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these partial forces increasing in number, but diminishing in intensity,
in the passage from any one group to the following; and being con-
nected with each other, within any such group, by simple numerical
ratios and angular relations.

(d). For example, the two forces ni, 1, 1,2 of the first group

are, rigorously, proportional to the numbers 1 and 3; the three forces

n2, 1, 2, 2, N2,3 of the second group are as the numbers 1, 2, 5; and

the four forces of the third group are proportional to 5, 9, 15, 35:

while the separate intensities of the first forces, in these three first

groups, have the expressions,


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.(e). All these partial forces are conceived to act at the moon; but
their directions may be represented by the respectively parallel unit-
lines Uni, 1, &c., drawn from the earth, and terminating on a great
circle of the celestial sphere (supposed here to have its radius equal to
unity), which passes through the geocentric (or apparent) places, ©
and D, of the sun and moon in the heavens.

(f). Denoting then the geocentric elongation OD of moon from sun
(in the plane of the three bodies) by + 0; and by 1, 2, and D1, D2,
Da, what may be called two fictitious suns, and three fictitious moons,
of which the corresponding elongations from O, in the same great



μδρ = 0, (3); δρόμ=0, (Ts)

which are independent of any hypothesis respecting the vibrations of

the ether.

(b). If op be next regarded as a displacement (or vibration), tan-

gential to the wave, and if dɛ denote the elastic force resulting, there
exists then, on Fresnel's principles, a relation between these two small
vectors; which relation may (with our notations) be expressed by
either of the two following equations,

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the function being of that linear, vector, and self-conjugate kind,
which has been frequently employed in these Elements.

(e). The fundamental connexion, between the functional symbol

, and the optical constants abc of the crystal, is expressed (p. 741,

comp. the formula (W3) in p. xlvi) by the symbolic and cubic equa-


(p+a ̄2) (p + b-2) (p + c-2) = 0 ;

of which an extensive use is made in the present Series.


(d). The normal component, μ-1Sμde, of the elastic force dɛ, is in-
effective in Fresnel's theory, on account of the supposed incompressi-
bility of the ether; and the tangential component, p-dp-μ-1Sμde, is
(in the same theory, and with present notations) to be equated to



2p, for the propagation of a rectilinear vibration (p. 737); we ob-
tain then thus, for such a vibration or tangential displacement, dp, the

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which is a Symbolical Form of the scalar Equation of the Index-Sur-
face, and may be thus transformed,

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(e). The Wave-Surface, as being the reciprocal (a) of the index-
surface (d), is easily found (p. 738) to be represented by this other
Symbolical Equation,


0 = Sp-1(p-2)-1p-1;
1 = Sp (p2 - 4-1)-1p.


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* This apparent exception arises (pp. 739, 740) from the circumstance, that
op and de have their directions generally fixed, in this whole investigation
(although subject to a common reversal by ±), when p and μ are given; whereas
du continues to be used, as in (a), to denote any infinitesimal vector, tangential to
the index-surface at the end of μ.

with others easily deduced, which may all be illustrated by the above-
cited Fig. 89.

(i). Among such deductions, the following equations (p. 740)
may be mentioned,

(Vvpv)2 + Svpv =0, (J6);

(Vwp-1w)2 + Swp ̄1w=0; (K6)

which show that the Locus of each of the two Auxiliary Points, u and
w, wherein the two vectors v and terminate (ƒ), is a Surface of
the Fourth Degree, or briefly, a Quartic Surface; of which two loci the
constructions may be connected (as stated in p. 741) with those of the
two reciprocal ellipsoids,

Spøp = 1, (Le), and Spp 1p=1;

p denoting, for each, an arbitrary semidiameter.


(j). It is, however, a much more interesting use of these two
ellipsoids, of which (by (W5), &c.) the scalar semiaxes are a, b, e for
the first, and a1, b-1, c-1 for the second, to observe that they may be
employed (pp. 738, 739) for the Constructions of the Wave and the
Index-Surface, respectively, by a very simple rule, which (at least for
the first of these two reciprocal surfaces (a)) was assigned by Fres-
nel himself.

(k). In fact, on comparing the symbolical form (A) of the equa-
tion of the Wave, with the form (H2) in p. xxxvii, or with the equa-
tion 412, XLI., in p. 683, we derive at once Fresnel's Construction :
namely, that if the ellipsoid (abc) be cut, by an arbitrary plane
through its centre, and if perpendiculars to that plane be erected at
that central point, which shall have the lengths of the semiaxes of
the section, then the locus of the extremities, of the perpendiculars so
erected, will be the sought Wave-Surface.

(1). A precisely similar construction applies, to the derivation of
the Index-Surface from the ellipsoid (able-1); and thus the two
auxiliary surfaces, (L) and (M6), may be briefly called the Generat-
ing Ellipsoid, and the Reciprocal Ellipsoid.


(m). The cubic (W3) in ø enables us easily to express (p. 741) the
inverse function (p+e)-1, where e is any scalar; and thus, by chang-
ing e top-2, &c., new forms of the equation (A) of the wave are
obtained, whereof one is,


0 = (p ̄1p)2 + (p2 + a2 + b2 + c2) Sp¢ ̈1p — a2b2c2 ;
with an analogous equation in μ (comp. the rule in (g)), to represent
the index-surface: so that each of these two surfaces is of the fourth
degree, as indeed is otherwise known.

(n). If either Spp-p or p2 be treated as constant in (Ne), the
degree of that equation is depressed from the fourth to the second;
and therefore the Wave is cut, by each of the two concentric quadrics,
Spp ph, (0%), p2 + 2 = 0,

in a (real or imaginary) curve of the fourth degree of which two quar-



tic curves, answering to all scalar values of the constants h and r, the
wave is the common locus.

(o). The new ellipsoid (0) is similar to the ellipsoid (M6), and
similarly placed, while the sphere (P) has r for radius; and every
quartic of the second system (n) is a sphero-conic, because it is, by the
equation (As) of the wave, the intersection of that sphere (Pe) with
the concentric and quadric cone,

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for the squares of its scalar semiaxes, and is therefore confocal with

the generating ellipsoid (L).

(p). For any point P of the wave, or at the end of any ray p, the

tangents to the two curves (n) have the directions of w and uw; so
that these two quartics cross each other at right angles, and each is a
common orthogonal in all the curves of the other system.

(2). But the vibration dp is easily proved to be parallel to w;
hence the curves of the first system (n) are Lines of Vibration of the
Wave and the curves of the second system are the Orthogonal Trajec-
toriest to those Lines.

(r). In general, the vibration dp has (on Fresnel's principles) the

direction of the projection of the ray p on the tangent plane to the

wave; and the elastic force dɛ has in like manner the direction of the

projection of the index-vector μ on the tangent plane to the index-

surface so that the ray is thus perpendicular to the elastic force

ARTICLE 423.-Mac Cullagh's Theorem of the Polar Plane,

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