrigues' transformation formula may be expressed in a very simple manner by means of quaternions; viz., we have

ix+jy+kz=(1+iλ+jμ+kv)−1(¿X+jY+kZ) (1+iλ+jμ+kv), where developing the function on the right-hand side, and equating the coefficients of i, j, k, we obtain the formulæ in question. A subsequent paper, Cayley, "On the application of Quaternions to the Theory of Rotation" (1848), relates to the composition of rotations.

Principal Axes, and Moments of Inertia. Article Nos. 142-163.

142. The theorem of principal axes consists herein, that at any point of a solid body there exists a system of axes Ox, Oy, Oz, such that

fyzdm=0, fzxdm=0, ƒxydm=0.

But this, the original form of the theorem, is a mere deduction from a general theory of the representation of the integrals

fx'dm, fy'dm, fz'dm, fyzdm, fzxdm, fxydm

for any axes through the given origin by means of an ellipsoid depending on the values of these integrals corresponding to a given set of rectangular axes through the same origin.

143. If, for convenience, we write as follows, M=fdm the mass of the body, and

A'=fx2dm, B'=f y3dm, C'=fz3dm, F'=f yzdm, G'=fzxdm, H'=fxydm,

and moreover

so that

A=f(y2+z2) dm, B=f(z2+x2) dm, C=f (x2+y3) dm,

F=-fyzdm, G=-fzxdm, H=·
=-fxydm*,

A=B'+C', B=C'+A', C=A'+ B', F‒‒F', G=—G', H=—H', then the ellipsoid which in the first instance presents itself for this purpose, and which Prof. Price has termed the Ellipsoid of Principal Axes, but which I would rather term the "Comomental Ellipsoid," is the ellipsoid

(A', B', C', F', G', H'(x, y, z)2=Mk1,

where k is arbitrary, so that the absolute magnitude is not determined. But it is more usual, and in some respects better to consider in place thereof the "Momental Ellipsoid" (Cauchy, "Sur les Moments d'Inertie," Exercices de Mathématique, t. ii. pp. 93–103, 1827),

(A, B, C, F, G, HXx, y, z)2=Mk1,

or as it may also be written,

(A'+B'+C')(x2+ y2+z2)—(A', B', C′, F', G', H'Xx, y, z)2=M%*,

which shows that the two ellipsoids have their axes, and also their circular sections coincident in direction.

144. And there is besides this a third ellipsoid, the "Ellipsoid of Gyration," which is the reciprocal of the momental ellipsoid in regard to the concentric sphere, radius k. The last-mentioned ellipsoid is given in magnitude, viz., if the body is referred to its principal axes, then putting A=Ma2, B=M¿2, C=Mc2, the equation of the ellipsoid of gyration is

x2 y2 z2 1.
a2 + b2

The axes of any one of the foregoing ellipsoids coincide in direction with the principal axes of the body, and the magnitudes of the axes lead very simply to the values of the principal moments A, B, C.

145. The origin has so far been left arbitrary: in the dynamical applications, this origin is in the case of a solid body rotating about a fixed point, the fixed point; and in the case of a free body, the centre of gravity. But the values of the coefficients (A, B, C, F, G, H), or (A', B', C', F', G', H'), corresponding to any given origin whatever, are very easily expressed in

* i have ventured to make this change instead of writing as usual F=fyzdm, &c.; as in most cases F=G=H=0, the formulæ affected by the alteration are not numerous.