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which is Fresnel's surface; hence the reciprocal polar of Lord Rayleigh's surface is the inverse of (4), and its equation is

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or 7o — {(B2 + C2) x2 + ( C2 + A2) y2 + ( A2 + B2) z2 } p•2

+B2 C2x2 + C2A2y2 + A2 B2z2 = 0.....(6)

where Aka, &c., k being the constant of inversion. The surface (6) is of the sixth degree, and therefore Lord Rayleigh's surface is of the sixth class.

The principal sections of (6) consist of a circle and the inverse of an ellipse with respect to its centre; and since the latter curve is a trinodal quartic and therefore of the sixth class, the principal sections of Lord Rayleigh's surface consist of a circle and a certain sextic curve of the fourth class. The complete section is therefore an improper curve of the eighth degree, which is the degree of the surface.

The circumstance that this surface is of the sixth class shows that three tangent planes can be drawn through the line TL to the half of the surface within the crystal; and, if all three planes were real, triple refraction would take place. This result does not appear to have been previously noticed, and it does not, as at first sight might appear, constitute an insuperable difficulty; for if the differences between the optical constants were sufficiently small, one of the tangent planes would be imaginary for all angles of incidence, in which case there would only be two real refracted rays. It is also possible that for proper values of these differences a certain range of angles of incidence might exist, for which all three planes were real, whilst for certain other ranges or definite angles of incidence one, two, or three tangent planes might be imaginary, and double refraction, single refraction, and extinction would be respectively produced.

6. If this theory is applied to uniaxal crystals, it leads to some very curious results. When the difference between the optical constants is small, the inverse of an ellipse with respect to its centre does not differ much from a circle. The centre is a real biflecnode, the tangents at which are imaginary, the other two nodes being at the circular points are imaginary, also the four double tangents and the four distinct stationary tangents are likewise imaginary. But, when the difference is such that the ratio of the polar to the equatorial axis is sufficiently small, the curve possesses a pair of real double

tangents parallel to the equatorial axis, having real points of contact, and four real stationary tangents. Hence the reciprocal curve, which is the meridian curve of Lord Rayleigh's surface, has a pair of nodes on the polar axis and four cusps. The surface has therefore two real conic nodes with real nodal cones on the polar axis, and two real cuspidal curves which are parallels of latitude symmetrically situated in each hemisphere. The form of the meridian curve

is shown in the figure. From this it follows that under suitable conditions uniaxal crystals would be capable of producing: first, ordinary external conical refraction; secondly, a ray of light issuing from a point on the cuspidal curve would after emergence consist of a single fan-shaped beam of light; thirdly, there would be one ordinary ray and two extraordinary rays for proper angles of incidence.

7. Triple refraction might, however, arise from a totally different cause. Every surface, except a quadric, possesses a determinate number of triple tangent planes; and when the surface is anautotomic the number was determined by Salmon* a great many years ago; but when the surface is autotomic every singular point or curve absorbs a certain number of these planest in much the same manner as a node on a plane curve absorbs a certain number of double and stationary tangents. If therefore the angle of incidence were chosen so that the line TL lay in a real triple tangent plane, triple refraction would take place if all three points of contact were real, and single refraction if one were real and the remaining two were imaginary. But as it may now be considered established that Fresnel's surface is the proper wave surface for biaxal crystals, it can be shown that such substances cannot produce triple refraction. The reciprocal polar of a triple tangent plane is a cubic node of the third species; that is to say, the nodal cone degrades into three planes passing through a point; but Fresnel's surface is a particular case of Kummer's 16-nodal quartic surface, and cannot therefore possess any cubic nodes; also, since it is its own reciprocal, it cannot possess any triple tangent planes.

*Trans. Roy. Irish Acad., vol. xxiii., p. 461; see also Basset, Geometry of Surfaces, chapter ix.

† See Quar. Jour, vol. xlii., p. 21

8. Surfaces possess six different species of singular tangent planes which may be denoted by the letters,,,, ..., w ̧. The planes, and ,, if they existed, would not give rise to any special optical phenomena;,, the triple tangent planes, have already been considered;, is a tangent plane whose point of contact is a tacnode on the section of the surface by the plane. And if Fresnel's surface possessed any real planes of this character, and the angle of incidence were chosen so that TL lay in one of them, the effect would be single refraction, owing to the coincidence of the two rays; and if the source of light were slowly moved, the two rays would begin to separate. But Fresnel's surface cannot possess any distinct planes, for the following reason:-The spinodal and flecnodal curves on a surface touch one another but do not intersect, and the points where they touch are the points. of contact of the planes ; but the curve of contact of every trope forms part of the spinodal curve, and therefore the latter consists of the 16 circles of contact of the 16 tropes, which together make up an improper curve of the 32nd degree. Each trope is a compound plane singularity, which includes among its constituents a certain number of the planes w ̧.

9. The plane touches the surface at a point which is a biflecnode on the section; and by Schubert's formula an anautotomic quartic surface possesses 600 of such planes. I have not examined whether Fresnel's surface possesses any distinct real planes of this character, but if they existed the optical effect would be similar to that produced by the planes

10. A quartic surface cannot possess a plane, unless it has a straight line lying in it, and Fresnel's surface possesses no such lines, but it is easy to see what optical effect would be produced. When a straight line lies in a surface of the th degree, the tangent plane along it is in general a torsal plane which touches the surface at n-1 distinct points. If, therefore, a quartic surface possessed a line lying in it, and the angle of incidence were chosen so that the line in question lay in the same plane as the line TL, the effect produced would be a special kind of triple refraction if all three points of contact were real, and single refraction if one were real and the other two imaginary.

*See Geometry of Surfaces, chapter ix.

Fledborough Hall,

Holyport, Berks.

L

EFFECT ON THE PRODUCT
WHEN ITS FACTORS ARE PERMUTED
IN EVERY POSSIBLE MANNER.

By G. A. MILLER.

ET s,, 8, ..., s, represent n operators of a group, and consider the continued product of these operators S=8,8,8 As $1, $,,..., are not necessarily commutative their continued product is generally dependent upon the arrangement of these factors. Two questions naturally arise. One of these is: whether the n! products, formed by the continued product of these n factors, when they are arranged in every possible way, may all be distinct. The second of these questions relates to the largest number of distinct orders which these n! products may have. That the former of these two questions must be answered in the affirmative is implied in the fact that the n! cyclic substitutions of degree n+1 are the products of then transpositions

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when these are arranged in every possible manner. n operators may be so selected and arranged that their continued product is an arbitrary one of n! distinct operators.

The question, as regards the largest number of different orders which such n! products may have, seems to be much more difficult to answer. It is easy to see that this number cannot exceed (n-1)!, since any cyclic permutation of all the factors cannot effect the order of their product; in fact, such a permutation can always be effected by transforming, but all the transforms of a given operator have the same order. When all the operators of such a product are of order 2, as they were in the example of the preceding paragraph, the order of the product is not affected by inverting the order of the factors. These facts may be stated as follows:

The order of the product of n factors is always an invariant as regards the cyclic substitution group on these factors, and the order of the product of n factors of order 2 is an invariant as regards the dihedral substitution group on these factors. In particular, the six products obtained by multiplying any three operators of order 2 in every possible way must have a common order.

VOL. XLII.

N

Having observed that the n! continued products of n factors cannot have more than (n-1)! distinct orders, we proceed to prove that, for every value of n, it is possible to find a set of n operators which have the property that the n! operators obtained by forming the continued product of the set of n operators, arranged in every possible manner, have (n-1)! different orders; that is, these n! products may be arranged in sets of n distinct operators so that the operators of any two distinct sets must have different orders. That this theorem is true for n=3 results directly from the following example of the product of three substitutions:

abc.ab.bc1, ab.abc.be=abc.

The following six products of four substitutions are intended to prepare the way for the proof of the general theorem as well as to illustrate that the theorem is true for n = 4:

ap hp cp 'dp.ab.ac.ad=apcp'.bp,' dp,

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If we let p=5=p, +1, p=11=p, +1, p, = 331 = p,' + 1, P=218461=p, +1, and replace each of the letters p,, P2 P3 P4 in the given products by a set of distinct letters whose number is equal to p', P P P respectively, the orders of the given six products are 4405898, 218477.331, 218797.11, 218803.5, 347.218461, 5.11.331.218461 respectively. As the first of these orders is not divisible by any one of the primes P1, P27 Pa P.; the second is divisible by none of them except P.; the third, fourth, and fifth are divisible only by P, P1, and P. respectively; while the sixth is divisible by all of them. It is clear that all of these orders are distinct.

In general, we select the primes P, P., P, so that

P1>n, p1>1+k,P12 P1=1+k2P1P21 ..., Px=1+kx-1 P1P ̧· · ·P\-19 and observe that none of these primes is a divisor of a sum of f them, 1 <k<n+1. From this it follows that the (n-1)! lucts obtained by multiplying the (n-1)! cyclic substi

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