MS 
 
THE COLLECTED AVORKS 
 
 JAMES MAC CULLAGH, LL.D 
 
DUBLIN UNIVERSITY PRESS SERIES 
 
 THE COLLECTED WORKS 
 
 OF 
 
 JAMES MAC CULLAGH, LLD,, 
 
 FELLOW OF TRINITY COLLEGE, 
 
 AND 
 
 PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF DUBLIN. 
 
 EDITED BY 
 
 JOHN H. JELLETT, B. D., 
 
 AND 
 
 SAMUEL HAUGHTON, CLK., M.D. 
 
 DUBLIN: HODGES, FIGGIS, & CO., GRAFTON-STREET. 
 LONDON : LONGMANS, GREEN, & CO., PATERNOSTER-ROW. 
 
 l88o. 
 
. 
 
 DUBLIN : 
 PRINTED AT THE UNIVERSITY PRESS. 
 
PREFACE. 
 
 THE present volume contains a complete collection of 
 the scientific works of the late Professor Mac Cullagh. 
 They have been reprinted for the most part from the 
 Proceedings and Transactions of the Royal Irish Aca- 
 demy, in which they originally appeared. Some few 
 have been taken from the Philosophical Magazine. 
 
 Prof. Mac Cullagh' s most important contributions 
 to Science were made in the departments of Physical 
 Optics and Geometry the first class being very much 
 the larger. The discrepancy is not, however, as great 
 as might at first sight appear. A considerable part 
 of his Optical researches, more especially those of 
 earlier date, really belong to the domain of Geometry ; 
 and, if they were so classed, the inequality between 
 the classes would be much reduced. Such a classifica- 
 tion, however, involving the separation of the purely 
 geometrical propositions from their physical applica- 
 tions, would be exceedingly inconvenient, and these 
 propositions have been allowed to remain in the con- 
 nexion in which the author placed them. 
 
 b 
 
iv Preface. 
 
 In his earlier Optical Memoirs, Prof. Mac Cullagh 
 aimed chiefly at elucidating, by means of geometrical 
 theorems, the physical theory of Fresnel. This is the 
 principal object of the Memoirs I. -IV. in the present 
 volume. In V. occurs the first notice of a problem 
 which subsequently occupied so large a space in Prof. 
 Mac Cullagh' s researches, namely, the investigation 
 of the laws according to which polarized light is re- 
 flected and refracted at the surface of a crystalline 
 medium. This problem is discussed at length in XI. 
 and XIV. In the former of these memoirs he deduces 
 a solution of the problem from certain assumed phy- 
 sical principles. In the second he seeks to establish 
 the theory upon a strictly mechanical basis by means of 
 the general dynamical equation of Lagrange. These, 
 which are the principal memoirs treating of the general 
 question, are supplemented by Memoirs XVI. XIX., 
 in which the same problem is discussed. Two other 
 important questions, namely, metallic reflexion and the 
 double refraction of quartz, which required a peculiar 
 mode of treatment, are considered in Memoirs VI., 
 VII., XV., XVII., XXI. 
 
 Prof. Mac Cullagh' s contributions to pure Geo- 
 metry, excluding, as has been said, all those theorems 
 which have been introduced by the author as auxiliary 
 to his Optical researches, form the second Part of the 
 present volume. The first of these is a Memoir on 
 
Preface. v 
 
 the Rectification of the Conic Sections, which (with 
 his first Optical Memoir) was communicated to the 
 Royal Irish Academy shortly after he obtained his 
 B. A. degree. This was followed, many years after- 
 wards, by an elaborate memoir, which may, indeed, be 
 fairly called a treatise: " On Surfaces of the Second 
 Order." In this memoir a new definition is given for 
 this class of surfaces analogous to the well-known 
 mode of defining curves of the second order by means 
 of a focus and directrix. 
 
 The articles contained in the third and fourth parts 
 of the present volume were not published during the 
 lifetime of the author. They are records of Courses 
 of Lectures on the subjects of Rotation and Attrac- 
 tion, given by Prof. Mac Cullagh. These records 
 were preserved by Professors Haughton and Allman, 
 and communicated by them to the Royal Irish Aca- 
 demy. 
 
 Two short Papers on Egyptian Chronology, which, 
 like most of Prof. Mac Cullagh' s writings, were origi- 
 nally communicated to the Royal Irish Academy, have 
 been printed at the end of this volume. 
 
CONTENTS, 
 
 PAET I. 
 
 PHYSICAL OPTICS. 
 
 PAGE 
 
 I. On the Double Refraction of Light in a Crystallized Medium, 
 
 according to the principles of Fresnel, .... 1 
 
 II. On the Intensity of Light when the Vibrations are Elliptical, . 14 
 
 III. Note on the subject of Conical Refraction, . . . .17 
 
 IV. Geometrical Propositions applied to the Wave Theory of Light, 20 
 Y. A Short Account of some Recent Investigations Concerning the 
 
 Laws of Reflexion and Refraction at the Surface of Crystals, 55 
 VI. Laws of Reflexion from Metals, ...... 58 
 
 VII. On the Laws of the Double Refraction of Quartz, . . .63 
 VIII. On the Laws of Reflexion from Crystallized Surfaces, . . 75 
 IX. On the Probable Nature of the Light Transmitted by the Dia- 
 mond and by Gold Leaf, 82 
 
 X. On the Laws of Crystalline Reflexion, 83 
 
 XI. On the Laws of Crystalline Reflexion and Refraction, . . 37 
 XII. On a new Optical Instrument, intended chiefly for the purpose 
 
 of making Experiments on the Light Reflected from Metals, 138 
 
 XIII. Laws of Crystalline Reflexion. Question of priority, . .140 
 
 XIV. An Essay towards a Dynamical Theory of Reflexion and Refrac- y ^ 
 
 tion, 415 
 
 XV. On the Optical Laws of Rock-Crystals, 185 
 
 XVI. On a Dynamical Theory of Crystalline Reflexion and Refraction : ^^ 
 
 (Supplement), . 
 
viii Contents. 
 
 XVII. Notes on Some Points in the Theory of Light, . . 7 43?- 
 
 XVIII. On the Prohlem of Total Reflexion, ..... 218 
 
 XIX. On the Dispersion of the Optic Axes, and of the Axes of 
 
 Elasticity in Biaxal Crystals, ..... 221 
 
 XX. On the Law of Double Refraction, ..... 227 
 
 XXI. On the Laws of Metallic Reflexion, and on the Mode of 
 
 making Experiments upon Elliptic Polarization, . . 230 
 XXII. On the attempt lately made by M. Laurent to explain, on 
 mechanical principles, the Phenomenon of Circular Pola- 
 rization in Liquids, ..... . . 249 
 
 XXIII. On Total Reflexion, ..... 250 
 
 PAET II. 
 
 GEOMETRY. 
 
 I. Geometrical Theorems on the Rectification of the Conic Sec- 
 tions, 255 
 
 II. On the Surfaces of the Second Order, 260 
 
 III. Note relative to the comparison of the Arcs of Curves, particu- 
 
 larly of Plane and Spherical Conies, 318 
 
 IV. Note on Surfaces of the Second Order, . 321 
 
 PAET III. 
 
 ROTATION. 
 
 I. On the Rotation of a Solid Body round a Fixed Point; being an 
 account of the late Professor Mac Cullagh's Lectures on that 
 subject. Compiled by the Rev. Samuel Haughton, Fellow of 
 Trinity College, Dublin, 329 
 
Contents. ix 
 
 
 
 PAET IV. 
 
 ATTRACTION. 
 
 PAGE 
 
 I. On a Difficulty in the Theory of the Attraction of Spheroids, . 349 
 II. On the Attraction of Ellipsoids, with a new demonstration of 
 Clairaut's Theorem, being an account of the late Professor 
 Mac Cullagh's Lectures on those subjects. Compiled by George 
 ' Johnston Allman, LL.D., of Trinity College, Dublin, . . 352 
 
 SUPPLEMENT. 
 
 EGYPTIAN CHEONOLOGY. 
 
 I. On the Chronology of Egypt, 373 
 
 II. On the Catalogue of Egyptian Kings, which is usually known by 
 
 the name of the Laterculum of Eratosthenes, .... 376 
 
I. ON THE DOUBLE REFRACTION OF LIGHT IN A CRYS- 
 TALLIZED MEDIUM, ACCORDING TO THE PRINCIPLES 
 OF FRESNEL. 
 
 [ Transactions of the Royal Irish Academy, VOL. xvi. Read June 21, 1830.] 
 
 THE mathematical difficulties under which the beautiful and 
 interesting theory of Fresnel has hitherto laboured are well 
 known, and have been regarded as almost insuperable. He tells 
 us, in his Memoir (see the Memoirs of the Royal Academy of 
 Sciences of Paris, torn. vii. p. 136), that the calculations, by 
 which he assured himself of the truth of his construction for 
 finding the surface of the wave, were so tedious and embarrass- 
 ing, that he was obliged to omit them altogether. A direct de- 
 monstration has since been supplied by M. Ampere (Annales de 
 Chimie et de Physique, torn, xxxix. p. 113) ; but his solution is 
 excessively complicated and difficult. 
 
 Judging from the simplicity and elegance of the results that 
 there must be some simple method of arriving at them, I have 
 been led to consider the subject with the attention which it 
 merits, and have succeeded in discovering a method by which 
 the whole may be explained with that simplicity which is cha- 
 racteristic of every theory that is founded in nature. 
 
 In the following Paper I shall give a brief view of this 
 method, sufficient to enable those who are acquainted with the 
 mechanical principles laid down in the original memoir of 
 Fresnel, to trace, at a glance, the connexion between the several 
 
 
2 The Double Refraction of Light in a Crystallized 
 
 parts of his theory. For this purpose it will be convenient to 
 premise the following Geometrical Lemmas : 
 
 1. If a, b, c, be the semiaxes of an ellipsoid, and a, |3, 7, 
 the angles which they make with a perpendicular from the centre 
 on a tangent plane, the square of the perpendicular will be 
 
 equal to 
 
 a 2 cos 2 o + b 2 cos 2 13 + c 2 cos 2 y. 
 
 Let a plane through the point of contact Q, and one of the 
 semiaxes OA, intersect the ellipsoid in 
 the ellipse AON, and the tangent plane 
 in the tangent QL, and draw QM per- 
 pendicular to OA ; then OA is a semi- 
 axis of the ellipse AON, and therefore is 
 a mean proportional between OH and 
 
 OL ; whence #_ = -, denoting OM by Fig. i. 
 
 x 
 
 x. But if p denote the length of the perpendicular from the 
 centre on the tangent plane at Q, the cosine of the angle 
 
 i) 
 which it makes with OA will be equal to =., and therefore 
 
 Similarly, 
 Hence 
 
 px . px 
 
 cos a = r-, ana a cos a = . 
 
 b cos |3 = -j-, and c cos y = . 
 
 cos 2 j3 + r cos 2 y = p 2 + + = p\ 
 
 . Cor. Since #, /, s, are as the cosines of the angles which OQ 
 makes with the semiaxes, it appears from the demonstration that 
 the cosines of the angles which the perpendicular to a tangent 
 plane makes with the semiaxes are, with respect to each other, 
 directly as the cosines of the angles which the semidiameter 
 through the point of contact makes with the semiaxes, and in- 
 versely as the squares of the semiaxes themselves. 
 
 2. If the semiaxes a, b, c, and a', b', c\ of two concentric 
 
Medium, according to the Principles of Fresnel. 3 
 
 ellipsoids coincide in direction, and be reciprocally proportional, 
 so that aa = W = cc = k z ; and if a semidiameter OR of the one 
 be cut perpendicularly in P by a plane which touches the other, 
 then will OR be inversely as OP, so that OP x OR will be 
 always equal to k 2 . 
 
 Let OR be a semidiameter of the ellipsoid whose semiaxes 
 are #', b', c f ; and let a, |3, 7, be the angles which it makes with 
 them ; then if #, y> s, be the co-ordinates of R, ye have 
 
 7* * &" * </ - 
 
 and therefore 
 
 \ ^"Q 
 1 cos 2 a cos 2 /3 cos 2 7 
 
 . I I ' ftr 
 
 '2 ?/2 '2 
 
 Fig. 2. 
 = (a 2 cos 2 a + b z cos 2 13 + c 2 cos 2 7.) 
 
 But by the preceding lemma, since OP is perpendicular to the 
 tangent plane at Q, we have 
 
 OP 2 - 2 cos 2 a + b 2 cos 2 13 + c 2 cos 2 7. 
 Hence 
 
 1 OP 2 
 
 and therefore 
 
 OP x OR = & 2 . 
 
 3. If through the point of contact Q the straight line OQ be 
 drawn to meet in JV the tangent plane at R, it will meet it at 
 right angles. 
 
 For the cosines of the angles made by OP with the semi- 
 axes are directly as the cosines of the angles made by OQ with 
 them, and inversely as the squares of a, 5, c (Lem. 1, Cor.) ; and 
 the cosines of the angles made with the semiaxes, by a perpen- 
 dicular to the tangent plane at R, are directly as the cosines of 
 the angles made with them by OP, and inversely as the squares 
 of a\ &', c', or as the same cosines and the squares of a, b, c, di- 
 rectly ; that is, simply, as the cosines of the angles made by OQ 
 
 B2 
 
 
4 The Double Refraction of Light in a Crystallized 
 
 with the semiaxes. Hence OQ coincides with the perpendicular 
 to the tangent plane at R. 
 
 4. If a perpendicular at to the plane POQ meet the sur- 
 face of the ellipsoid abc in q, then will OQ and Oq be the semi- 
 axes of the section QOq made by a plane passing through 
 them. 
 
 For the tangent plane at Q and the plane QOq are perpen- 
 dicular to POQ, and therefore the intersection of the two former, 
 which is a tangent to the ellipse QOq at Q, is perpendicular to 
 OQ ; whence OQ is one semiaxis, and Oq the other. 
 
 If the perpendicular Oq meet the other ellipsoid in r, then 
 OH and Or will be the semiaxes of the section ROr made by a 
 plane passing through them; for (by lem. 3), the straight line 
 OQN is perpendicular to the tangent plane at R. 
 
 5. In a straight line, at right angles to any diametral section 
 QOq of the ellipsoid abc, let OTand OF be taken equal to OQ 
 and Oq, the semiaxes of the section, and imagine the double sur- 
 face which is the locus of all the points T and V ; then if OS 
 be perpendicular to the plane which touches this surface in T, 
 and OP to that which touches the ellipsoid in Q, the lines OP 
 and OS will be equal and perpendicular to each other, and the 
 four, OP, OQ, OS, OT, will lie in the same plane, which will be 
 at right angles to Oq. 
 
 By the preceding lemma it is evident that Oq is perpendicular 
 to the plane POQ; and since OT is perpen- 
 dicular to the plane QO^, it follows that OP, 
 OQ, OT, are in the same plane at right 
 angles to Oq. In the surface which is the 
 locus of T, and in the plane TOq, let a point 
 T, be taken indefinitely near to T ; then the 
 plane of the section at right angles to OT, 
 will pass through OQ, and will have one of 
 its semiaxes (that to which OT, is equal) 
 indefinitely near to OQ, and therefore differ- 
 ing from OQ by an indefinitely small quan- 
 tity of the second order, adopting, for brevity, the language of 
 
Medium, according to the Principles of FresneL 5 
 
 infinitesimals. To see this, it is only necessary to recollect that, 
 by the property of maxima and minima, the semiaxis differs 
 from any semidiameter indefinitely near to it, such as OQ, by an 
 indefinitely small quantity of the second order. Hence, since 
 OT is equal to OQ, and OT, to the above-mentioned semiaxis, 
 it follows that OT and OT, differ by an indefinitely small quan- 
 tity of the second order, and that therefore the angle OTT t is 
 ultimately a right angle : consequently the tangent to the curve 
 in which the plane TOq intersects the locus of T is perpendi- 
 cular to the plane PQOT. But the tangent plane at T passes 
 through this tangent, and therefore the perpendicular OS to the 
 tangent plane must lie in the plane PQOT. 
 
 Again, let a point T f indefinitely near to T, and in the plane 
 PQOT, be taken in the surface which is the locus of T 7 , and let 
 the plane of the section which is perpendicular to OT' intersect 
 the plane PQOT in the straight line OQ' which meets the ellip- 
 soid in Q'. Then that semiaxis of the section to which T f is 
 equal will be indefinitely near to OQ', and will therefore differ 
 from it by an indefinitely small quantity of the second order. 
 Hence, since OT is equal to OQ, the angle OTT' will be ulti- 
 mately equal to OQQ' ; and therefore, TS and QP being tan- 
 gents, the angles OTS and OQP are equal. But OT= OQ, and 
 the angles P and 8 are right ; therefore 08 = OP, and the angle 
 SOT = POQ ; whence SOP = TOQ = a right angle. 
 
 Similarly, if one perpendicular be let fall from on a plane 
 touching the locus in V 9 and another on the plane touching the 
 ellipsoid in <?, it may be proved that the two perpendiculars are 
 equal and at right angles to each other, and that, with the lines 
 OF and Oq, they lie in a plane at right angles to OQ. 
 
 6. An ellipsoid being cut by any plane through its centre, 
 the difference between the squares of the reciprocals of the semi- 
 axes of the section is proportional to the rectangle under the 
 sines of the angles which the plane of the section makes with 
 the planes of the two circular sections of the ellipsoid. (See 
 Fresnel's Memoir, p. 150.) 
 
6 The Double Refraction of Light in a Crystallized 
 
 Let the plane which cuts the ellipsoid intersect its circular 
 sections in the lines OR, OS', and 
 let the principal section AOC of the 
 ellipsoid cut the circular sections in 
 OR and OS ; then OR, OS, OR', 
 OS', will be all equal to the mean 
 semiaxis OB, and hence the semiaxes 
 OA, OC f of the section AOC' will 
 bisect the acute and obtuse angles 
 made by OR and OS'. Let a plane 
 through OB and OA intersect the 
 principal plane ACO in the line OT. 
 Then, by the nature of the ellipse, we 
 have, in the ellipse AOC', 
 
 _1_ 1 _ / 1 1 
 
 OR 2 OA' 2 ~ \OC' 2 OA 2 
 
 Fig. 4. 
 
 and in the ellipse BOT, 
 1 1 
 
 
 sin 2 BOA, 
 
 r 2 \OR or 
 
 Hence, observing that OB and OR are equal, we have 
 1 1/1 1 
 
 OC' 2 OA' 2 
 But the ellipse OAC gives 
 
 J_ 1 / 1 J^_ 
 
 OR 2 OT 2 ~\OC* OA 2 
 
 = fJ_ J_ 
 
 Therefore 
 1 1 
 
 - x sin ROT sin SOT. 
 
 1 
 
 OA 2 \OC 2 
 because OB and OR are equal. 
 
 sn 
 
Medium, according to the Principles of FresneL 7 
 
 Imagine a sphere described from the centre with a radius 
 equal to OB, and passing through R' and $', and cutting AO 
 in P. The sides BP and PR' of the spherical triangle BPR' 
 subtend the angles BOA and A' OR' at the centre ; and its 
 angle PBR' is equal to the angle ROT-, whence the sines of the 
 sides being proportional to the sines of the opposite spherical 
 angles, it follows that 
 
 sm A OR 
 
 is equal to the sine of the spherical angle BR'P, which is the 
 angle made by the section A'OC' with the plane of the circular 
 section BOR. Similarly, by means of the spherical triangle 
 BPS', it may be shown that 
 
 . 
 -, sin SOT 
 
 sm A OS 
 
 is equal to the sine of the angle made by the plane A'&6' with 
 the plane of the circular section BOS. Therefore, since the 
 angles AOR f and AOS f are equal, it follows that 
 
 multiplied by the product of the sines of the angles which the 
 plane A'OC' makes with the planes of the two circular sections. 
 
 I shall now demonstrate a geometrical construction for find- 
 ing the magnitude and direction of the elastic force arising from 
 a displacement in any direction a construction which, with the 
 help of the preceding lemmas, will lead us immediately to all 
 the conclusions established by FresneL 
 
 Let be the position of a point in the medium when 
 quiescent, and let three rectangular axes passing through it, and 
 fixed in space, be taken for the axes of co-ordinates. For a small 
 displacement in the direction of x, let the elastic forces, excited 
 in the directions of x, y, s, be a, b, c ; for an equal displacement 
 in the direction of y let the forces be #', b', ,c ; and for the same 
 
8 The Double Refraction of Light in a Crystallized. 
 
 in the direction of s let them be a", I", c". Then, if a point 
 receive an equal displacement in a direction 01 making with 
 OX, OY, OZ, the angles a, |3, 7, the forces in the direction of 
 x, y, z (denoting then by X, F, Z, respectively), will be 
 
 X - a cos a + a cos j3 + a" cos 7, 
 T = b cos a + b' cos j3 + 6" cos 7, 
 
 Z = C COS a + c' COS j3 + c" COS 7, 
 
 as follows from considering (see Fresnel's Memoir, p. 82) that 
 the force arising from a displacement in any direction is the 
 resultant of the forces arising from the three displacements in 
 the directions of x, y, z, which are the statical components of 
 that displacement. But since (p. 90) the force in] the direction 
 of one of the axes, arising from a displacement in the direction 
 of another, is equal to the force in the direction of the latter, 
 arising from an equal displacement in the direction of the^_f ormer, 
 it follows that a' = b, a" = c, b" = c' ; and hence 
 
 X = a cos a + b cos ]3 + c cos 7, 
 Y = b cos a + b' cos )3 + c f cos 7, 
 Z = c cos a + c cos |3 + c" cos 7. 
 
 Let ax* + b'y* + c"z* + %c'yz + 2czx + %bxy - 1 be the equation 
 of a surface of the second order. Let 01 intersect it in /, the 
 co-ordinates of / being #', y' ', z' ; then the equation of the tan- 
 gent plane at / will be, by the known formulae 
 
 (ax + by + cz) x + (bx + b'tf + cV) y + (ex + cy + c"z') z = \ . 
 
 Put 0/=r, and let the tangent plane intersect OX, OY, OZ, 
 in the points P, Q, R ; then from this equation we have 
 
 - = ax' + by' + cz' = r (a cos a + b cos /3 + c cos 7) = rX, 
 
 bx + b'y + c'z = r (b cos a + b' cos )3 + c cos 7) = rY 9 
 
 OQ 
 
 _ = <#' + c'y + c"z = r (c cos a + c cos j3 + c" cos 7) = rZ. 
 OH 
 
Medium , according to the Principles of Fresnel. 9 
 
 Now if p be the length of the perpendicular let fall from 
 on the tangent plane, the cosines of the angles which this per- 
 pendicular makes with the axes of co-ordinates will be equal to 
 
 P P P 
 OP OQ 1 OR' 
 
 respectively, that is, to prX, prY, prZ\ and since the sum of the 
 squares of these cosines is equal to unity, we have 
 
 pr v 2 + F 2 + Z z = 1. 
 
 Hence it appears that the perpendicular let fall from on 
 the tangent plane is parallel to the direction of the resultant 
 elastic force, and that the magnitude of the resultant is ex- 
 
 pressed by . 
 J pr 
 
 From this conclusion we may, with the greatest facility, 
 deduce several corollaries. 
 
 1. Since the elastic force is supposed finite, whatever be the 
 direction of the displacement, it is manifest that the above-men- 
 tioned surface of the second order must be an ellipsoid, and that 
 when the displacement is in the direction of any of the three 
 axes of the ellipsoid, the elastic force excited will be in the 
 direction of the same axis, because the tangent plane will be 
 perpendicular to it. Hence the remarkable consequence, that 
 there are always three axes of elasticity at right angles to each 
 other. (Memoir, p. 93.) Also the elasticities arising from equal 
 displacements in the directions of the three axes are inversely 
 as the squares of the axes ; and hence, the positions of the axes 
 and the elasticities in their respective directions being given, the 
 ellipsoid may be constructed. 
 
 2. The ellipsoid being thus constructed, the direction of the 
 elastic force, arising from a displacement in the direction of any 
 of its semidiameters, will be parallel to the normal at the extre- 
 mity of that semidiameter ; and for equal displacements the 
 magnitude of the force will be inversely as the rectangle under 
 the semidiameter and the perpendicular from the centre on the 
 tangent plane at its extremity. If the displacements are pro- 
 
io The Double Refraction of Light in a Crystallized 
 
 portional to the semidiameters, the elastic forces will be both 
 parallel and proportional to the normals; for the normal, ter- 
 minated by any of the principal planes, is inversely as the per- 
 pendicular on the tangent plane. 
 
 3. If the elastic force be resolved into two, one parallel and 
 the other perpendicular to the direction of the displacement, the 
 former will be inversely as the square of the semidiameter in 
 the direction of the displacement. 
 
 4. If the ellipsoid be cut by a plane through its centre, and 
 if the elastic force arising from a displacement in the direction 
 of either axis of the section be resolved parallel and perpendi- 
 cular to that axis, the part perpendicular to the axis will also be 
 perpendicular to the plane of the section. For (by Lem. 4) the 
 plane passing through one axis and the perpendicular to the 
 tangent plane at its extremity, is perpendicular to the other axis, 
 and therefore to the plane of the section. But if the displace- 
 ment be in the direction of any other diameter of the section, 
 the elastic force, resolved perpendicularly to that diameter, will 
 be oblique to the plane of the section. 
 
 To apply these things to the double refraction of light 
 in a crystallized medium, imagine the ellipsoid to be described 
 as above, and let it be cut through its centre by a plane 
 parallel to that of a plane wave of light incident on the crystal ; 
 then if the vibrations of the light be parallel to either of the 
 axes of the section, the plane containing the direction of the 
 vibrations and that of the elastic force arising from them will 
 be perpendicular to the plane of the wave (No. 4, preceding) ; 
 and therefore, according to Fresnel's theory, the direction of the 
 vibrations will remain parallel to itself, while the wave is pro- 
 pagated. But if the light be common light, or if it be polarized, 
 and the plane of polarization be not perpendicular to either of 
 the axes of the section, the wave will be divided into two others 
 having the directions of their vibrations parallel to the semiaxes 
 of the elliptic section, and their planes of polarization perpendi- 
 cular to them : their velocities of propagation measured in a 
 direction perpendicular to their plane will be different, and 
 
Medium, according to the Principles of Fresnel. 1 1 
 
 will be in the sub-duplicate ratio of the elasticities in the direc- 
 tion of their respective vibrations, and therefore (by No. 3) in- 
 versely as the semiaxes of the section to which those vibrations 
 are parallel. 
 
 If a wave be propagated in all directions from an origin 
 within a crystal, its surface will at each instant be touched by 
 the simultaneous position of a plane wave which passed through 
 at the instant when the former began to be propagated 
 (Memoir, p. 127). Hence, to find the surface of the double 
 wave in a crystal, let the above-mentioned ellipsoid be cut by 
 any plane through its centre 0, and imagine two other planes 
 parallel to this section, and at distances from it which are third 
 proportionals to its semiaxes, OR, Or, and any given line k : the 
 double surface which touches these planes in all their positions 
 will be the surface of the wave. 
 
 Now conceive another concentric ellipsoid, having the direc- 
 tions of its semiaxes the same, but their lengths a, b, c, inversely 
 proportional to those of the former, so that the rectangle under 
 any coinciding pair of semiaxes is equal to k* : then if a plane, 
 touching this second ellipsoid in Q, cut OR perpendicularly in 
 P, the line OP will be a third pro- 
 portional to OJ?and k (Lem. 2) ; and 
 if Or intersect the second ellipsoid 
 in q, the semiaxes of the section 
 QOq will be OQ and Oq (Lem. 4). 
 Draw OT perpendicular to QOq 
 and equal to OQ, and conceive the 
 surface which is the locus of the 
 point T to be described, and a tan- Fi s- 5 - 
 
 gent plane, to which the line OS is drawn perpendicular, to be 
 applied at the point T. Then OS will be perpendicular to the 
 plane ROr and equal to OP (Lem. 5) : and hence the point T 
 always lies in the surface of the wave. Similar things may be 
 proved with respect to the other semiaxis Or of the ellipse ROr. 
 Hence we deduce the following construction for the surface of 
 the wave : 
 
12 The Double Refraction of Light in a Crystallized 
 
 " Describe an ellipsoid whose axes are in the directions of the 
 axes of elasticity, the squares of their lengths being directly as 
 the elasticities in their respective directions ; cut the ellipsoid by 
 a plane through its centre, as QOq, and in a perpendicular to 
 that plane take two portions OT and V equal to the semiaxes 
 OQ and Oq of the section. The double surface which is the locus 
 of the points T and Fis the surface of the wave." 
 
 As to the planes of polarization of the rays belonging to the 
 two parts of the wave, Fresnel has shown how to find them by 
 means of a surface which he calls the surface of elasticity. But 
 it is desirable to be able to find them by means of the same 
 ellipsoid which serves to find the surface of the wave. Now TS, 
 parallel to OR, is the direction of the vibrations of the ray OT, 
 and the tangent plane at Q is perpendicular to OR, and there- 
 fore parallel to the plane of polarization of the ray OT. In like 
 manner the plane of polarization of the ray V is parallel to 
 the tangent plane at q. Hence the planes of polarization of two 
 different rays, having a common direction, are parallel to the 
 planes which touch the ellipsoid at the extremities of the semi- 
 axes of the diametral section perpendicular to their common 
 direction. 
 
 When two of the axes are equal, the ellipsoid becomes a 
 spheroid, and the crystal is said to be uniaxal, the double refrac- 
 tion being regulated by the third axis which 
 is perpendicular to their plane. Let A. OB 
 be a section of the spheroid through the 
 third axis OA, which is its axis of revolu- 
 tion ; take OK = OB, and OA' = OA, and 
 let the ellipse A OK and the circle BOB' 
 revolve about OB" as an axis ; they will 
 describe a surface compounded of a sphe- Fig. 6. 
 
 roid and sphere, which will in this case be the surface of the 
 double wave. " For if OH be the direction of a ray, and if a 
 plane perpendicular to OM cut the ellipse AOB in OQ, and the 
 equator of the spheroid in Oq, the lines OQ and Oq, of which 
 the latter is equal to OB, will be the semiaxes of the section 
 
Medium, according to the Principles of Fresnel. 13 
 
 QOq. Taking, therefore, OT and V equal to OQ and OB, the 
 locus of Twill evidently be the circle BOB' \ and the locus of 
 Twill be the ellipse A' OB', since the angle B'OT' is equal to 
 the angle BOQ. 
 
 In the general case, OT and OF, which are equal to OQ and 
 Oq, represent the velocities of the two different sorts of rays 
 having a common direction in the crystal ; and two right lines, 
 which lie in the plane of the greatest and least axes of the ellip- 
 soid, and are perpendicular to its two circular sections, are 
 called the optic axes. It appears, therefore, by the 6th Lemma, 
 that the difference of the squares of the reciprocals of the velo- 
 cities of the two rays, having a common direction in the crystal, 
 is proportional to the product of the sines of the angles which 
 that direction makes with the optic axes. This is the celebrated 
 law of M. Biot, to which he was led by analogy, and which he 
 afterwards found to agree with that previously laid down by 
 Dr. Brewster. 
 
 Such, in their simplest form, are the principal features of the 
 Mechanical Theory of Double Refraction, invented by the late 
 M. Fresnel a theory which would do honour to the sagacity of 
 Newton, and which gives us ample reason to regret that the life 
 of its author was not longer spared, to enrich with further dis- 
 coveries his favourite science. Of him it may be said, as New- 
 ton said of Cotes and apparently with much greater reason 
 that if he had lived longer, we should have known something at 
 last of the laws of nature. 
 
II. ON THE INTENSITY OE LIGHT WHEN THE VIBRA- 
 TIONS ARE ELLIPTICAL. 
 
 [Edinburgh Journal of Science, April, 1831.] 
 
 ACCORDING to the opinions commonly received, the intensity of 
 light, in the undulatory hypothesis, is proportional to the -vis 
 viva, which again is proportional to the square of the greatest 
 velocity. Now the greatest velocity will be the same in an 
 ellipse and a right line which have the same period, if the 
 greater axis of the former be equal to the whole extent of the 
 latter ; so that in elliptic vibrations the intensity would be in- 
 dependent of the minor axis, which is far from being true. I 
 would propose the integral Sv*dt so remarkable for its mecha- 
 nical properties as the measure of the intensity, the integral 
 being extended to the whole time of a vibration. This gives 
 precision to the notion of vis viva, and leads, moreover, to an 
 elegant result ; for if a and b denote the semiaxes of the ellipse, 
 and T the time of vibration, the integral, by an easy calculation, 
 
 2?r 2 
 will be found equal to (a? + # 2 ), showing that for the same 
 
 colour the intensity is proportional to the sum of the squares of 
 the semiaxes, and that for different colours it increases- with the 
 rapidity of the vibrations, as it would be natural to suppose 
 a priori. 
 
 This theorem assigns very simply the reason why two por- 
 tions of light polarized at right angles do not interfere ; but to 
 
On the Intensity of Light, &c. 15 
 
 show this it will be necessary to lay down the following general 
 rule for compounding rectilinear vibrations having the same 
 period, whatever be the difference of their origin and direc- 
 tion : 
 
 Let A A and J?^ (Fig. 7), bisecting each other at 0, represent 
 in extent and direction the vibrations 
 to be compounded, and suppose C 
 and D to be two simultaneous posi- 
 tions of the moving molecule, which 
 it would have in virtue of each vi- 
 bration singly. Complete the paral- 
 lelograms OE and OP, and through 
 P describe an ellipse having for 
 its centre, and touching the sides of Fig. 7. 
 
 the parallelogram OE ; this ellipse will represent the resulting 
 vibration; it will have the same period as the compound one, 
 and equal areas will be described in equal times about its centre. 
 
 To apply this construction to the case proposed, it is neces- 
 sary to show that when OA and OB are constant and at right 
 angles to each other, the intensity of the elliptic vibration, or 
 the sum of the squares of the semiaxes, is independent of the 
 difference of origin, or of the position of the points C and D. 
 
 Now in an ellipse, when a perpendicular from the centre on a 
 tangent makes an angle $ with the major axis, the square of its 
 length is equal to a 2 cos 2 + b 2 sin 2 0. If be the angle which 
 the major axis makes with OA, it will make its complement 
 with OB, and we shall have 
 
 OA* = a 2 cos 2 + b 2 sin 2 0, 
 
 OB* = a* sin 2 $ + b 2 cos 2 0, 
 and therefore 
 
 OA 2 + OB 2 = a 2 + b 2 . 
 
 Hence the intensity is independent of the difference of origin, 
 and therefore the rays do not interfere. 
 
 This remarkable circumstance is commonly accounted for by 
 
16 On the Intensity of Light, &c. 
 
 observing, that since the component velocities are at right angles 
 to each other, the square of the actual velocity must be the sum 
 of their squares ; but this proves the proposition only when the 
 greatest velocities are simultaneous, which happens only in the 
 cases of a complete accordance or of a difference of a semi- 
 undulation in the interfering portions. 
 
 It may be observed also, that the method usually given for 
 finding the intensity of the vibration resulting from two or more 
 rectilinear vibrations proceeds upon no certain grounds. All the 
 vibrations may be reduced to two in rectangular directions, the 
 expression for the square of the velocity in each of their direc- 
 tions consisting of two parts, one of which is constant, and the 
 other depends on the cosine of an arc increasing proportionally 
 to the time : the square of the resultant velocity, therefore, con- 
 sists also of a constant part, and a part depending similarly on 
 the time ; and it is assumed that the intensity of the resulting 
 vibration is proportional to the former, which is true only in 
 the very particular cases just mentioned, if the intensity be 
 measured by the square of the greatest velocity. 
 
 This assumption, however, gives a correct result ; and the 
 reason that it does so is obvious from the principle laid down in 
 the commencement ; for if the expression for the square of the 
 velocity be multiplied by the differential of the time and inte- 
 grated, the variable parts will vanish when the integral is ex- 
 tended to the whole time of a vibration. 
 
III. NOTE ON THE SUBJECT OF CONICAL REFRACTION. 
 
 [From the Philosophical Magazine, VOL. in., 1833.] 
 
 WHEN Professor Hamilton announced his discovery of Conical 
 Eefraction, he did not seem to have been aware that it is an ob- 
 vious and immediate consequence of the theorems published by 
 me, three years ago, in the Transactions of the Royal Irish Aca- 
 demy, vol. xvi., pt. ii., p. 65, &c. The indeterminate cases of my 
 own theorems, which, optically interpreted, mean conical refrac- 
 tion, of course occurred to me at the time ; but they had nothing 
 to do with the subject of that Paper ; and the full examination 
 of them, along with the experiments they might suggest, was 
 reserved for a subsequent essay, which I expressed my intention 
 of writing. Business of a different nature, however, prevented 
 me from following up the inquiry. 
 
 I shall suppose the reader to have studied the passage in 
 pp. 75, 76, of the volume referred to. He will see that when 
 the section of either of the two ellipsoids employed there is a 
 circle, the semiaxes answering to OR, Or, and to OQ, Oq, in 
 the general statement* are infinite in number, giving of course 
 an infinite number of corresponding rays. And this is conical 
 refraction. I shall add a few words on the two cases : 
 
 1. When ROr is a circle, any two of its rectangular radii 
 may be taken for OR and Or. The line OS and the tangent 
 plane perpendicular to it at S are fixed ; but the point of con- 
 
 * The right line Oqr is perpendicular to the plane of the figure, and intersects 
 the two ellipsoids in q and r. 
 
 C 
 
1 8 Note on the Subject of Conical Refraction. 
 
 tact Tis variable, for the plane EOS in which it lies changes 
 
 with OR. Thus we get a curve 
 
 of contact on the tangent plane 
 
 of the wave surface, and a cone 
 
 of rays OT derived from the same 
 
 incident ray. The vibrations of 
 
 any ray OT are in the line TS 
 
 passing through the fixed point S, 
 
 as follows from a general remark 
 
 in the place referred to. 
 
 The three right lines OQ, Or, OT, are at right angles to each 
 other, and a geometer will observe that the first two of them are 
 confined to given planes. For Or is always in the plane of the 
 circle EOr ; and the point Q must be in a given plane, because 
 the line OP, perpendicular to the plane that touches the ellip- 
 soid in Q, is in a given plane JKOr. 
 
 2. When QOq is a circle, the points Tand V coincide in a 
 nodal point n, where the two sheets of the wave surface cross 
 each other. At this point there are an infinite number of tan- 
 gent planes, for OQ and Oq are now indeterminate. The same 
 refracted ray On may therefore be derived from any one of an 
 infinite number of incident rays, and its polarization will differ 
 accordingly ; for the vibrations are in the line nS drawn from 
 the node to the foot of the perpendicular OS on the tangent 
 plane. The ray On, however, is always accompanied by another, 
 but variable, refracted ray. 
 
 The lines, OP, Oq, OS, are at right angles to eacn other, and 
 the first two of them are confined, as before, to given planes. 
 For Oq is in the plane of the circle QOq ; and OP, being perpen- 
 dicular to the tangent plane at Q, must lie in a given plane. 
 These given planes are parallel to two principal tangent planes 
 passing through n, and touching the circle and ellipse that com- 
 pose the wave section in the plane of the nodes : whence it is 
 easy to see that every nodal tangent plane intersects the two 
 principal tangent planes in lines that are constantly at right 
 angles ; for these lines are parallel to OP and Oq. 
 
Note on the Subject of Conical Refraction. 19 
 
 The examination of both cases is completed by the following 
 theorem : 
 
 When three right lines at right angles to each other pass 
 through a fixed point, in such a manner that two of them are 
 confined to given planes, the plane of these two, in all its posi- 
 tions, touches the surface of a cone whose sections parallel to the 
 given planes are parabolas ; while the third right line describes 
 another cone, whose sections parallel to the same planes are 
 circles. 
 
 The application is obvious. "We see that the curve of contact 
 in the first case is a circle. The points S in the second case are 
 also in a circle. 
 
 NOTE ON THE ABOVE, ADDRESSED BY PROFESSOR MAC CuLLAGH TO THE 
 
 EDITORS OF THE PHILOSOPHICAL MAGAZINE. VOL. in. 1833. 
 
 THE introductory part of my Note which appeared in your last Number 
 was written in haste, and I have reason to think it may not be 
 rightly understood. You will therefore allow me to add a few obser- 
 vations that seem to be wanting. 
 
 The principal thing pointed out in the Paper published some time 
 ago in the Transactions of the Royal Irish Academy is a very simple 
 relation between the tangent planes of Fresnel's Wave Surface and the 
 sections of two reciprocal ellipsoids. Now this relation depends upon 
 the axes of the sections, and therefore naturally suggested to me the 
 peculiar cases of circular sections in which every diameter is an axis. 
 Thus a new inquiry was opened to my mind. And accordingly, with- 
 out caring just then to obtain final results, which seemed to be an 
 easy matter at any time, I expressed in conversation my intention of 
 returning to the subject of Fresnel's theory in a supplementary Paper. 
 The design was interrupted, and I was prevented from attending to it 
 again, until I was told that Professor Hamilton bad discovered cusps 
 and circles of contact on the wave surface. This reminded me of the 
 cases of circular section, and the details given in my last note were 
 immediately deduced. 
 
 c2 
 
IV. GEOMETRICAL PROPOSITIONS APPLIED TO THE 
 WAYE THEORY OF LIGHT. 
 
 [Transactions of the Royal Irish Academy, VOL. xvu. Read June 24, 1833.] 
 
 PART I. GEOMETRICAL PROPOSITIONS. 
 
 1. THEOREM I. Conceive a curved surface B to be generated 
 from a given curved surface A in the following manner : having 
 assumed a fixed origin 0, apply a tangent plane at any point Q 
 of the given surface, and perpendicular to this plane draw a 
 right line OPR cutting the plane in P, 
 and terminated in R, so that OP and OR 
 may be reciprocally proportional to each 
 other, their rectangle being equal to a Fig. 9 - 
 
 constant quantity &*, and let all the points R taken according 
 to this law generate the second surface B. Then the relation 
 between these two surfaces, and between the points Q and R, will be 
 reciprocal ; that is to say, if a tangent plane be applied at the 
 point R of the second surface, a perpendicular ON to- this plane 
 will pass through the point Q of the first surface, and ON and 
 OQ will be reciprocally proportional to each other, the rect- 
 angle under them being also equal & 2 . 
 
 2. To prove this theorem, take a point q, in the tangent plane 
 of the surface A, and near the point of contact Q (Fig. 9). 
 Through q let several other planes be drawn touching the sur- 
 face A in points Q', Q", Q'", &c., and draw the perpendiculars 
 OP'R', OP"R", OP"'K", &o., according to the same law as 
 OPR. The points R, R', R",R'", &c., will thus be upon the 
 second surface J5, and they will moreover be all in the same 
 
Geometrical Propositions. 2 i 
 
 plane ; for from any one of them R let Rn be drawn perpendi- 
 cular to the right line Oq and meeting Ogmn; then, on account 
 of the similar right-angled triangles OP'q and Ontf, the rect- 
 angle nOq will be equal to the rectangle R'OP, or to the 
 constant quantity k\ so that the point n, or the foot of the per- 
 pendicular let faU upon Oq 9 will be the same for aU the points 
 R, R', R", R m , &c., and consequently all these points will lie in 
 a plane cutting the right line Oqn perpendicularly in , so as to 
 make the rectangle nOq equal to k\ Now, while the point Q 
 remains fixed, let the point q approach to it without limit in the 
 tangent plane at Q ; and the points R, R", R m , &e., will in like 
 manner approach without limit to the fixed point R ; the plane 
 which contains all those neighbouring points having for its limit- 
 ing position the tangent plane at R. Also the point n will 
 ultimately coincide with N. It follows, therefore, that the tan- 
 gent plane at R cuts the right line OQ perpendicularly in N, so 
 as to make the rectangle NOQ, equal to k*. 
 
 3. Corollary. If any point Q upon the surface A should be 
 a point of intersection, where the surface admits an infinite 
 number of tangent planes, the perpendiculars from upon these 
 planes will form a conical surface having for its vertex. In 
 OQ take, as before, a point N, so that ON x OQ = k\ and let a 
 plane passing through N at right angles to OQ cut the conical 
 surface. The intersection will be a certain curve. From the 
 preceding demonstration it is evident that every point of this 
 curve belongs to the surface B, and that the plane which touches 
 this surface at any point of the curve cuts OQ perpendicularly 
 in J^; or, in other words, that the same plane touches the surface 
 B through the whole extent of the curve. 
 
 4. Two surfaces related to each other like A and B in the 
 preceding theorem may be called reciprocal surfaces, and points 
 like Q and R reciprocal points ; the radii OQ and OR may like- 
 wise be termed reciprocal. A familiar example of such surfaces 
 is afforded, as I have shown on a former occasion,* by two ellip- 
 
 * Transactions of the Royal Irish Academy, Vol. xvi., pt. ii., pp. 67, 68. Supra, p. 3. 
 
2 2 Geometrical Propositions applied 
 
 soids having a common centre at the point 0, and their semi- 
 axes coincident in direction, and connected by the relation 
 aa f = bb f = cc = k* ; where a, b, c, are serniaxes of one ellipsoid in 
 the order of their magnitude, a being the greatest ; and a', b', c' ', 
 those of the other ellipsoid, a' being the least. The mean semi- 
 axes b and b' coincide, and the circular sections of both ellipsoids 
 pass through the common direction of b and b'. 
 
 5. It has also been shown with regard to those ellipsoids, 
 that if Q and R be reciprocal points on the surfaces of abc and 
 a'b'cf respectively, and if a right line Oqr, perpendicular to the 
 plane QOJft, cut the first ellipsoid in q and the second in r, the 
 lines OQ and Oq will be the semiaxes of the section made in the 
 ellipsoid abc by a plane passing through them ; and the lines 
 OR and Or,' in like manner, will be the semiaxes of the section 
 made in the other ellipsoid a' b' c' by the plane in which they lie. 
 
 6. It may further be remarked, that if the radius OQ in one 
 of the reciprocal ellipsoids describe a plane, the corresponding 
 radius OR will describe another plane. For the planes touching 
 the ellipsoid abc in the points Q will all be parallel to a certain 
 right line, and therefore the perpendiculars OR to these tangent 
 planes will all lie in a plane perpendicular to that right line. 
 These two planes, containing the reciprocal radii, may, for 
 brevity, be called reciprocal planes. 
 
 When two reciprocal radii lie in a principal plane, at right 
 angles to a semiaxis of the ellipsoids, it is evident that two planes 
 intersecting in this semiaxis, and passing through the reciprocal 
 radii, are reciprocal planes. 
 
 7. THEOREM II. If three right lines at right angles to each 
 other pass through a fixed point 0, so that two of them are con- 
 fined to given planes, the plane of these two, in all its positions, 
 touches the surface of a cone whose sections parallel to the given 
 planes are parabolas ; while the third right line describes ano- 
 ther cone, whose sections parallel to the given planes are circles. 
 
 Let the plane of the figure (Fig. 10), supposed parallel to one 
 of the given planes, be intersected by the other given plane in 
 the right line MN '; and let OQ be perpendicular to the latter 
 
to the Wave Theory of Light. 23 
 
 plane, while OP is perpendicular to the former and to the plane 
 of the figure, so that PQ being joined will meet MN at right 
 angles in P. Let OA, OB, OC, be the three perpendicular lines, 
 of which OA is parallel to the plane 
 of the figure ; this plane will be inter- 
 sected by the plane of OA and OB in 
 a right line BT parallel to OA, and 
 therefore perpendicular to both OB 
 and OP, and to the plane BOP, and 
 to the line BP. Thus the angle PBT 
 is always a right angle, and therefore 
 BT always touches the parabola whose 
 focus is P and vertex R ; or, which 
 
 comes to the same thing, the plane AOBT always touches the 
 cone which has for its vertex, and the parabola for its section. 
 Again, since OB, OP, OC, are all at right angles to OA, 
 they are in the same plane, and therefore the points B, P, C, 
 are in the same straight line ; and as BOC is a right angle, the 
 rectangle under BP and PC is equal to the square of the per- 
 pendicular OP ; but QO R is also a right angle, and therefore 
 QP x PR = OP 2 ; whence BP x PC = QP x PR, and therefore 
 the points B, R, C, Q, are in the circumference of a circle, so 
 that the angle at C is a right angle, being in the same segment 
 with the angle at R. Thus the point C describes the circle whose 
 diameter is PQ, and OC describes the cone of which this circle 
 is the section. 
 
 8. Of the two right lines OP and OQ perpendicular to the 
 given planes, one is also perpendicular to the plane of the section. 
 That one is OP. Its extremity P is the focus of the parabola. 
 The extremities of both are the extremities of the diameter PQ 
 of the circle. The vertex of the parabola is the point R, where 
 the diameter of the circle intersects that given plane to which 
 the plane of section is not parallel. 
 
 9. THEOREM III. In a straight line at right angles to any 
 diametral section QOq of an ellipsoid abc whose centre is 0, let 
 OT and OF" be taken respectively equal to OQ and Oq, the semi- 
 axes of the section, and imagine the double surface which is the 
 
24 Geometrical Propositions applied 
 
 locus of all the points T and V\ then if 08 be perpendicular to 
 the plane which touches the surface in T, and OP to the plane 
 which touches the ellipsoid in Q, the lines OP and 08 will be 
 equal and perpendicular to each other, and the four straight 
 lines OP, OQ, 08, OT, will lie in the same plane at right 
 angles to Oq 
 
 10. This theorem is taken from a former communication to 
 the Academy.* The surface to which it relates, being the wave 
 surface of FRESNEL, is one of frequent occurrence in optical in- 
 quiries, and it is therefore desirable to give it a distinctive name 
 not derived from any physical hypothesis. I shall call it a 
 biaxal surface, from the circumstance implied in its construction, 
 and adopted as the definition on which the preceding theorem is 
 founded namely, that any pair of its coincident diameters are 
 equal to the two axes of a central section made in the generating 
 ellipsoid abc, by a plane perpendicular to the common direction 
 of the two diameters. The name, perhaps, may appear the more 
 appropriate, as it reminds us of the place which the surface holds 
 in the optical theory of biaxal crystals. 
 
 11. THEOREM IY. The biaxal surfaces generated by two 
 reciprocal ellipsoids are themselves reciprocal. 
 
 For if Q and R (Fig. 11) be reciprocal points on the two 
 ellipsoids, abc and a'b'c', a tangent plane at Q will cut OR pen- 
 pendicularly in P ; a tangent plane at R 
 will cut OQ perpendicularly in JV; and 
 the rectangles ROP and NOQ will be 
 equal to each other and to F (Art. 4). 
 Also if the straight line Oqr, at right angles 
 to the plane of the figure, cut the first ellip- 
 soid in q and the second in r, then (5) the 
 elliptic section QOq will have OQ and Oq 
 for its semiaxes, and the lines OR and Or Fig. 11. 
 
 will be the semiaxes of the other section ROr. Draw, therefore, 
 in the plane of the figure, the right lines OIL and OSM pen- 
 pendicular to the right lines OQN and OPP, making OT, OL, 
 
 * Transactions of foe Royal Irish Academy , Vol. xvi., pt, ii., pp. 67, 68. Supra, p. 4. 
 
to the Wave Theory of Light. 25 
 
 08, OM, equal to OQ, ON, OP, OR, respectively ; the angles, 
 at 8 and L being of course right angles. Then it is evident 
 that the point T is on the biaxal surface generated by the ellip- 
 soid abc, because OT is perpendicular to the plane of the ellipse 
 QOq and equal to the semiaxis OQ', and by Theorem III. it 
 appears that 08 is perpendicular to the tangent plane at T. In 
 like manner, the point M is on the biaxal surface generated by 
 the other ellipsoid a'b'c, and OL is perpendicular to the tangent 
 plane at M. Moreover, the rectangles MO 8 and LOT, being 
 equal to the rectangles EOF and NOQ, are each equal to & 2 . 
 Hence the proposition is manifest. 
 
 12. As the ellipsoid whose semiaxes are a, b, c, may be called 
 the ellipsoid abc, so the biaxal surface generated by this ellipsoid 
 may be called the biaxal abc ; and that which is generated by the 
 ellipsoid dlJc' may be called the biaxal a'b'c. 
 
 13. PROPOSITION Y. To find what properties of biaxal sur- 
 faces are indicated by the cases wherein one of the two sections 
 QOq, ROr, in the preceding theorem, is a circle. 
 
 Case 1. "When QOq is a circular section of the ellipsoid abc, 
 the points T and V (9), in the description of the biaxal surface 
 abc, coincide in a single point n. At this point there are an in- 
 finite number of tangent planes; because the semiaxes of the 
 circular section QOq being indeterminate, any two perpendicular 
 radii of the circle may take the place of OQ, Oq, in the general 
 construction. The point n is therefore a point of intersection (3), 
 where the two biaxal sheets cross each other, and it may be called 
 a nodal point, or simply a node. As OQ always lies in the plane 
 of the circle QOq, the line OR, which is reciprocal to OQ, must 
 lie (6) in a given plane reciprocal to the plane of the circle. 
 And as Oq lies in the plane of the circle, we have three right 
 lines OR, Oq, 08, which are at right angles to each other, and 
 of which the first two are confined to given planes. Therefore 
 by Theorem II. the third line 08 describes a cone whose sections 
 parallel to the given planes are circles. Now, T8 or in the 
 present case n8 is parallel to the fixed plane which contains 
 OR, and therefore the point 8 describes a circle ; or, in other 
 
2 6 Geometrical Propositions applied 
 
 words, the feet of the perpendiculars 08, let fall from on the 
 nodal tangent planes, occupy the circumference of a circle pass- 
 ing (8) through the nodal point. 
 
 14. Parallel to the plane of the circle and to its reciprocal 
 plane, conceive two planes passing through the node, and call 
 them the principal tangent planes at n. The plane of the circle 
 and its reciprocal plane are intersected in the right lines Oq, OR, 
 by the plane qOR, which is parallel to a tangent plane at n. 
 Consequently this tangent plane at n intersects the two principal 
 tangent planes in lines that are parallel to Oq, OR ; and as Oq, 
 OR are perpendicular to each other, it follows that every nodal 
 tangent plane intersects the two principal tangent planes in lines 
 that are at right angles. 
 
 Hence again, the nodal tangent planes touch (7) the surface 
 of a cone whose sections, parallel to the principal tangent planes, 
 are parabolas. As this cone touches the biaxal surface all round 
 the point n, it may be called the nodal tangent cone. 
 
 15. Case 2. "When ROr is a circular section of the ellipsoid 
 a'b'c', any two perpendicular radii of the circle may be taken for 
 OR, Or : and because OR = V, and OR x OP = k* = IV, we have 
 OP or OS equal to b, the mean semiaxis of the ellipsoid abc. 
 Hence OS is given both in position and length ; for it is perpen- 
 dicular to the fixed plane ROr, and it is equal to b. Now, a 
 plane cutting OS perpendicularly at S is a tangent plane to the 
 biaxal abc ; and we have just seen that this tangent plane re- 
 mains the same, whatever pair of rectangular radii are taken for 
 OR, Or. But the point of contact T is variable, for the plane 
 ROS in which it lies changes with OR. Therefore as OR re- 
 volves, the point T describes a curve of contact on the tangent 
 plane of the biaxal abc. 
 
 The lines OR, Or, are in the fixed plane ROr; and as OQ 
 is reciprocal to OR, it lies in a fixed plane reciprocal to the plane 
 Ror (6). Therefore the first two of the three perpendicular right 
 lines Or, OQ, OT, are confined to fixed planes. Hence the third 
 line T describes a cone, whose sections parallel to these planes 
 are circles. But the tangent plane is parallel to the fixed plane 
 
to the Wave Theory of Light. 27 
 
 ROr, and its intersection with T describes the curve of contact. 
 Therefore the curve of contact is a circle passing (8) through the 
 point 8. 
 
 16. "We have examined the two cases of circular section with 
 reference only to the biaxal abc. If we examine the same cases 
 with regard to the second biaxal ab'c', we shall find that their 
 indications are reversed; the supposition which gives a node 
 upon one biaxal, giving a circle of contact on the other : and 
 that the node and the circle, thus corresponding, are so related, 
 that a line drawn from to the node passes through the circum- 
 ference of the circle, cutting the plane of the circle perpendicu- 
 larly ; whilst every line drawn from through the circumference 
 of the circle is perpendicular to some nodal tangent plane. 
 
 These things are evident on looking at the figure. For when 
 ROr is a circle, it is plain that the point M is a node of the 
 biaxal ab V, since OM is perpendicular to the plane of the circle 
 ROr and equal to its radius OR. But we have already seen (15) 
 that when ROr is a circle, the other biaxal abc has a circle of 
 contact, whose plane is perpendicular to OM at the point 8 of its 
 circumference. The line OTL is perpendicular, in general (11), 
 to a tangent plane at Jf, and therefore perpendicular, in the 
 present case, to a nodal tangent plane ; whilst the point T, 
 through which it passes, is on the circle of contact. It is also 
 evident that OT x OL = k\ 
 
 We have here an example of the general remark in the corol- 
 lary of Theorem I. 
 
 17. The section made in the biaxal surface abc, by any of 
 the principal planes of its generating ellipsoid, consists of an 
 ellipse and a circle. 
 
 For, let the plane QOq pass through one of the semiaxes, a, 
 and let it revolve round this semiaxis, while the right line OTV 
 (9), perpendicular to the plane QOq, revolves about in the 
 plane of the semiaxes , c. Then the semiaxis a of the ellopsoid 
 will always be one of the semiaxes of the ellipse QOq] and if 
 T be equal to this semiaxis, the point T will describe a circle 
 with the radius a about the centre 0. The other semiaxis of the 
 
2 8 Geometrical Propositions applied 
 
 ellipse QOq is that semidiameter of the principal ellipse be which 
 lies in the' intersection of the plane be with the plane QOq ; and 
 as V is equal and perpendicular to this semidiameter, the point 
 V describes an ellipse equal to be, but turned round through a 
 right angle, so that the greater axis of the ellipse described by V 
 coincides in direction with the less axis of the ellipse be. As the 
 radius a of the circle is greater (4) than both the semiaxes b, c, 
 of the ellipse, the circle will lie wholly without the ellipse. 
 
 In like manner, the section made in the biaxal surface by 
 the plane ab consists of a circle with the radius c, and an ellipse 
 with the semiaxes a, b ; and as the radius of the circle is less than 
 both the semiaxes of the ellipse, the circle lies wholly within the 
 ellipse. 
 
 18. But when the section lies in the plane of the greatest 
 and least semiaxes # , c, the circle and ellipse, of which it is com- 
 posed, intersect each other. For the radius b of the circle is less 
 than one semiaxis of the ellipse ac and greater than the other. 
 Leaving the ellipse ac in the position which it has as a section of 
 the ellipsoid abc, if we describe the circle b with the centre and 
 radius b, the ellipse and the circle will cut each other in four 
 points at the extremities of two diameters ; and planes, passing 
 through these diameters and through the semiaxis b of the ellip- 
 soid, will evidently be the planes of the two circular sections of 
 the ellipsoid. Now, turning the ellipse ac round through a right 
 angle (17), the circle and the ellipse in its new position will con- 
 stitute the section of the biaxal sur- 
 face, and will cut each other (Fig. 12) 
 in four points n at the extremities of 
 two diameters nOn, nOn, which are 
 perpendicular to the two former dia- 
 meters, and therefore perpendicular 
 to the planes of the two circular 
 sections. Consequently, the biaxal 
 surface has four nodes at the four Fi S- 12 - 
 
 points n. These nodes, it is manifest, are alike in all their pro- 
 perties ; and they are the only points common to the two biaxal 
 
to the Wave Theory of Light. 29 
 
 sheets, since the points Tand V (9), in the description of the 
 biaxal surface, cannot coincide unless the section Q,0q, perpen- 
 dicular to OTV, be a circle. 
 
 19. The plane of the greatest and least semiaxes, a, c 9 of the 
 generating ellipsoid, may be called the plane of the nodes ; and 
 the two diameters nOn, nOn, passing through the nodes, may 
 be called the nodal diameters. 
 
 At one of the nodes n (Fig. 12) draw tangents nf, nk, to the 
 ellipse and the circle that compose the biaxal section ; and 
 through draw Op perpendicular to On, cutting the circle in p. 
 Then as On is perpendicular to the plane of a circular section of 
 the ellipsoid abc, this circular section will have Op for its radius, 
 and its circumference will cross that of the ellipse ac (belonging 
 to the ellipsoid) in the point p. A line touching the ellipse ac 
 at p will be parallel to every plane that touches the ellipsoid in a 
 point of the circular section, and will therefore (6) be perpendi- 
 cular to the plane which is reciprocal to the plane of the circular 
 section. But the tangent at p is perpendicular to the tangent nf, 
 since the two tangents would coincide if the ellipse ac were turned 
 round (18) through a right angle, the pointy then falling upon 
 n. Hence the circular section and its reciprocal plane are 
 parallel to the tangents nk, nf\ and therefore two planes perpen- 
 dicular to the plane of the figure, and passing through these tan- 
 gents, are the planes that we have called (14) the principal 
 tangent planes at n. 
 
 20. Produce Op to meet nf in v, and conceive a parabola 
 having its focus at 0, its vertex at v (8), and its plane perpendi- 
 cular to the plane of the figure. A cone, with its vertex at n 
 and this parabola for its section, is (14) the nodal tangent cone. 
 
 Draw Of perpendicular to nf at /, and meeting nk in k. The 
 perpendiculars let fall from upon the nodal tangent planes 
 form a cone, of which the circles described in planes perpendi- 
 cular to the figure upon the diameters nf, nk, are sections (8) . 
 On the other biaxal surface a'b'c' there is (16) a circle of contact 
 whose plane is perpendicular to On. This circle of contact is 
 (16) another section of the cone last mentioned. 
 
30 Geometrical Propositions applied 
 
 21. To the circle b and to the principal section ac of the 
 ellipsoid abc conceive a common tangent d'i' to be drawn, in a 
 quadrant adjacent to that which contains the node n, and let it 
 touch the circle in d' and the ellipse ac in i'. A radius Od', 
 drawn through the point d' to meet the ellipsoid a'b'c' in the 
 point d", will be reciprocal to the radius Oi', because it is perpen- 
 dicular to a tangent at i', and it will be equal in length to b', be- 
 cause Od" x Qd' = k* = bb', and Otf = b ; whence Od" = b. There- 
 fore Od" is in a circular section of the ellipsoid a'b'c . Two planes 
 perpendicular to the plane of the figure, and passing through 
 the reciprocal radii Od", Oi', are (6) reciprocal planes, and we 
 have seen that the first of them makes a circular section in the 
 ellipsoid a'b'c'. They are therefore (15) the fixed planes in the 
 second case of Prop. V. 
 
 22. Now draw di a common tangent to the circle b and 
 ellipse ac composing the biaxal section, and let it touch the circle 
 in d and the ellipse in i. The lines Od, Oi, are of course perpen- 
 dicular to the lines Od', Oi', and therefore perpendicular to the 
 fixed planes just mentioned. Hence the line Od and the point 
 d are the same as the fixed line 08 and the point 8 in the second 
 case of Prop. V. The plane of the circle of contact is therefore 
 perpendicular to Od at the point d (15) ; and the points d and i, 
 where its plane intersects the right lines Od, Oi, perpendicular 
 to the fixed planes, are (8) the extremities of a diameter. 
 
 These things agree with the obvious remark, that the points 
 of contact d and i must be points of the circle of contact ; and 
 that di must be a diameter, because the plane of the circle is per- 
 pendicular to the plane of the figure, and this latter plane divides 
 the biaxal surface symmetrically. 
 
 As the circle and ellipse may have a common tangent oppo- 
 site to each node, there are four circles of contact in planes per- 
 pendicular to the plane of the nodes.* 
 
 23. The biaxal surface belongs to a class that may be called 
 
 * The curves of contact on biaxal surfaces, and the conical intersections or nodes, 
 were lately discovered by Professor Hamilton, who deduced from these properties 
 a theory of conical refraction, which has been confirmed by the experiments of 
 
to the Wave Theory of Light. 31 
 
 apsidal surfaces, from the manner in which they are conceived to 
 be generated. 
 
 Let G be a given surface, and a fixed origin or pole. If 
 a plane passing through cut the surface G, the curve of inter- 
 section will in general have several apsides A, A.', A." ' , &c., where 
 the lines OA, OA, OA", &c., are perpendicular to the curve. 
 Through the point conceive a right line perpendicular to the 
 plane of the curve, and on this perpendicular take from the 
 distances Oa, Oa', Oa", &o., respectively equal to the apsidal dis- 
 tances OA, OA', OA", &c. Imagine a similar construction to be 
 made in every possible position of the intersecting plane passing 
 through 0, and the points a, a', a", &o., will describe the dif- 
 ferent sheets of an apsidal surface. 
 
 The apsidal surface has a centre at the point 0, because the 
 lengths Oa, Oa', Oa", &o., may be measured on the perpendicular 
 at either side of the intersecting plane. 
 
 Referring* to the demonstration of Theorem III., it will be 
 seen to depend only on the supposition that the point Q is an 
 apsis of the section made by the plane QOq ; or, which is the 
 same thing, that OQ is a position wherein the radius vector from 
 
 to the curve of section is a maximum or a minimum. Hence 
 we have the following general theorem : 
 
 24. PROP. YI. THEOREM. If tangent planes be applied at 
 corresponding points A, a, on the surface G and the apsidal sur- 
 face which it generates, these tangent planes will be perpendi- 
 cular to each other and to the plane of the points 0, A, a. 
 
 This is equivalent to saying that perpendiculars from on 
 the tangent planes are equal to each other, and lie in the plane 
 of the lines OA, Oa. 
 
 25. If Q and R be reciprocal points on two reciprocal surfaces, 
 of which Q is the fixed origin or pole, the tangent plane at Q 
 
 Professor Lloyd. See Transactions of the Royal Irish Academy, Vol. xvii., part i., 
 pp. 132, 145 ; and the present Paper, Art. 55-58. 
 
 The indeterminate cases of circular section at least the case of the nodes had 
 occurred to me long ago ; but having neglected to examine the matter attentively, 
 
 1 did not perceive the properties involved in it (13). 
 
 * Transactions of the Royal Irish Academy, Vol. xvi., part ii., p. 68. 
 
Geometrical Propositions applied 
 
 will be (1) perpendicular to OR and to the plane QOIl. Let a 
 plane also perpendicular to the plane QOR pass through OQ, 
 cutting the surface to which the point Q belongs in a certain 
 curve, and the tangent plane at Q in a tangent to this curve. 
 The tangent is evidently perpendicular to 0Q, and therefore the 
 point Q is an apsis of the curve. 
 
 In like manner, the point R is an apsis of the section made 
 in the other surface by a plane passing through OR and perpen- 
 dicular to the plane QOR. 
 
 26. From these observations, and from Prop. VI., it appears 
 that if the points Q, R, in the figure of Theorem IV., be reciprocal 
 points on any two reciprocal surfaces, and if the same construc- 
 tion be supposed to remain, the points T and M will be points 
 on the apsidal surfaces generated by these reciprocal surfaces, and 
 the tangent planes at T and M will be perpendicular to the lines 
 OH and OT respectively. Also the rectangles LOT and MOS 
 will be equal to A*. Hence we have another general theorem : 
 
 PROP. VII. THEOREM. The apsidal surfaces generated by 
 two reciprocal surfaces are themselves reciprocal. 
 
 27. A very simple example of apsidal surfaces, with nodes 
 and circles of contact, may be had by supposing the generatrix 
 G to be a sphere, and the pole to be within the sphere, between 
 the surface and the centre (7. 
 
 It is evident that the apsidal surface in this case will be one 
 of revolution round the right line OC 
 as an axis. Therefore taking for the 
 plane of the figure (Fig. 13) a plane 
 passing through OC and cutting the 
 sphere in a great circle of which the 
 radius is OS, let a plane at right 
 angles to the figure revolve about 0, 
 cutting the circle CS in the points 
 A, A'. The section of the sphere 
 made by the revolving plane will have 
 only two apsides A, A\ with respect Fig. 13. 
 
 to the point 0, except when the plane is perpendicular to OC. 
 
to the Wave Theory of Light. 33 
 
 Hence, if we draw the right line Oaa' perpendicular to A OA', 
 taking 0#, Oa', always equal to OA, OA f , the points a, a' ', will 
 describe a section of the apsidal surface. This section will evi- 
 dently consist of two circles C'S', C"S", equal to the circle C8 9 
 and having their centres (7, (7", on the opposite sides of in a 
 right line C' OC" perpendicular to 0(7; the distances 0(7, 00', 
 00" being equal. The circles <7'S', 0"S", intersect in two 
 points n 9 n, on the line 0(7, and have two common tangents di, 
 d'i' 9 which are bisected at right angles by OC in the points c, c. 
 
 28. Now let the circles C'S', (7"$", with their common tan- 
 gents, or only one of the circles with the half tangents, revolve 
 about the axis 0(7, and we shall have the apsidal surface with 
 nodes at n 9 n' 9 and with circles of contact described by the 
 radii cd, c'd'. 
 
 The section of the sphere, by a plane passing through at 
 right angles to On 9 is a circle of which is the centre. If there- 
 fore we suppose that the point n answers to a in Prop. VI., the 
 apsis A corresponding to n will be indeterminate, and the posi- 
 tion of the tangent plane at n will also be indeterminate, which 
 ought to be the case at a node. 
 
 The surface reciprocal to the sphere, the pole being at 0, is 
 evidently a surface of revolution about the axis 0(7 (it is easily 
 shown to be a spheroid having a focus at 0) ; and the section of 
 this reciprocal surface, by a plane perpendicular to the axis at 0, 
 is a circle of which is the centre. This circumstance indicates 
 (15) that on the apsidal surface there is a curve of contact, whose 
 plane is parallel to the plane of circular section ; which agrees 
 with what we have already seen. 
 
 29. When the point is without the sphere, the axis 0(7 
 will pass between the circles (7'$', C"S" 9 without intersecting 
 either of them. The apsidal surface, described by the revolution 
 of one of these circles about 0(7, will be a circular ring. The 
 nodes have disappeared ; but the circles of contact still exist, as 
 is evident. 
 
34 Geometrical Propositions applied 
 
 PART II. ON THE WAVE THEORY OF LIGHT. 
 
 30. Some of the foregoing propositions lead to a simple trans- 
 formation of the theory of light. 
 
 In this theory, the surface of waves, or the wave surface, is a 
 geometrical surface used to determine the directions and veloci- 
 ties of refracted or reflected rays, being the surface of a sphere 
 in a singly refracting medium ; a double surface, or a surface of 
 two sheets, in a doubly refracting medium ; a surface of three 
 sheets on the supposition of triple refraction ; and having always 
 a centre round which it is symmetrical. The radii of the wave 
 surface, drawn from its centre in different directions, repre- 
 sent the velocities of rays to which they are parallel. 
 
 31. We shall consider particularly the case of a doubly re- 
 fracting crystal, with two plane faces parallel to each other, and 
 surrounded by a medium of the common kind wherein the con- 
 stant velocity is V: supposing, for the sake of clearness, that 
 the crystal refracts more powerfully than the surrounding 
 medium, so that the velocities in the crystal are less than the 
 velocity V. 
 
 A ray 8 / 0, falling on the first surface of the crystal at the point 
 0, is partly reflected according to the common law of reflection, 
 and partly refracted. The two refracted rays pass on to the 
 second surface, where each of them is divided by internal re- 
 flection into a pair, the two reflected pairs being parallel to each 
 other ; while the two emergent rays one from each refracted 
 ray are parallel to each other and to the incident ray S'O. 
 The directions of the rays within the crystals are usually found 
 by the following construction : 
 
 32. Describe a wave surface of the crystal, having its centre 
 at the point of incidence. By the nature of the wave surface, 
 a right line OTU, drawn from the point 0, will in general cut 
 this surface in two points T, 7, on the same side of ; and a 
 ray passing through the crystal in a direction parallel to OTU 
 will have one of the two velocities represented by the radii OT, 
 
to the Wave Theory of Light. 
 
 35 
 
 Fig. 14. 
 
 U, taking a line of a certain length k to represent the uniform 
 velocity V in the external medium. "With the centre and a 
 radius OS equal to this line k describe a sphere. As the veloci- 
 ties in the crystal are supposed to be 
 less than V, the wave surface will lie 
 wholly within this sphere. Let the 
 plane of the figure (Fig. 14) be the 
 plane of incidence, perpendicular to f 
 the parallel faces of the crystal, and 
 intersecting the first face in the right 
 line FA. Through the point S, 
 where the incident ray S'O, produced 
 through the crystal, cuts the surface 
 of the sphere, draw SI at right angles 
 to OS and meeting FA in the point /. A right line perpendi- 
 cular to the plane of the figure, and passing through this point /, 
 we shall call the right line I. 
 
 33. Through the right line I draw two planes touching the 
 two sheets of the wave surface, on the side remote from the inci- 
 dent light, in the points T, T', which will lie within the sphere 
 (32) ; then the incident plane wave, perpendicular to OS, will 
 be refracted into two plane waves parallel to these two tangent 
 planes ; and the lines OT, T', will be the directions of the re- 
 fracted rays along which the refracted waves are propagated. 
 The lengths OT, OT ', represent the velocities with which the 
 light moves along the rays ; and of course the normal velocities, 
 which are the velocities of the refracted waves, are represented 
 by the perpendiculars OG, OH, let fall from on the two tan- 
 gent planes at T, T'. These two perpendiculars OG, OH, evi- 
 dently lie in the plane of the figure ; but the points T, T', in 
 general, do not lie in this plane. 
 
 34. Again, through the right line / draw two other planes 
 touching the wave surface, at the side of the incident light, in 
 the points t, t' . The rays OT, OT', arriving at the second sur- 
 face of the crystal, will each be divided by internal reflection 
 into two rays parallel to Ot, Of ; and these four reflected rays, 
 
 D2 
 
36 Geometrical Propositions applied 
 
 arriving at the first surface, will each be divided, by a new re- 
 flection, into two rays parallel to T, OT' ; and so on, for any 
 number of reflections. Any of the rays emerging at the lirst 
 surface, after internal reflections, is parallel to the ray Os pro- 
 duced by ordinary reflection at the point of incidence ; and any 
 ray emerging at the second surface is parallel to the incident 
 ray S'OS. 
 
 35. This construction may be changed into another that will 
 be found more convenient both in theory and practice. 
 
 Through 8 draw Sit perpendicular to 01, and meeting OG, 
 OH, produced, in the points P, M. Then as the angles at G 
 and R are right angles, the points J, R, G, P, are in the circum- 
 ference of a circle, and therefore OP x OG = 01 x OR = OS 2 = k* ; 
 and similarly, M x OH = k*. If then we take for the fixed 
 origin, or pole, and k* for the constant rectangle (Theorem I.), 
 and describe the surface which is reciprocal to the wave surface, 
 it is evident that the points P and M will be points of the sur- 
 face so described, and that OT, OT', will coincide in direction 
 with perpendiculars let fall from on planes touching the sur- 
 face at P and M, and will be inversely proportional to these 
 perpendiculars. It follows in the very same manner, that if per- 
 pendiculars Og, Oh, let fall from on the tangent planes at t, t', 
 be produced to meet SR in the points p, m, these points will also 
 be on the surface reciprocal to the wave surface. 
 
 In the present case, it is manifest that this reciprocal surface 
 lies wholly without the sphere 08. 
 
 36. The surface reciprocal to the wave surface, the pole 
 being at 0, we shall call the surface of refraction. 
 
 It is hardly necessary to observe that the surface of refrac- 
 tion has a centre at the point 0, round which it is symmetrical ; 
 that it is a sphere in a singly refracting medium, a double sur- 
 face in a doubly refracting medium, and a surface of three sheets 
 if we suppose a case of triple refraction. 
 
 37. In the case that we are considering, let the figure 
 (Fig. 15) represent a section made in the double surface of re- 
 fraction and its attendant sphere by the plane of incidence. 
 
to the Wave Theory of Light. 
 
 37 
 
 Through the point S, where the incident ray $'#, prolonged, 
 cuts the circular section of the sphere, draw SH perpendicular to 
 the face of the crystal, or to FA ; and let SR, produced, cut the 
 circle again in the point s. Then Os is the direction of the ray 
 given by ordinary reflection at the first surface of the crystal. 
 
 Fig. 15. 
 
 Produce the right line Sits both ways, to cut the surface of re- 
 fraction in the points P, M, behind the crystal, and in the points 
 p, m, before it ; and conceive planes to touch the surface of re- 
 fraction at the points P, Jf, p, m. Suppose also that perpendi- 
 culars OP 7 , OJT, Op', Om f , are let fall from upon these tangent 
 planes, and that they intersect the planes in the points P', M' , 
 p', m', respectively. 
 
 Then from the preceding observations (33, 34, 35), it is 
 manifest that OP', OJf' , are the directions of the rays into 
 which S f is divided by refraction ; that each of these refracted 
 rays, on arriving at the second surface of the crystal, is divided 
 by internal reflection into two rays parallel to Op', Om' ; and that 
 each of the four reflected rays, on arriving at the first surface, is 
 again divided by reflection into two rays parallel to OP', OJ/', 
 
3 8 Geometrical Propositions applied 
 
 and so on. In general, every ray going into the crystal from 
 the first surface, whether after refraction or after any even 
 number of internal reflections, is parallel either to OP f or to OM '; 
 and every ray returning from the second surface of the crystal 
 after any odd number of internal reflections, is parallel either to 
 Op' or to Om '. Thus the direction of every ray in the interior 
 of the crystal is the same as the direction of some one of the four 
 lines OP', OM ', Op', Om' ; and the velocity of the ray is in- 
 versely as the length of this line ; so that the velocity of the ray 
 OM', for example, or of any ray parallel to OM ', is to the velo- 
 city V as OS is to OM'. The little plane waves that, keeping 
 always parallel to themselves, move along these rays, are respec- 
 tively perpendicular to the lines OP, OM, Op, Om\ and the 
 lengths of these lines are inversely as the velocities of the waves 
 estimated in directions perpendicular to their planes ; so that the 
 velocity of the wave which moves along the ray OM', or along 
 any parallel ray, is to the velocity V as OS is to OM. 
 
 38. The ray OP', and all the rays parallel to it, are perpen- 
 dicular to the plane which touches at P the surface of refraction ; 
 and the waves which move along these rays are perpendicular to 
 the right line OP. Any ray of this set may be called a ray P, 
 and any of the waves a wave P. In like manner, the rays M,p, m, 
 are rays that are perpendicular to the tangent planes at the 
 points M, p, m, respectively ; and the waves M, p, m, are the 
 waves that belong to these rays, and that have their planes re- 
 spectively perpendicular to the right lines OM, Op, Om.- The 
 rays P, M, all come from the first surface of the crystal ; the 
 rays p, m, from the second. 
 
 As the ordinates RP, Rp, are greater than the ordinates 
 RM, Rm, so the rays P, p, are more refracted or more reflected 
 than the rays M, m. The former rays may therefore be said to 
 be plus refracted, or plus reflected, and the latter to be minus re- 
 fracted, or minus reflected. Or for the convenience of naming 
 the rays P, p, may be called plus rays ; and the rays M, m, 
 minus rays. The waves P, p, in like manner, may be termed 
 phis waves, and the waves M, m, minus waves. 
 
to the Wave Theory of Light. 39 
 
 For a medium of the common kind, or a singly refracting 
 medium, we may use the letters 8 and s. Thus the incident ray 
 S'OS, or any ray emerging parallel to OS from the second sur- 
 face of the crystal, may be marked by the letter S ; while the 
 ray Os produced by common reflection, or any ray emerging pa- 
 rallel to Os from the first surface, may be denoted by the letter s. 
 39. The course of a ray through the crystal may now be 
 easily expressed. A ray SMps, for example, is a ray (8) inci- 
 dent on the crystal, undergoing minus refraction (M) at the 
 first surface, plus reflection (p) at the second, and emerging (s) 
 from the first surface in a direction parallel to Os. Of this ray 
 the part within the crystal is Mp. A. ray SPS is a ray plus re- 
 fracted, and then emerging in a direction parallel to that of 
 incidence. A ray SPpMS is a ray plus refracted at the first 
 surface, then plus reflected at the second surface, then minus 
 reflected at the first surface, and finally emerging from the second 
 surface in a direction parallel to that of incidence. Its path 
 within the crystal is PpM. 
 
 These examples indicate the general method of expressing 
 the path of a ray. 
 
 Suppose light to be moving in the same direction and with 
 the same velocity along two proximate parallel rays, so that it 
 is at the point A in one ray when it is at the point B in the 
 other ; and through the points A and B conceive two planes 
 perpendicular to the common direction of the rays. These planes 
 are either coincident, or maintain a constant distance. In the 
 first case, the rays are said to be in complete accordance. In the 
 second case, the constant distance between the planes is called 
 the interval between the portions of light composing the rays, or 
 the interval between the waves that move along the rays. 
 
 We proceed to find the lengths of these intervals in the case 
 of rays emerging parallel to each other, at either side of the 
 crystal that we have been hitherto considering. 
 
 41. Let the tangent planes at P, M, p, m, intersect the plane 
 of the figure (Fig. 15) in the right lines PP,, MM t , pp t , mm,, 
 which of course are tangents to the section of the surface of re- 
 
4O . Geometrical Propositions applied 
 
 fraction represented in the figure ; let a perpendicnlar at to the 
 face of the crystal cut these tangents in the points P /? Jf,, p 4 , m 4 ; 
 and let the lines OP", OM", Op", Om', respectively parallel to 
 PP,, MM pp mm ^, cut the line SJRs in the points P", M", p" , m". 
 The length of the path which a ray P describes within the 
 crystal is equal to the thickness 9 of the crystal divided by 
 the cosine of the angle P / OP / which the path of the ray makes 
 with a perpendicular to the faces of the crystal ; and the velocity 
 
 O Sf 
 of P is equal to V x - -, (37) : dividing therefore the length 
 
 of the path by the velocity, we find that the time in which a ray 
 P crosses the crystal is equal to 
 
 ex op / 
 
 Fx OS* cosP'OP/ 
 But as OP' is perpendicular to the tangent plane at P, we have 
 
 ^- _ = OP = PP" 
 cosP'OP, ^ 
 
 Therefore the time is equal to ^= ^-^ . Similarly, the times in 
 
 y x c/o 
 
 which rays M, p, m, pass from one surface of the crystal to the 
 other are equal to 
 
 6 x MM" x pp" x mm" 
 
 42. Now suppose the path of a ray P to be projected per- 
 pendicularly on a right line having any proposed direction in 
 space. Through conceive a right line OL parallel to the pro- 
 posed direction, and meeting in L the tangent plane at P. The 
 length of the projection is equal to the length of the path multi- 
 plied by the cosine of the angle POL which the ray P makes 
 
 with OL ; that is, the projection is equal to - . But 
 
 cos Jr (JJr 
 
 because OP is perpendicular to the tangent plane at P, we have 
 
 OP* 
 cos POL = ~, and cos FOP, 
 
to the Wave Theory of Light. 41 
 
 therefore 
 
 cos POL PP", 
 
 COB POP, ^YjTTj. OFT *X ' 
 
 Hence the projection is equal to 9 TPF\ - 
 
 /: ^pRNl! / 
 
 If the path of a ray P be projected on the incident- ray OS, 
 then producing OS to meet PP X in /, we see, by what has just 
 been proved, that the length of the projection is equal to 
 
 PP" SP" 
 
 01 OS ' 
 
 by similar triangles. In like manner, the projections of the 
 paths of rays M, p, m, on the direction of the incident ray OS, 
 are equal to 
 
 OS ' OS' OS ' 
 
 respectively. 
 
 43. Let each rectilinear path be measured in the direction 
 in which the light moves along it ; and according as the 
 direction so measured makes an acute or an obtuse angle with 
 the direction OS, measured from to 8, let the projection of 
 the path on OS be reckoned positive or negative. Then if 
 SPmMpMS be any ray entering the crystal at 0, and emerging 
 from its second surface at E, and if a perpendicular El be let 
 fall from E upon OS, meeting OS in /, the distance 01, from 
 to the foot of this perpendicular, will evidently be equal to 
 the algebraic sum of the projections of the paths P, m, M, p, M, 
 contained within the crystal, taking each projection with its 
 proper sign. It is obvious that the projections of the P and M 
 rays are always positive. And as the lines Op', Omf the 
 directions of the rays p, m lie in planes which are respectively 
 perpendicular to pp fl mm fl or to Op", Om", it is easy to see that 
 these directions make acute or obtuse angles with 08, according 
 as the points p", m", lie below the point S or above it ; that is, 
 the projections are positive or negative according as the points 
 p", m", lie without the circle OS towards P, M, or within the 
 
42 Geometrical Propositions applied 
 
 circle. Therefore the distance 07, in the case of the figure, i 
 equal to 
 
 ^ (SP" - 8m" + 811" - Sp" + SM") . 
 
 44. If the paths of rays P, M 9 p, m, be projected on th 
 direction Os of the ordinarily reflected ray, the lengths of thei 
 projections will be 
 
 sP" sM" sp" sm" 
 
 "OS' "OS' "OS' ~Q8' 
 
 respectively. The projections upon Os of the raysj*?, m, will b< 
 always positive ; and the projections of the rays P, J/, will b< 
 positive or negative according as the points P", Jf", lie abov< 
 the point s or below it ; that is, according as the points P", M" 
 lie without the circle OS towards p and m, or within the circle 
 So that if SPmMps be a ray entering the crystal at anc 
 emerging from the first surface at e, and if a perpendiculai 
 ei be let fall from e upon Os, the distance Oi, from the point C 
 to the foot of this perpendicular, or the algebraic sum of the 
 projections of the paths P, m, M, p, contained within the 
 crystal, will be equal to 
 
 in the case of the figure. 
 
 45. Let us imagine that the light in the incident ray S'O, 
 instead of being interrupted at by the crystal, had continued 
 to move with the same velocity V in the same right line OS, 
 leaving the point at the moment when the refracted light 
 enters the crystal at 0. Comparing the light in this imaginary 
 ray with that in a ray emerging parallel to it from the second 
 surface of the crystal, after an even number of internal reflec- 
 tions, we shall find that the emergent is behind the imaginary 
 ray, and that the interval between them (40) or the retar- 
 dation of the former may be derived very easily from the 
 letters that designate that ray. Let SPmMpMS be any such 
 
to the Wave Theory of Light. 43 
 
 *ay. The sum of the distances of the point 8 from each of the 
 )oints marked by the letters (PmMpM) that denote (39) the 
 )art of the ray contained within the crystal, is proportional to 
 he interval of retardation, that interval being equal to 
 
 -- (SP + 8m + SM + 8p + SM). 
 G/o 
 
 For if from the point E> where the last internal ray M 
 smerges from the second surface of the crystal, a perpendicular 
 El be let fall upon 08, meeting 08 in /, the time of describing 
 11 with the velocity V would (43) be 
 
 (SP" - Sm" + SM" - 8p" + SM"}. 
 
 3ut (41) the actual time of describing the broken path PmMpM 
 
 s 
 
 
 
 OS 
 
 (PP' + mm" + MM" + pp" + MM") ; 
 
 .nd, on inspecting the figure, this time is seen to be greater 
 han the time of describing O/, by 
 
 (SP + Sm + SM + Sp + 8M), 
 
 Fx 08 
 
 r by the time in which the line 
 
 -^ (8P + Sm + SM + Sp + SM) 
 
 rould be described with the velocity V. Consequently, at the 
 loment when the light in the ray SPmMpMS emerges at the 
 oint E from the second surface of the crystal, the light in the 
 aiaginary uninterrupted ray OS will have passed the point / 
 y an interval equal to the line just mentioned ; and as the two 
 ays afterwards have the same velocity and parallel directions, 
 tiis interval is the retardation of the emergent ray. 
 
 46. The rays emerging from the first surface after any odd 
 umber of internal reflections are to be compared with the 
 
44 Geometrical Propositions applied 
 
 ordinarily reflected ray Os to which they are parallel, the lig 
 in Os, which moves with the velocity F", being supposed to lea 
 at the moment when the refracted light enters the crystal 
 0. The mode of proceeding in this case is exactly similar 
 that in the last, and the interval is determined in the same WE 
 using s in place of 8 ; the retardation of the ray SPmMps, j 
 example, of which the part PmMp is contained within t 
 crystal, being equal to 
 
 r\ 
 
 (sP + sm + sM + sp)* 
 
 OS 
 
 47. It is remarkable that the preceding demonstration in r 
 wise depends upon the supposition that the planes perpendicul 
 to the rays P, M, p, m, are tangent planes to the surface 
 refraction at the points P, M, p, m. If we had supposed a: 
 planes different from the plane of the figure to pass throui 
 the points P, M, p, m, and the rays to coincide in the directi 
 with perpendiculars let fall from upon these planes, and 
 have velocities inversely proportional to the lengths of t 
 perpendiculars, the intervals of retardation would have remain 
 unchanged. Hence the retardations are the same as if the lir 
 OP, OM, Op, Om, were the directions of the rays in passi: 
 through the crystal, as will appear by conceiving the plar 
 that we have spoken of to be perpendicular to these lines. 
 
 If the incident ray S'O were refracted in the ordinary w 
 
 OP 
 
 with an index equal to 7^, it would take the direction OP : 
 
 (/o 
 
 it were refracted, in like manner, with the index -^, it won 
 
 Oo 
 
 take the direction OM ; and if the two rays, thus ordinari 
 refracted, were to emerge from the second surface of the crysl 
 in directions parallel to OS, it is evident, from what has be 
 said, that they would be in complete accordance, respective] 
 with the rays SPS and SMS. 
 
 * The change of phase, which may take place at a surface of the crystal, is i 
 here considered as affecting the intervals. 
 
to the Wave Theory of Light. 45 
 
 If the surface of refraction should happen to have a node 
 V, which is a point of intersection where it admits an infinite 
 lumber of tangent planes (3), let the direction of the incident 
 ay S'OS be chosen, so that the right line US perpendicular 
 o the face of the crystal, being produced below 8, may pass 
 hrough JV, and we shall have a cone of refracted rays formed 
 >y the perpendiculars let fall from upon the tangent planes 
 ^ N ; all of which rays, on emerging parallel to OS from the 
 econd surface of the crystal, will be in complete accordance 
 rith one another. For we have just seen that if the ray S'OS 
 vere supposed to emerge after being refracted in the ordinary 
 
 ON 
 
 ray with an index equal to -y^, it would be in complete 
 
 Accordance with any ray of the cone. 
 
 48. The interval between any two rays emerging at the 
 ame side of the crystal is the difference of their retardations. 
 n taking the difference, the letters that are common to the 
 tames of the two rays may be left out. Thus the ray SPmMS 
 s behind the ray 8P8 by the interval 
 
 r\ 
 
 Che line - Pp is the interval between the rays SMS and 
 
 , or between the reflected ray Os and the ray SPps, 
 Jid so on. 
 
 49. The retardations of the two refracted rays SPS and 
 
 Q 
 3MS, emerging without internal reflection, are ^ SP and 
 
 ^ SM respectively. The difference of these is 6 -^ . Con- 
 yo Oo 
 
 equently, when the two refracted rays have emerged from the 
 econd surface in directions parallel to the incident ray, the 
 ight in the plus emergent ray is behind the light in the minus 
 
 smergent ray by an interval equal to -^ . Or, in other 
 
 C/o 
 
 vords, the incident plane wave, perpendicular to OS, produces 
 
46 Geometrical Propositions applied 
 
 two emergent waves parallel to each other and to the incid( 
 wave, moving along the emergent rays with equal velocities 
 
 and preserving the distance - between their planes, 1 
 
 minus wave being foremost. If 08, the radius of the sphere, 
 taken for unity, PM will be a number generally a very sm 
 fraction and the interval will be the thickness of the crys 
 multiplied by this number. 
 
 50. Suppose the right line PMR, remaining always perpe 
 dicular to the face of the crystal, to describe a cylindrical si 
 face, with the condition that the part PM, intercepted betwe 
 the two sheets of the surface of refraction, shall remain of a cc 
 stant length ; the point R will then describe, on the surface 
 the crystal, a curve whose radii OR are the sines (to the radi 
 08} of the angles of incidence of a cone of rays ; and every r 
 S'O of this cone, when refracted by the crystal, will afford t 1 
 emergent rays, or two waves, having the same given inters 
 between them. Lines drawn from the eye parallel to the sic 
 of this cone are the emergent rays belonging to a ring, wh 
 rings are made to appear, in any of the usual ways, on trai 
 mitting polarized light through the plate of crystal. In nomii 
 conformity to this, we see that the line PM describes a ring 
 constant breadth between the two sheets of the surface of refre 
 tion. The ring described by supposing pm to remain consta 
 corresponds to the interval between two rays p and m reflect 
 at the same point of the second surface of the crystal, and th 
 emerging at the first. The other intercepts Pp, Mm, Pm, Bi 
 are proportional (48) to intervals like those in Newton's rings 
 to the intervals, namely, between the reflected ray Os and t 
 rays 8Pps, SMms, SPms, SMps, emerging at the first surfa 
 after one reflection within the crystal ; or to the intervals betwei 
 rays that are twice reflected in the crystal and the rays tran 
 mitted without reflection. 
 
 51. The general investigation of the figure of a geometric 
 ring does not distinguish between the different intercepts, ai 
 will therefore include all the rings PM, pm, Pp, Mm, Pm, Mj 
 
to the Wave Theory of Light. 47 
 
 so that it will be sufficient to contemplate any one of them, as 
 PM, of which the breadth PM is equal to a given line /. 
 
 The points P and M describe, in general, similar and equal 
 curves of double curvature, which may be called ring-edges, as 
 being the edges of the ring ; and if we imagine the surface of 
 refraction, carrying these curves along with it, to be shifted 
 either way, in a direction parallel to PJf, through a distance equal 
 to 7, it is clear that the new position of one of the ring-edges 
 will exactly coincide with the first position of the other, and that 
 therefore the curve of the latter ring-edge will be given by the 
 intersection of the two equal surfaces in these two positions. 
 Let U = where V is a function of x, y, z, and given quanti- 
 ties be the equation of the surface of refraction in its original 
 position ; and, the axes of co-ordinates being fixed, suppose that 
 by the shifting of the surface the co-ordinates of a point assumed 
 on it are diminished by the given lines/, g, h, which are the 
 projections of the given line / on the axes of x, y, z, respectively. 
 Then the equation of the surface in its new position will be had by 
 substituting x +/, y + g, z + h, for x, y, z, in the equation U= 0, 
 which will thus become U + V= 0, where V is the increment of 
 U produced by the substitution. These two equations com- 
 bined are equivalent to the equations U = 0, V = 0, which are 
 therefore the equations of one of the ring- edges. If the surface 
 had been shifted the opposite way, in a direction- parallel to PM, 
 the intersection would have been the other ring-edge, whose 
 equations are therefore deducible from those already found, by 
 changing the signs of/, g, h. 
 
 52. If the equation of the surface of refraction be trans- 
 formed, so that the plane of xy may coincide with the face of the 
 crystal, and the axis of z be perpendicular to it, the origin of 
 co-ordinates being at the centre 0, no change will be produced 
 in x or in y by the motion of the surface, because PM, the direc- 
 tion of the motion, is now parallel to the axis of z ; but z will be 
 diminished or increased by J; and, accordingly, if U' = Q be the 
 equation of the surface in its first position, when the centre 
 is at 0, and if U' become U' + V when z becomes z + 7, the 
 
48 Geometrical Propositions applied 
 
 equation of the surface in its second position, when the centre 
 has moved through a distance equal to / along the axis of s, will 
 be U f + V = ; and these two equations combined will give 
 U' = 0, V' = 0, for the equations of one of the ring-edges. The 
 equations of the other ring-edge are deduced from these by 
 changing the sign of I. 
 
 The projection of each of the ring-edges on the plane xy is 
 the curve traced by the point R on the surface of the crystal 
 (50). This curve may be called a ring-trace. Its equation is 
 obtained by eliminating z between the equations of a ring-edge ; 
 and as the result must be the same whether / be taken posi- 
 tive or negative, the equation of the ring-trace, when found by 
 this general method, will contain only even powers of /. The 
 radii drawn from to the points R of the ring-trace are (50) 
 the sines (to the radius 08) of the angles of incidence or emer- 
 gence of the rays that form an optical ring, the rays that come 
 from this ring to the eye being parallel to the sides of the cone 
 described by the right line S'OS, while the point R describes the 
 ring-trace. 
 
 53. It is evident that tangents to the ring-edges, at the 
 points P and M, are parallel to each other, and therefore parallel 
 to the intersection of two planes touching the surface of refrac- 
 tion at P and M, because these tangent planes pass through the 
 tangents. But the directions OP', OM f , are perpendicular to 
 the tangent planes, and therefore the plane P'OJLT, containing the 
 two rays, is perpendicular to the intersection of the tangent planes, 
 and of course perpendicular to the parallel tangents. Hence 
 the plane P^OM' intersects the face of the crystal in a right line 
 perpendicular to the projection of the parallel tangents on the 
 face of the crystal. As this projection is a tangent to the curve 
 described by jR, it follows that the normal to the ring-trace at 
 the point R is parallel to the line joining the points in widen 
 the two refracted rays cut the second surface of the crystal. 
 
 In like manner, taking any two consecutive rays (P and m), 
 having a common extremity on one surface of the crystal, the 
 line joining the points where these rays cut the other surface is 
 
to the Wave Theory of Light. 49 
 
 parallel to the normal at the point R of the ring-trace which is 
 described when the intercept (Pni) between the letters that mark 
 the rays is supposed to remain constant. 
 
 54. In all that precedes we have made no supposition about 
 the surface of refraction except that it is a surface of two sheets ; 
 and if we supposed it to have three sheets, the conclusions would 
 be easily extended to this hypothesis. 
 
 In the theory of FRESNEL, the wave surface is* a biaxal 
 whose generating ellipsoid has its centre at the point 0, and its 
 semiaxes parallel to the three principal directions of the crystal, 
 the length of each semiaxis being equal to OS divided by one 
 of the principal indices of refraction. The surface of refraction 
 is reciprocal to the wave surface, and is (11) therefore another 
 biaxal generated by an ellipsoid reciprocal to the former, having 
 its centre at the same point 0, and the directions of its semiaxes 
 the same as before, the rectangle under each coincident pair of 
 semiaxes being equal to k z or OS*. Hence the semiaxes of the 
 ellipsoid which generates the biaxal surface of refraction are 
 equal in length to OS multiplied by each of the three principal 
 indices. This biaxal surface is of course to be substituted for 
 the surface of refraction in the preceding observations. 
 
 55. When the line RS, produced below S, passes through a 
 node N of the biaxal surface of refraction, the points P, Jf, 
 coincide in the point N, and the interval PM vanishes. At the 
 point N there are an infinite number of tangent planes, and the 
 perpendiculars from on these tangent planes give a cone of 
 refracted rays whose sections we have already shown how to 
 determine (20). All the rays in this cone, on arriving at the 
 second surface of the crystal, emerge parallel to the incident ray 
 OS ; and if the rays in the emergent cylinder be cut by a plane 
 perpendicular to their common direction, they will all arrive at 
 this plane at the same instant, because the interval PM vanishes. 
 See Art. 47. 
 
 56. Suppose fig. 12 to be a section of the wave surface. The 
 
 * Transactions of the Royal Irish Academy, VOL. xvi., p. 76 (supra, p. 11). 
 
 E 
 
50 Geometrical Propositions applied 
 
 right line Od will pass through N\ and the circle of contact, 
 described on the diameter di in a plane perpendicular to the 
 right line OdN, will be a section of the refracted cone. Now it 
 will be recollected* that, in general, the vibrations of a ray OT, 
 which goes to any point T of the wave surface, are parallel to 
 the line which joins the point Twith the foot of the perpendi- 
 cular let fall from on the tangent plane at T. In the present 
 case, the perpendicular is the same for all the rays of the re- 
 fracted cone, and its extremity coincides with the point d : so 
 that the line dT, drawn from d to any point T of the circle of 
 contact, is parallel to the vibrations of the ray T which passes 
 through T. Conceive, therefore, a plane perpendicular to ON 
 at the nodal point N. This plane will cut the refracted cone in 
 a circle whose circumference will pass through N\ and a line 
 NT', drawn from the node to any other point T f of the circum- 
 ference, will be the direction of the vibrations in a ray OT f which 
 crosses the circle at this point. The plane of polarization is 
 perpendicular to the direction of the vibrations. 
 
 57. The transverse section of the emergent cylinder is always 
 a very small ellipse, affording a hollow pencil of parallel rays in 
 complete accordance (55). If the crystal be thin, this ellipse 
 will be of evanescent magnitude. Hence the line OS will be 
 the direction of a line drawn from the eye to the centre of the 
 rings commonly observed (50) with polarized light ; or it will 
 be what is called the apparent direction of one of the optic axes. 
 The diameter passing through N will be the direction of the 
 optic axis within the crystal. There are therefore two optic 
 axes, parallel to the two nodal diameters (19) of the surface of 
 refraction. 
 
 As ON is equal to the mean semiaxis of the generating 
 ellipsoid, or to the mean index of refraction, when OS is unity, 
 it follows that the apparent direction of an optic axis is the 
 direction of an incident ray, which, if refracted in the ordinary 
 way, with an index equal to the mean index of refraction, would 
 pass along a nodal diameter of the surface of refraction. 
 
 * Transactions of the Koyal Irish Academy, VOL. xvi., p. 76 (supra, p. 12). 
 
to the Wave Theory of Light. 51 
 
 58. We have seen (15) that there is a circle of contact on 
 the biaxal surface of refraction. If an incident ray S'OS be 
 taken, cutting the sphere in .8, so that the line US produced 
 may pass through the circumference of this circle, it is manifest 
 that the direction of the refracted ray will be the same through 
 whatever point II of the circumference the line US may pass, 
 because that direction is perpendicular to the tangent plane at 
 II, which is in fact the plane of the circle itself. If, therefore, 
 the line US move parallel to itself along the circumference of 
 the circle, cutting the sphere in a series of points 8 9 every inci- 
 dent ray S'OS which passes through a point S so determined 
 will be refracted into two rays, of which one will have a fixed 
 direction in the crystal, being perpendicular to the plane of the 
 circle of contact, and therefore coinciding (16) with nOn, one of 
 the nodal diameters of the wave surface. But though the direc- 
 tion On of the refracted ray is fixed, its polarization changes 
 with the incident ray from which it is derived ; for if n be the 
 point in which the line ES, corresponding to any position of 
 the incident ray, crosses the circle of contact, the vibrations of 
 the refracted ray On will be contained in the plane of the lines 
 On, OH, and will be perpendicular to On. Conceive a circle 
 described on the diameter nf in a plane perpendicular to the 
 figure (Fig. 12). This circle, and the circle of contact on the 
 surface of refraction, are (20) sections of the same cone. Let FT 
 therefore be the point at which On, in any position of the inci- 
 dent ray, crosses the circumference of the circle nf\ and the line 
 ttn, drawn to the node of the wave surface, will be the corre- 
 sponding direction of the vibrations in the ray On. 
 
 59. With regard to the general law of polarization in the 
 theory of FRESNEL, it may be observed, that if the ellipsoid abc 
 which generates the biaxal surface of refraction be cut by a 
 plane perpendicular to OP, the vibrations of the ray P will be 
 parallel to the greater axis of the section, and therefore the plane 
 of polarization will pass through OP and the less axis ; whence 
 it is easy to show that the plane of polarization of a ray P bisects 
 one of the angles made by two planes intersecting in OP and 
 
 E2 
 
5 2 . Geometrical Propositions applied 
 
 passing through the nodal diameters of the surface of refraction ; 
 the bisected angle being that which contains the least semiaxis 
 c of the generating ellipsoid. The plane of polarization of the 
 ray p is found in like manner. But for the rays Jf, m, the 
 angle to be bisected is that which contains within it the greatest 
 semiaxis a. 
 
 If OP" be perpendicular to a tangent plane at P, the vibra- 
 tions of the ray P will be perpendicular to OP, and will lie in 
 the plane POP'. A similar remark applies to the rays M,p, m. 
 
 60. When two semiaxes #, b, of the ellipsoid abc become 
 equal, it changes into a spheroid aac described by the revolution 
 of the ellipse ac about the semiaxis c ; and the biaxal aac, gene- 
 rated by this spheroid, is* composed of a sphere whose radius is a, 
 and a concentric spheroid ace described by the revolution of the 
 ellipse ac about the semiaxis a ; so that, the diameter of the 
 sphere being equal to the axis of revolution of the spheroid, the 
 two surfaces touch at the extremities of the axis. This combina- 
 tion of a sphere and a spheroid is the surface of refraction for 
 uniaxal crystals. In these crystals, therefore, the refracted ray 
 whose direction is determined by the intersection of the right line 
 US with the surface of the sphere follows the ordinary law of a 
 constant ratio of the sines, and is called the ordinary ray ; whilst 
 the other, whose variable refraction is regulated by the intersec- 
 tion of ftS with the spheroid, is called the extraordinary ray. And 
 hence uniaxal crystals are usually divided into the two classes 
 of positive and negative, according to the character of the extra- 
 ordinary ray ; being called positive when it is the plus ray, and 
 negative when it is the minus ray. The first case evidently 
 happens when the spheroid is oblate, and therefore lies without 
 the sphere described on its axis ; the second, when the spheroid 
 is prolate, and therefore lies within the sphere. The second case 
 (which is that of Iceland spar) may be supposed to be repre- 
 sented in the figure (Fig. 15), where the elliptic section of the 
 spheroid, made by a plane of incidence oblique to the axis, lies 
 
 * Transactions of the Royal Irish Academy, YOL. xvi., p. 77 (supra, p. 12). 
 
to the Wave Theory of Light. 53 
 
 within the circular section of the sphere, and the minus raj is 
 of course the extraordinary one. 
 
 61. Let PJf, preserving a constant length J, move parallel 
 to itself between the surfaces of the uniaxal sphere and spheroid, 
 so as to form a ring (50). Then supposing the spheroid, with 
 the ring-edge described on it by the point _3f, to remain fixed, 
 imagine the sphere, carrying the ring edge P along with it, to 
 move parallel to PJf, from P towards Jf, through a distance 
 equal to J, and the two ring-edges will exactly coincide. 
 
 Hence the uniaxal ring-edge is the intersection of a sphere 
 and a spheroid, the diameter of the sphere being equal to the 
 axis of revolution of the spheroid, and the line joining their 
 centres being perpendicular to the faces of the crystal and equal 
 to the breadth / of the ring. And the projection of this inter- 
 section, on a plane perpendicular to the line joining the centres 
 of the sphere and the spheroid, is the uniaxal ring-trace. 
 
 62. The biaxal ring-edge is (51) the intersection of two equal 
 biaxal surfaces similarly posited, the line joining their centres 
 being perpendicular to the faces of the crystal and equal to the 
 breadth of the ring. And the projection of this intersection, on 
 a plane perpendicular to the line joining the centres of the sur- 
 faces, is the biaxal ring-trace* 
 
 * In applying the general theory (51, 52) to biaxal rings, it is necessary to 
 know the equation of a biaxal surface, which may be found in the following 
 manner : Let r, r', r", be three rectangular radii of the generating ellipsoid abc, 
 the two latter being the semiaxes of the section made by a plane passing through 
 them; so that if from the centre two distances OT, 0V, equal to /, r", be taken 
 on the direction of r, the points T and V will belong (9) to the biaxal surface ; and 
 let a plane parallel to the plane of r', r", and touching the ellipsoid, cut the direc- 
 tion of r at the distance p from the centre. Then if r make the angles a, ft, 7, 
 with the semiaxes a, b, c, we shall have, by the nature of the ellipsoid, 
 
 1 cos 2 a cos 2 cos 2 y 
 ^ = ~tf~ ~V r ~ + ~^~ ' 
 
 p* = # 2 cos 2 a + i 2 cos 2 b + c 2 cos* -y. 
 Now, since the sum of the squares of the reciprocals of three rectangular radii of 
 
54 Geometrical Propositions, &c. 
 
 an ellipsoid is constant, as well as the parallelepiped described on three conjugate 
 semidiameters, we have the equations 
 
 1 1 1 1 1 1 
 
 1 _a i cos 2 a + 2 cos j8 + c 2 cos 2 7 _ 
 r' 2 r" = * i 2 c 2 
 
 Whence it appears that /, /', are the values of p in the equation 
 
 in which p denotes indifferently either semidiameter, OT or OF, of the biaxal sur- 
 
 face. Therefore putting for M and N their values, and writing -, -, -, instead of 
 
 P P P 
 
 cos a, cos j8, cos 7, and # 2 + y z + z 3 instead of p 2 , we obtain, for the equation of the 
 biaxal surface, 
 
 This is the equation of the surface of refraction for a biaxal crystal in which 
 a, b, c, are (54) the three principal indices of refraction, taking OS the radius of 
 the sphere to be unity. The left-hand member of the equation is therefore the ex- 
 pression supplied by FRESNEL for the function 7 in Art. 51. 
 
 "When the faces of the crystal are parallel to any of the principal planes of the 
 ellipsoid to the plane of xy for example the nature of the ring -trace may be 
 found very easily. For if the difference of the two values of z, deduced from the 
 preceding equation of the surface of refraction, be put equal to a constant quantity 
 J, the result, when cleared of radicals, will be an equation of the fourth degree in 
 x and y, which will be the equation of the corresponding ring-trace. This is a case 
 that occurs frequently in practice ; the crystal being often cut with its faces per- 
 pendicular to the axis of x or of z, because these lines bisect the angles made by the 
 optic axes. 
 
55 ) 
 
 V. A SHORT ACCOUNT OF SOME EECENT INVESTIGA- 
 TIONS CONCERNING THE LAWS OF REFLEXION AND 
 REFRACTION AT THE SURFACE OF CRYSTALS. 
 
 [fifth Report of the British Association, 1835.] 
 
 To understand the nature of the general problem which a com- 
 plete theory of double refraction requires to be solved, let it be 
 supposed that a ray of light is reflected and refracted at the 
 separating surface of an ordinary medium and a doubly refract- 
 ing crystal, the light passing out of the former medium into the 
 latter. This limited view of the subject is taken merely for the 
 sake of clearness of conception ; since we might suppose that both 
 media are crystallized, without increasing the difficulty of the 
 problem. The question, it is obvious, naturally divides into 
 two distinct heads. The first relates to the laws of the propa- 
 gation of light in the interior of either of the two media, before 
 or after it has passed their separating surface ; and this part of 
 the subject has been fully treated, according to their different 
 methods, by MM. Fresnel and Cauchy. The second division of 
 the subject had been left completely untouched. It relates to 
 the more complex consideration of what takes place at the sepa- 
 rating surface of the media, the laws according to which the 
 light is there divided between the reflected and refracted rays, 
 including a determination of the attendant circumstances indi- 
 cated by the wave theory, with regard to the vibrations in the 
 reflected and refracted rays. In the case above mentioned, when 
 the incident light is polarized, there are four things to be deter- 
 
56 Laws of Reflexion and Refraction 
 
 mined, namely, the magnitude and direction of the reflected vi- 
 bration, with the magnitudes of the two refracted vibrations. The 
 four conditions necessary for this determination are furnished 
 by two new laws, which could not be easily stated without en- 
 tering too much into detail. The results applied to determine 
 the polarizing angle of a crystal, in different azimuths of the 
 plane of reflexion, agree very closely with the admirable experi- 
 ments of Sir David Brewster on Iceland spar. In the course of 
 these experiments it was observed that the polarizing angle re- 
 mained the same when the crystal was turned half round (through 
 an angle of 180) ; although the inclination of the refracted 
 rays to the axis of the crystal was thereby greatly changed. 
 This remarkable fact is a consequence of theory. After some 
 complicated substitutions in the primary equations, the value of 
 the polarizing angle is found to contain only even powers of the 
 sine or cosine of the azimuth of the plane of reflexion, and there- 
 fore a change of 180 in the azimuth produces no change in the 
 polarizing angle. 
 
 The two new laws above mentioned, on which the theory 
 depends, occurred to the author in the beginning of last Decem- 
 ber ; but, owing to an oversight in forming one of the equations, 
 they were not fully verified until the beginning of June. 
 
 In this theory it is supposed that the vibrations are parallel 
 to the plane of polarization, according to the opinion of M. Cau- 
 chy. This is contrary to the views of Fresnel, whose theory 
 of double refraction obliged him to adopt the hypothesis that 
 the vibrations are perpendicular to the plane of polarization. It 
 is further supposed that the density of the vibrating ether is 
 the same in both media ; and the hypothesis of a constant den- 
 sity in different media, which was found necessary for the 
 theory, seems to accord, better than the supposition of a varying 
 density, with the phenomena of astronomical aberration. 
 
 If we conceive the three principal indices of refraction for 
 the crystal to become equal, we shall obtain the solution of a 
 very simple case of the general problem with which we have 
 been occupied the case of an ordinary refracting medium such 
 
at the Surface of Crystals. 5 7 
 
 as glass. This simple case, it is well known, was solved by 
 Fresnel. The foregoing theory leads to a simple law, expressing 
 all the particulars of the case, but differing with regard to the 
 magnitude of the refracted vibration from the formulae of 
 Fresnel. The law may be stated by saying that the refracted 
 vibration is the resultant of the incident and reflected vibrations ; 
 the first vibration being the diagonal of a parallelogram, of 
 which the other two vibrations are the sides, just as in the com- 
 position of forces. The plane of the parallelogram is the plane 
 of polarization of the refracted ray. It is to be remembered, 
 that the vibrations in each ray are perpendicular to the ray itself, 
 and parallel to its plane of polarization. 
 
 This simple case has been considered by M. Cauchy in a 
 short Paper inserted in the Bulletin Universal, torn. xiv. ; but it 
 does not seem to have been observed by anyone that his solu- 
 tion is erroneous. His formula for light polarized parallel to 
 the plane of reflexion is that which belongs to light polarized 
 perpendicular to the plane of reflexion, and vice versa. 
 
VI. LAWS OF REFLEXION FEOM METALS. 
 
 [Proceedings of the Royal Irish Academy, VOL. i., p. 2. Read Oct. 24, 1836.] 
 
 THE author observes that the theory of the action of metals 
 upon light is among the desiderata of physical optics, whatever 
 information we possess upon this subject being derived from 
 the experiments of Sir David Brewster. But, in the absence of 
 a real theory, it is important that we should be able to represent 
 the phenomena by means of empirical formulae ; and, accord- 
 ingly, the author has endeavoured to obtain such formulae by 
 a method analogous to that which Fresnel employed in the case 
 of total reflexion at the surface of a rarer medium, and which, 
 as is well known, depends on a peculiar interpretation of the 
 sign \/ - 1. For the case of metallic reflexion, the author 
 assumes that the velocity of propagation in the metal, or the re- 
 ciprocal of the refractive index, is of the form 
 
 m (cos x + \/ ~ 1 s i n x) ' 
 
 without attaching to this form any physical signification, but 
 using it rather as a means of introducing two constants (for 
 there must be two constants, m and %, for each metal) into 
 Fresnel's formulas for ordinary reflexion, which contain only 
 one constant, namely, the refractive index. 
 
 Then if i be the angle of incidence on the metal, and i' the 
 angle of refraction, we have 
 
 sin i' = m (cos x + \/ ~ 1 sin x) sin *> (1) 
 
Laws of Reflexion from Metals. 59 
 
 and therefore we may put 
 
 cos i f = m' (cos \ - y - 1 sin x ') cos , (2) 
 
 if 
 
 m* cos 4 / = 1 - 2m* cos2 x sin 2 / + m* sin 4 /, (3) 
 
 and 
 
 m 2 sin2v sin 2 / 
 
 tan 2 X = 1 5 4 (4) 
 
 1 - m 2 cos 2 X sm 2 
 
 Now, first, if the incident light be polarized in the plane of 
 reflexion, and if the preceding values of sin /', cos /', be substi- 
 tuted in Fresnel's expression 
 
 sin (*' - 
 
 sin (/ + /')' 
 
 for the amplitude of the reflected vibration, the result may be 
 reduced to the form 
 
 a (cos 8 - */ - 1 sin ), 
 if we put 
 
 tan i// = (6) 
 
 tan 8 = tan 2i// sin ( x + x '), (7) 
 
 a 2 = 1 ~ sin 2$ cos ( x + \) , g , 
 
 1 + sin 2t// cos ( x + x ')" 
 
 Then, according to the interpretation, before alluded to, of 
 -t/~i, the angle 8 will denote the change of phase, or the retar- 
 dation of the reflected light ; and a will be the amplitude of the 
 reflected vibration, that of the incident vibration being unity. 
 The values of m' 9 x ', for any angle of incidence, are found by 
 formulae (3), (4), the quantities m 9 x , being given for each metal. 
 The angle x ' is very small, and may in general be neglected. 
 
60 Laws of Reflexion from Metals. 
 
 Secondly, when the incident light is polarized perpendicu- 
 larly to the plane of reflexion, the expression 
 
 tan (i- - i') 
 tan (t + 0' 
 
 treated in the same manner, will become 
 
 a' (cos V - v/" 1 "! sin S'), (9) 
 
 if we make 
 
 tan i//' = mm', (10) 
 
 tan X = tan 2^' sin ( x - x ') , (11) 
 
 /a = 1 ~ sin W cos ( x - x) . 
 1 + sin 2f cos ( x - x ') ' 
 
 and here, as before, & will be the retardation of the reflected 
 light, and a' the amplitude of its vibration. 
 
 The number M= may be called the modulus, and the 
 m 
 
 angle x the characteristic of the metal. The modulus is some- 
 thing less than the tangent of the angle which Sir David 
 Brewster has called the maximum polarizing angle. After two 
 reflexions at this angle a ray originally polarized in a plane in- 
 clined 45 to that of reflexion will again be plane-polarized in a 
 plane inclined at a certain angle (which is 17 for steel) to the 
 plane of reflexion ; and we must have 
 
 tan - ^. (13) 
 
 a" 
 
 Also, at the maximum polarizing angle we must have 
 
 S' - S = 90. (14) 
 
 And these two conditions will enable us to determine the con- 
 stants H and x for any metal, when we know its maximum 
 polarizing angle and the value of ; both of which have been 
 
Laws of Reflexion from Metals. 
 
 61 
 
 found for a great number of metals by Sir David Brewster. 
 The following Table is computed for steel, taking M = 3|, 
 X = 54. 
 
 t 
 
 S 
 
 8' 
 
 or 
 
 a' 2 
 
 i (a 2 i a' 2 ) 
 
 
 
 27 
 
 27 
 
 0-526 
 
 0-526 
 
 0-526 
 
 30 
 
 23 
 
 31 
 
 0-575 
 
 0-475 
 
 0-525 
 
 45 
 
 19 
 
 38 
 
 0-638 
 
 0-407 
 
 0-522 
 
 60 
 
 13 
 
 54 
 
 0-729 
 
 0-308 
 
 0-518 
 
 75 
 
 7 
 
 98 
 
 0-850 
 
 0-240 
 
 0-545 
 
 85 
 
 2 
 
 152 
 
 0-947 
 
 0-491 
 
 0-719 
 
 90 
 
 
 
 180 
 
 1-000 
 
 1-000 
 
 1-000 
 
 The most remarkable thing in this Table is the last column, 
 which gives the intensity of the light reflected when common 
 light is incident. The intensity decreases very slowly up to a 
 large angle of incidence (less than 75), and then increases up to 
 90, where there is total reflexion. This singular fact, that the 
 intensity decreases with the obliquity of incidence, was dis- 
 covered by Mr. Potter, whose experiments extend as far as an 
 incidence of 70. Whether the subsequent increase which ap- 
 pears from the Table indicates a real phenomenon, or arises from 
 an error in the empirical formulae, cannot be determined without 
 more experiments. It should be observed, however, that in these 
 very oblique incidences Fresnel's formulae for transparent media 
 do not represent the actual phenomena for such media, a great 
 quantity of the light being stopped, when the formulae give a 
 reflexion very nearly total. 
 
 The value ' - S, or the difference of phase, increases from 
 to 180. When a plane-polarized ray is twice reflected from 
 a metal, it will still be plane-polarized if the sum of the values 
 of & - B for the two angles of incidence be equal to 180. 
 
 It appears from the formulae that when the characteristic x 
 is very small, the value of ' will continue very small up to the 
 
62 Laws of Reflexion from Metals. 
 
 neighbourhood of the polarizing angle. It will pass through 
 90, when mm' = 1 ; after which the change will be very rapid, 
 and the value of & will soon rise to nearly 180. This is exactly 
 the phenomenon which Mr. Airy observed in the diamond. 
 
 Another set of phenomena to which the author has applied 
 his formulae are those of the coloured rings formed between 
 a glass lens and a metallic reflector ; and he has thus been 
 enabled to account for the singular appearances described by 
 M. Arago in the Memoires d'Arctieil, torn, iii., particularly the 
 succession of changes which are observed when common light 
 is incident, the intrusion of a new ring, &c. But there is one 
 curious appearance which he does not find described by any 
 former author. It is this. Through the last twenty or thirty 
 degrees of incidence the first dark ring, surrounding the central 
 spot, which is comparatively bright, remains constantly of the 
 same magnitude ; although the other rings, like Newton's rings 
 formed between two glass lenses, dilate greatly with the obli- 
 quity of incidence. This appearance was observed at the same 
 time by Professor Lloyd. The explanation is easy. It depends 
 simply on this circumstance (which is evident from the Table), 
 that the angle 180 - S', at these oblique incidences, is nearly 
 proportional to'cos i. 
 
 As to the index of refraction in metals, the author conjec- 
 tures that it is equal to . 
 
 cos x 
 
.VII. ON THE LAWS OP THE DOUBLE REFRACTION OF 
 
 QUARTZ. 
 
 [Transactions of the Royal Irish Academy, VOL. xvn. Read Feb. 22, 1836.] 
 
 THE singular laws of the double refraction of quartz, which have 
 been discovered by the successive researches of Arago, Biot, 
 Fresnel, and Airy, are known merely as so many independent 
 facts ; they have not been connected by a theory of any kind. 
 I propose, therefore, to show how these laws may be explained 
 hypothetically, by introducing differential coefficients of the 
 third order into the equations of vibratory motion. 
 
 Suppose a plane wave of light to be propagated within a crys- 
 tal of quartz. Let the co-ordinates #, y, s, of a vibrating mole- 
 cule be rectangular, and take the axis of z perpendicular to the 
 plane of the wave, and the axis of y perpendicular to the axis of 
 the crystal. Let us admit that the vibrations are accurately in 
 the plane of the wave, and of course parallel to the plane of xy. 
 Then, using and TJ to denote, at any time t y the displacements 
 parallel to the axes of x and y respectively, we shall assume the 
 two following equations for explaining the laws of quartz : 
 
 
 The peculiar properties of this crystal depend on the con- 
 stant (7. When (7=0, the third differentials disappear, and 
 
64 On the Laws of the Double Refraction of Quartz. 
 
 the equations are reduced to the ordinary form, in which state 
 they ought to agree with the common equations for uniaxal 
 crystals. Hence, putting a for the reciprocal of the ordinary 
 index, b for the reciprocal of the extraordinary, and for the angle 
 made by the axis of z with the axis of the crystal, we must have 
 
 A = a\ B = a z - (a* - b 2 ) sin 2 0, (3) 
 
 supposing the velocity of propagation in air to be unity. 
 
 Now, from the nature of equations (1) and (2), the vibra- 
 tions must be elliptical. In fact, if we put 
 
 -=- (st - 2) , r] = q sin \-f (st - s) (4) 
 
 = cos 
 
 where p, q, s, I are constant quantities, the differential equations 
 will be satisfied by assigning proper values to s and to the ratio 
 
 -. For, after substituting in equations (1) and (2) the values 
 
 of the partial differential coefficients obtained by differentiat- 
 ing formulae (4), we shall find that every term of each equa- 
 tion will have the same sine or cosine for a factor : omitting 
 
 therefore, the common factors, and making - = A*, we shall get 
 the following equations of condition : 
 
 s 2 = A - Ck, (5) 
 
 Subtracting these, we have 
 
 which, by formulae (3), becomes 
 
 (8) 
 
 Let us now interpret these results. It is obvious, from for- 
 
On the Laws of the Double Refraction of Quartz. 65 
 
 mulse (4) , that s is the velocity of propagation for a wave whose 
 length is /, and that each vibrating molecule describes a little 
 ellipse whose semiaxes p and q are parallel to the directions of x 
 and y. But the number k, expressing the ratio of the semiaxes, 
 has two values, one of which is the negative reciprocal of the 
 other, as appears by equation (8) ; and each value of k has a cor- 
 responding value of s determined by equation (5) or (6) . Hence 
 there will be two waves elliptically polarized, and moving with 
 different velocities, the ratio of the axes being the same in both 
 ellipses ; but the greater axis of the one will coincide with the 
 less axis of the other. The difference of sign in the two values 
 of k shows that if the vibration be from left to right in one wave, 
 it will be from right to left in the other. These laws were dis- 
 covered by Mr. Airy. 
 
 The law by which the ellipticity of the vibrations depends on 
 the inclination 0, and on the colour of the light, is contained in 
 equation (8). The value of the constant will be determined 
 presently. In the mean time we may observe, that C denotes a 
 line, whose length is very small, compared with the length of a 
 wave. 
 
 When = 0, the light passes along the axis of the crystal. 
 In this case we have k z = 1, and k = 1 ; which shows that there 
 are two rays, circularly polarized in opposite directions. The 
 value of s for each ray may be had from equation (5) or (6), by 
 putting + 1 and - 1 successively for k. Calling these values s' 
 and s", we find 
 
 (9) 
 
 *'" = 2 + 27T , s" = a 1 + (10) 
 
 I \ d II 
 
 Suppose a plate of quartz to have two parallel faces perpen- 
 dicular to the axis, and conceive a ray of light, polarized in a 
 given plane, to fall perpendicularly on it. The incident rectilinear 
 vibration may be resolved into two opposite circular vibrations, 
 
66 On the Laws of the Double Refraction of Quartz. 
 
 which will pass through the crystal with different velocities ; and 
 which, after their emergence into air, will again compound a 
 rectilinear vibration, whose direction will make a certain angle p 
 with that of the incident vibration : so that the plane of polariza- 
 tion will appear to have been turned round through an angle 
 equal to /o, called the angle of rotation. This angle may be de- 
 termined by means of the preceding formulae. Putting for the 
 thickness of the crystalline plate, the circularly polarized wave 
 whose velocity is ' will pass through it in the time 
 
 0_0f 7rC\ 
 7 " a \ tfl) ; 
 
 and the wave whose velocity is s" in the time 
 
 0_ / 7rC\ 
 s"~a\ a* I/ 
 
 Therefore, if S be the difference of the times, we have 
 
 But, since the velocity of propagation in air is supposed to 
 be unity, the time and the space described are represented by the 
 same quantity ; and therefore 8, which is evidently a line, de- 
 notes the distance between the fronts of the two circularly pola- 
 rized waves, when they emerge into air. The waves being at 
 this distance from each other, if we conceive, at the same depth 
 in each of them, a molecule performing its circular vibration, 
 and carrying a radius of its circle along with it, the two radii 
 will revolve in contrary directions, and will always cross each 
 other in a position parallel to the incident rectilinear vibration. 
 Now let two series of such waves be superposed, so as to agitate 
 every molecule by their compound effect, and it is evident that, 
 when the radius vector of one component vibration attains the 
 position just mentioned, the radius vector of the other will be se- 
 
 parated from it by an angle equal to , where X is the length of 
 
On the Laws of the Double Refraction of Quartz. 67 
 
 a wave in air. The resultant rectilinear vibration will bisect this 
 angle ; and therefore ^, the angle of rotation, will be equal to 
 
 y-. Hence, substituting for S its value, and observing that /, 
 the length of a wave in quartz, is equal to #A, we find 
 
 2- 2 C6 
 p " a*\* ; 
 
 which gives the experimental law of M. Biot, that the angle of 
 rotation is directly as the thickness of the crystal, and inversely 
 as the square of the length of a wave for any particular colour. 
 By changing the sign of (7, we should have an equal rotation in 
 the opposite direction. And here we may remark, that C may 
 be made negative in all the preceding equations, its magnitude 
 remaining. There are two kinds of quartz, the right-handed 
 and left-handed, distinguished by the sign of C. 
 
 The angle of rotation, 'for a given colour and thickness, is 
 known from M. Biot's experiments. We can therefore find the 
 value of C by means of the last formula ; and substituting this 
 value in equation (8), we shall be able to compute k when and 
 / are given. Now it happens that Mr. Airy,* by a very inge- 
 nious method of observation, has determined the values of k in 
 red light for two different values of <f> ; and of course we must 
 compare these observed values of k with the independent results 
 of theory. As Mr. Airy's experiments were made upon red 
 light, we shall select, for the object of our calculations, the red 
 ray which is marked by the letter C in the spectrum of Fraun- 
 hofer. For this ray, Fraunhofer has given the length A, which, 
 expressed in parts of an English inch, is equal to '0000258 ; and 
 M. Eudberg has found a = '64859, b = 64481. Moreover, from 
 the experiments of M. Biot, we may collect that the arc of rota- 
 tion, produced by the thickness of a millimetre, is something 
 more than 19 degrees for the ray we have chosen ; so that the 
 fraction % may be taken to express nearly the length of that 
 
 * Transactions of the Cambridge Philosophical Society, Vol. iv., p. 205. 
 
68 On the JLaivs of the Double Refraction of Quartz. 
 
 arc in a circle whose radius is unity. We have, therefore, 
 
 = -03937 inch, and P = '333. Substituting these values in the 
 formula 
 
 I 7T0 
 
 derived from (12), we find 
 
 from which it appears that C is about twenty thousand times less 
 than the millionth part of an inch. 
 
 Again, since a 2 - b* = '00489, we have 
 
 so that equation (8) becomes 
 
 n 2 0. = l. (13) 
 
 The results of this formula are compared with Mr. Airy's 
 experiments in the following Table, in which the less root is 
 taken for k, and its sign is neglected. 
 
 Values of <p. 
 
 Values of Jc. 
 
 Observed. 
 
 Calculated. 
 
 6 15' 
 8 54' 
 
 tan 16 38' = -2897 
 tan 8 56' = -1582 
 
 2980 
 1579 
 
 The angles 0, in the first column, are deduced from the ob- 
 served inclinations of the rays in air to the axis of the crystal ; 
 and as k was observed to be somewhat different for the ordinary 
 and extraordinary rays, its mean values are given in the second 
 column. The exact coincidence between these and the calculated 
 values is, perhaps, in some degree accidental ; but a less perfect 
 agreement would be sufficient to confirm the theory. 
 
On the Laws of the Double Refraction of Quartz. 69 
 
 The magnitude of k varies considerably with the colour of 
 the light, increasing from the red to the violet, while the coeffi- 
 cient of sin 2 . k, in formula (13) diminishes. If we take the 
 violet ray If, for example, this coefficient will be 159. But it 
 would be useless to make any more calculations, as we have no 
 experiments with which they might be compared. 
 
 The figure of the wave surface yet remains to be examined. 
 
 Eliminating k between formulae (5) and (6), we obtain the 
 equation 
 
 ( - A) (s> -S). 4^ ^, (14) 
 
 which expresses the nature of the surface, * being a perpendi- 
 cular from the origin on a tangent plane. From this equation 
 it follows that the two values of s can never become equal in 
 quartz, as they do in other crystals ; for if we solve the equation 
 for s 2 , and put the radical equal to zero, we shall get the condi- 
 tion 
 
 (A - BY + 167T 2 ^ = 0, 
 
 which cannot be fulfilled, since the quantity which ought to 
 vanish is the sum of two squares. The two sheets, or nappes, of 
 the wave surface are therefore absolutely separated. 
 
 It is commonly assumed that one of the rays is refracted 
 according to the ordinary law ; but this is not the case, since 
 neither of the values of s is constant. However, the fay which 
 has the greater velocity (a being greater than b) may still, for 
 convenience, be called the ordinary ray. Of the two roots of 
 equation (8) , the one, k 09 whose numerical value (supposing not 
 to vanish) is less than unity, corresponds to this ray. When C 
 is positive, k is negative ; and when C is negative, k is positive : 
 therefore in both kinds of quartz, by formulse (5) and (6), we 
 have s 2 > A, and s e 2 < B ; denoting by s and s e the respective 
 velocities of propagation of the ordinary and extraordinary waves. 
 Hence, if we conceive a sphere of the radius a, with its centre at 
 the origin, and a concentric prolate spheroid, whose semiaxis of 
 
7O On the Laws of the Double Refraction of Quartz. 
 
 revolution is also equal to , and parallel to the axis of the 
 crystal, while the radius of its equator is equal to &, the ordi- 
 nary nappe of the wave surface will fall entirely without the 
 sphere, and the extraordinary nappe entirely within the spheroid, 
 whether the crystal be right-handed or left-handed. With re- 
 spect to the little ellipse in which the vibrations are performed, 
 and of which the semiaxes parallel to x and y are represented by 
 p and q respectively, it is evident that p > q f or the ordinary 
 wave, since & < 1 ; and that p < q for the extraordinary wave. 
 When (7 vanishes, the minor axis of each ellipse also vanishes, 
 and the rays become plane-polarized, the ordinary vibrations 
 being then parallel to the direction of #, and the extraordinary 
 parallel to that of y. This is exactly what ought to happen on 
 the supposition that the vibrations of a plane-polarized ray* are 
 parallel to its plane of polarization a supposition which was 
 kept in view in framing the fundamental equations (1) and (2). 
 To show, with precision, how the two kinds of quartz are to 
 be distinguished by the sign of (7, we must give definite direc- 
 tions to the axes of co-ordinates. To this end, let us imagine the 
 plane of xy to be horizontal, and a circle to be described in it 
 with the origin for its centre ; and let the north, east, and 
 south points of this circle be marked respectively with the letters 
 N, E, S. Let the direction of + x be eastward, from to E ; 
 that of + y northward, from to JV; and that of -t- 2 vertically 
 downwards ; the progress of the light through the crystal being 
 
 * On this point there are two very different opinions. Fresnel supposed, as is 
 well known, that the vibrations of a plane -polarized ray are perpendicular to its 
 plane of polarization ; whereas, according to M. Cauchy, whom I have followed, 
 they are parallel to that plane. I am induced to adopt the latter supposition, be- 
 cause I have succeeded, by means of hypotheses which are grounded on it, in dis- 
 covering the laws of reflexion from crystallized surfaces ; laws which include, as a 
 particular case, those discovered by Fresnel for ordinary media. The hypotheses 
 alluded to, along with some of their results, are published in the London and Edin- 
 burgh Philosophical Magazine, Vol. vin., p. 103, in a letter to Sir David Brewster 
 (supra, pp. 75, et seq.} See also Vol. vn., p. 295, of the same Journal (supra, 
 pp. 55, et seq.} I hope soon to offer the Academy a detailed account of my researches 
 on this subject. 
 
On the Laws of the Double Refraction of Quartz. 7 i 
 
 also downwards, and the plane of the wave moving parallel, as 
 before, to the plane of xy. Then the crystal will be right-handed 
 or left-handed, according as C is positive or negative. For, if C 
 be positive, will be negative, and formulae (4) will become, by 
 exhibiting the sign of , 
 
 5 = jp cos (st -a ) , 17 = - k p sin ~ (*tf - 2) , (15) 
 I ; If- ; 
 
 for the ordinary vibration ; and 
 
 rt = q sin - (s - z) , (16) 
 I * ; 
 
 for the extraordinary vibration. Now if we suppose the arc 
 (st - 2) either to vanish, or to be a multiple of the circum- 
 
 t 
 
 ference, the molecule will be at the east point of its vibration ; 
 and upon increasing the time a little, the value of i? will become 
 negative in (15), and positive in (16), so that the movement will 
 be towards the south in the first case, and towards the north in 
 the second. Therefore, when C is positive, the ordinary vibra- 
 tion takes place in the direction NES, or from left to right, and 
 the extraordinary in the direction SEN, or from right to left, 
 supposing a spectator to look in the direction of the progress of 
 the light. It may be shown, in like manner, that when C is ne- 
 gative, the ordinary and extraordinary vibrations are in the 
 directions SEN and NES, or from right to left and from left to 
 right respectively. Now if a plane-polarized ray be transmitted 
 along the axis of the crystal, the plane of polarization will be 
 turned in the direction of the ordinary vibration, because this 
 vibration, being propagated more quickly, will be in advance of 
 the other, upon emerging from the crystal. Hence, the rotation 
 is from left to right when C is positive, and from right to left 
 when C is negative ; and the crystal is called right-handed in 
 the first case, and left-handed in the second. 
 
 We have all along supposed that C is a constant quantity, 
 and the agreement of our results with experiment proves that 
 
72 On the Laws of the Double Refraction of Quartz. 
 
 this supposition is at least very nearly true in the neighbourhood 
 of the axis. It is probable, however, not only that (7 varies with 
 0, but that it becomes different in equations (1) and (2) ; that 
 is to say, it is probable that the following equations 
 
 (17) 
 
 _ 7? _ r" 
 d? ~ dz* d**' 
 
 in which C f is a little different from (7, would be more correct 
 than those which we have assumed. Indeed Mr. Airy's experi- 
 ments seem to indicate that C' is greater than C ; for he found, 
 as we have already said, that the ratio of the axes of the little 
 ellipse described by a vibrating molecule is somewhat different 
 for the two rays, being more nearly a ratio of equality for the 
 ordinary than for the extraordinary ray. Now if we set out 
 from equations (17), instead of (1) and (2), and proceed in all 
 respects as before, we shall arrive at the formula 
 
 -) sin $.* = (I 8 ) 
 
 C' 
 
 instead of formula (8). The quantity -^- will be greater than 
 
 
 
 unity, if C' be greater than C, and the value of A* will be greater 
 than before. This seems to be the explanation of the -difference 
 between the ratios observed by Mr. Airy. 
 
 It may be proper to state briefly the considerations which led 
 to the foregoing theory. Beginning with the simple case of a 
 ray passing along the axis, the first thing to be explained was 
 the law of M. Biot, that the angle of rotation varies inversely as 
 the square of / or of A. Now it was remarked by Fresnel, who 
 first resolved the phenomena of rotation into the interference of 
 two circularly polarized waves, that the interval 8 between these 
 waves, at their emergence from the crystal, must be inversely as 
 /, if the angle of rotation be inversely as the square of I. This re- 
 
On the Laws of the Double Refraction of Quartz. 73 
 
 mark suggested* to me the idea of adding, to the equations of the 
 common theory, terms containing the third differential coefficients 
 of the displacements ; for it was evident that such additional 
 terms would give, in the value of s 2 , a part inversely proportional 
 to /. It was also evident that the third differential coefficient 
 of should be combined with the second differential coefficients 
 of j, and the third of rj with the second of , in order that, after 
 substitutions such as we have indicated in deducing formulae (5) 
 and (6), the sines or cosines might disappear by division, and 
 that thus the value of s 2 might be independent of the time, as 
 it ought to be. This kind of reasoning led me to assume the 
 equations 
 
 for the case of a ray passing along the axis of quartz ; and then, 
 substituting in these equations the values of the differential co- 
 efficients obtained by differentiating the formulae 
 
 = p cos -8t-z , rj = p sn -=- 
 if ; 
 
 which express a circular vibration (from right to left, or from 
 left to right, according to the sign of the second p) , the result 
 was 
 
 S * = a' + ^ C 
 
 I 
 
 from (19), and 
 
 S > = ' D 
 
 * " The singular relation between the interval of retardation [S] and the length 
 of the wave [7] seems to afford the only clue to the unravelling of this difficulty." 
 " Eeport on Physical Optics," by Professor Lloyd (" Fourth Report of the British 
 Association," p. 409). It was in reading this Report that Fresnel's remark, about 
 the relation between 8 and I, first came to my knowledge. 
 
74 On the Laivs of the Double Refraction of Quartz. 
 
 from (20) ; which showed that D = - (7, since the values of s, 
 corresponding to the same circular vibration, ought to be equal. 
 The transition from this simple case to that of a ray inclined at 
 a given angle to the axis was easily made, by taking into ac- 
 count the doubly refracting structure of the crystal. This was 
 done by supposing 5 and jj parallel to the principal directions in 
 the plane of the wave, arid by changing # 2 , in equation (20), into 
 a* - (a? - b 2 ) sin 2 < ; and thus the fundamental equations (1) and 
 (2) were obtained. 
 
( 75 ) 
 
 VIII. OK THE LAWS OF REFLEXION FROM CRYSTAL- 
 LIZED SURFACES. 
 
 [From the Philosophical Magazine, VOL. VIH., 1835.] 
 
 To SIR DAVID BREWSTER. 
 
 DEAR SIR I have great pleasure in sending you an account 
 of the laws by which I conceive that the vibrations of light are 
 regulated when a ray is reflected and refracted at the separating 
 surface of two media ; especially as the only guide which I had, 
 in my inquiry after these laws, was your Paper on the action of 
 crystallized surfaces upon light, published in the Philosophical 
 Transactions for the year 1819. The observation which I found 
 there, that the polarizing angle was the same for a given plane 
 of incidence, " whether the obtuse angle of the rhomb [of Iceland 
 spar] was nearest or furthest from the eye, or whether it was to 
 the right or left hand of the observer," disappointed me at first, 
 being contrary to what I had anticipated from principles ana- 
 logous to those which had been employed by Fresnel in the 
 problem of reflexion from ordinary media. I then sought other 
 principles, and the observation is now a result of theory. 
 
 Assuming, as a basis for calculation, that Fresnel's law of 
 double refraction is rigorously true, I have been obliged to 
 make an essential change in his physical ideas. Conceive an 
 ellipsoid whose semiaxes are parallel to the three principal direc- 
 tions of the crystal, and equal respectively to its three principal 
 indices of refraction, and let a central section of the ellipsoid 
 be made by a plane parallel to the plane of a wave passing 
 
7 6 On the Laws of Reflexion 
 
 through the crystal. The section will be an ellipse, and the 
 wave will be polarized by the crystal in a plane parallel to either 
 semiaxis of this ellipse, the index of refraction for the wave 
 being equal to the other semiaxis. This is Fresnel's law of 
 double refraction ; and the theory which led him to it makes it 
 necessary to admit that the vibrations of the wave are perpendi- 
 cular to its plane of polarization ; whereas, according to the 
 views which I have adopted, the vibrations of the wave are pa- 
 rallel to its plane of polarization, and to one semiaxis of the 
 elliptic section, while its index of refraction is equal to the other 
 semiaxis. These views nearly agree with the theory of M. Cauchy, 
 according to whom the vibrations of polarized light are parallel 
 to its plane of polarization, but inclined at small angles to the 
 plane of the wave in crystallized media, instead of being exactly 
 parallel to the latter plane, as I have supposed them to be. Be- 
 sides, the theory of M. Cauchy, founded on the six equations of 
 pressure in a crystallized medium, implies the existence of a 
 third ray of feeble intensity, and for the other two rays gives a 
 law somewhat different from that of Fresnel. Being obliged, in 
 order to account for your experiments, to abandon the physical 
 ideas of Fresnel, and to approximate towards those of M. Cauchy, 
 I was embarrassed by this .third ray ; and wishing to get rid of 
 it, as well as of the slight deviations from the symmetrical law 
 of Fresnel, I adopted the expedient of altering the equations of 
 pressure, in such a way as to make them afford only two rays, 
 and give a law of refraction exactly the same as Fresnel's. The 
 equations which I found to answer this purpose are the follow- 
 ing : 
 
 0- 8jF 
 
 dx dy 
 
from Crystallized Surfaces. 77 
 
 + I** 
 
 dy) 
 
 In these equations, the axes of co-ordinates are perpendicular 
 to each other, and parallel to the principal directions of the crys- 
 tal ; x, y, z are the co-ordinates of a vibrating molecule at the 
 time t ; , 17, are the components of the displacement of the 
 same molecule at the same time ; a, 6, c are the three principal 
 indices of refraction out of the crystal into an ordinary medium 
 in which the velocity of light is equal to F"; and p is the density 
 of the ether, which density I suppose to be the same in all media. 
 The quantities A, F, E are the components, parallel to the axes 
 of x, y, 2, respectively, of the pressure upon a plane perpendi- 
 cular to the axis of x ; F, B, D are the components of the pres- 
 sure upon a plane perpendicular to the axis of y ; and E, Z), C 
 the components of the pressure upon a plane perpendicular to 
 the axis of z. The values of D, E, .Fare the same as those given 
 by M. Cauchy ; but the values of A, B, C are different from 
 his, and much simpler. By introducing into the equations of 
 M. Cauchy the condition that the vibrations shall be performed 
 without any change of density, the resulting values of A, B, C 
 might be shown to agree nearly with those given above. The 
 six pressures, A, B, (7, D, E, F, being known, it is easy to find 
 the pressure upon a plane making any given angles with the 
 axes of co-ordinates. 
 
 These things being premised, it is time to mention the laws, 
 or rather hypotheses, which I have imagined for discovering the 
 relations that exist, as to direction and magnitude, among the 
 vibrations in each ray, when reflexion and refraction take place at 
 the separating surface of two media, whether crystallized or not. 
 In stating the two very simple laws that have occurred to me 
 
78 On the Laws of Reflexion 
 
 for this purpose, it will be convenient, when the first medium is 
 an ordinary one, to suppose that the incident light is polarized. 
 Then, by the first law, the vibrations in one medium are equivalent 
 to those in the other ; that is to say, if the incident and reflected 
 vibrations be compounded, like forces acting at a point, their 
 resultant will be the same, both in length and direction, as the 
 resultant of the refracted vibrations similarly compounded. By 
 the second law, the lateral pressure upon the separating surface is 
 the same in both media ; the lateral pressure being understood to 
 mean the pressure in a direction perpendicular to the plane of 
 incidence. 
 
 As it would engage us too long to follow these laws into 
 detail, I shall merely state some of the results which I have ob- 
 tained from them, for the case of a uniaxal crystal into which 
 the light passes out of an ordinary medium. 
 
 Imagine the surface of the crystal to be horizontal, and call 
 the point of incidence /. With the centre /and any radius, con- 
 ceive a sphere to be described, cutting in the point Z a vertical 
 line IZ drawn through the centre, z. 
 and let a radius ZP, parallel to the 
 axis of the crystal, meet the surface 
 of the sphere 'in P. Let the great 
 circle ZOEloe the plane of incidence, 
 containing both the direction 10 of 
 the ordinary refracted ray produced 
 backwards, and the direction IE of 
 a normal to the extraordinary wave ; 
 and draw the great circles PZ, PO, PE. The angle Zwill be the 
 azimuth of the plane of incidence. Let ZO = 0, ZE = 0', P0= */>, 
 PE = i/,', the angle ZOP = 0, and the angle ZEP = 0'. CaU the 
 angle of incidence i, and suppose b to be the reciprocal of the ordi- 
 nary refractive index and a the reciprocal of the extraordinary. 
 
 Each of the refracted rays, in turn, may be made to disappear, 
 by polarizing the incident ray in a certain direction assigned by 
 theory. When the extraordinary ray disappears, the reflected 
 ray is polarized in a plane inclined to the plane of incidence at 
 
from Crystallized Surfaces. 79 
 
 an angle /3 determined by the formula 
 
 Q-I y| 2 * 
 
 tan j3 = cos (e 4 0) tan + 2 (# 2 - 2 ) sin sin ^ cos ^ - p - r . (2) 
 
 When the ordinary ray disappears, the plane of polarization 
 of the reflected ray is inclined to the plane of incidence at an 
 angle )3 determined by the formula 
 
 - tan j3' = cos (i + 0') cotan & 
 
 ,_ cos 20' . , sin 2 ?' 
 
 sin cos 
 
 . r r< . 
 
 sin 9 sm( . -0 ) 
 
 And when the angles /3, j3 r , become equal, the plane of polariza- 
 tion of the reflected ray becomes independent of the plane of po- 
 larization of the incident ray ; and the angle of incidence i, at 
 which this equality takes place, is the polarizing angle of the 
 crystal. Hence we have the equation of condition 
 
 cos (i + 0) tan 9 + 2 (a 2 - 2 ) sin 9 sin i// cos 
 
 sin (i <p 
 
 orm 2 r/ m*n^ i 
 
 + cos (i -f d/) cotan 9' + (a 2 - b*) . -& sini// GOBI!/ -^r-. ; 
 
 sin a sin(e 7 
 
 to be fulfilled at the polarizing angle. 
 
 Since i + 0, in this equation, is nearly equal to a right angle, 
 
 put i + (j> = y + S, and 8 will be a small quantity. Draw PR 
 
 an arc of a great circle perpendicular to ZOE, and let ZR = p, 
 PR - q. Then we shall find from equation (4), after various 
 substitutions and reductions, 
 
 JTcos 2 q (cos 2 - cos ; where K= 
 
 In deducing this value of S, the approximations were made 
 with a tacit reference to the case of reflexion in air from a com- 
 mon rhomb of Iceland spar. The coefficient K, in this case, is 
 equal to about nine degrees, and the resulting numerical values of 
 
8o On the Laws of Reflexion 
 
 the polarizing angles in various azimuths agree very well with 
 your experiments. You will perceive that the value of 8 is the 
 same in supplementary azimuths, which explains the observation, 
 cited in the beginning of my letter, relative to the equality of 
 the polarizing angles at opposite sides of the perpendicular IZ 
 in a given plane of incidence. 
 
 When the point JR falls upon 0, we have 8 = 0, and i + <j> 
 equal to a right angle. Hence, when the cotangent of ZR is 
 equal to the ordinary index, the tangent of the polarizing angle 
 is equal to the same index. This theorem, though deduced from 
 an approximate equation, might be shown to be exact. 
 
 When the axis of the crystal lies in the plane of incidence, we 
 may obtain an exact expression for the polarizing angle. The 
 condition of polarization then becomes 
 
 cos ( + f) - (< - V) sin f cos f . '* - ; (6) 
 
 bill \^i ({) J 
 
 from which, by the proper substitutions, we obtain the following 
 
 expression : 
 
 . ' . 1 - a 2 cos 2 A - b 2 sin 2 A 
 
 sm 2 * = -- - ; (7) 
 
 i - a 2 b 2 
 
 where A denotes the complement of ZP, or the inclination of the 
 axis to the face of the crystal, and i is the polarizing angle. 
 This formula, in a shape somewhat different, was communicated, 
 above a year ago, to Professor Lloyd, who has noticed it, in con- 
 nexion with your Paper, in his " Eeport on Physical Optics." 
 When a and b become equal, the formula gives your law of the 
 tangent for ordinary media. 
 
 The foregoing results show that, when a ray is polarized by 
 reflexion from a crystal, the plane of polarization deviates from 
 the plane of incidence, except when the axis lies in the latter 
 plane ; and that the deviation may be made very great by 
 placing the crystal in contact with a medium whose refractive 
 power is nearly equal to that of the crystal itself ; for when i is 
 nearly equal to or to $', the divisor sin (i - <j>) or sin (i - $') is 
 
Laws of Reflexion from Crystallized Surfaces. 81 
 
 very small, and therefore tan |3 or tan |3' is very great. But 
 this remark is of no value whatever in explaining the very sin- 
 gular phenomena which you have observed in the extreme case 
 just mentioned ; nor can I imagine any reason why there should 
 be a deviation, as there was in some of your experiments, when 
 the axis lies in the plane of incidence, since everything is then 
 alike on both sides of this plane. Indeed the whole of this 
 subject, which occupies the latter part of your Paper of 1819, is 
 very extraordinary and interesting ; and I was glad to hear that 
 you had resumed the investigation of it, and made many experi- 
 ments which have not been published. 
 
 I wish you would publish them. They seem to be of great 
 importance in the present state of optical science. 
 
 I am, dear Sir, ever truly yours, 
 
 J. MAC CTTLLAGH. 
 
 TBIN. COLL., DUBLIN, Dec. 22, 1835. 
 
IX. ON THE PEOBABLE NATURE OF THE LIGHT TRANS- 
 MITTED BY THE DIAMOND AND BY GOLD LEAP. 
 
 [Proceedings of the Royal Irish Academtj, VOL. I. p. 27- Read Jan. 9, 1837-] 
 
 PROFESSOR MAC CTJLLAGH made a verbal communication on the 
 probable nature of the light transmitted by the diamond and by 
 gold leaf. He conceives that as there is a change of phase caused 
 by reflexion from these bodies, so there is also a change of phase 
 produced by refraction ; the change being different according as 
 the incident light is polarized in the plane of incidence, or in the 
 perpendicular plane. Consequently, if the incident ray be po- 
 larized in any intermediate plane, the refracted ray should be 
 elliptieally polarized ; and on examining the light transmitted 
 by gold leaf, this was found to be the case. Of course the same 
 thing is true of the light which enters the other metals, and 
 which is subsequently absorbed. The same remark explains the 
 appearance of double refraction in specimens of the diamond 
 which give only a single image ; and it is likely that other 
 precious stones will be found to possess similar properties. Mr. 
 Mac Cullagh has obtained a general formula for the difference 
 of phase between the two component portions of the refracted 
 light one polarized in the plane of incidence, and the other 
 perpendicular to it. He finds from this formula, that the dif- 
 ference of phase, which is nothing at a perpendicular incidence, 
 increases until it becomes equal to the characteristic at an inci- 
 dence of 90; and when the light emerges into air, the difference 
 of phase is doubled. The formula has not yet been submitted 
 to the test of experiment. 
 
83 
 
 X. ON THE LAWS OF CRYSTALLINE REFLEXION. 
 
 [From the Philosophical Magazine, VOL. x., 1837.] 
 
 IN a Number of PoggendorfFs Annalen (No. 6, for 1836), which 
 reached Dublin late in November, there are some remarks by 
 M. Seebeck on a Paper of mine which appeared in the last 
 February Number of this Journal (vol. viii. p. 103). That 
 Paper contains a general theory of reflexion at the surfaces of 
 crystallized media ; and M. Seebeck, in comparing the results 
 with his own experiments, has fully confirmed some of my 
 formulae, while he has shown that others are defective. I have 
 therefore been obliged to revise my theory, and I have ascer- 
 tained that it was vitiated by the introduction of a certain re- 
 lation among the quantities denominated pressures, which, 
 following the example of M. Cauchy, I had supposed to be 
 concerned in the problem. This relation I had observed to 
 hold in the case of singly refracting media, and I concluded, 
 without any other reason, that it would hold good generally. 
 But though it led to the correct formula for the polarizing 
 angles in different azimuths, it was nevertheless arbitrary and 
 unfounded ; and therefore it is now banished entirely from the 
 investigation, the place which it occupied being supplied by the 
 natural and simple law of the preservation of vis viva, while 
 everything else remains as before. I hope the imperfection of 
 my first essay will be excused, when it is considered that the 
 erroneous proposition bears but a small proportion to the whole 
 theory; and, moreover, that the general problem, which I under- 
 took to resolve, is one that has not been attempted by any other 
 
84 On the Laws of Crystalline Reflexion. 
 
 person, although the want of a solution has long been felt. The 
 difficulties which we have to deal with, in entering upon this 
 problem, are not mere mathematical difficulties, but difficulties 
 arising from the want of first principles; and, in physical 
 questions of this kind, where we must, at the outset, have re- 
 course to conjecture, in order to supply the very principles of 
 our reasoning, it can hardly be expected that the whole truth 
 should be divined at once. I think, however, that I have now 
 obtained a true mechanical theory; and if so, it will help to 
 decide, not only the question immediately before us, but also 
 the other much-disputed, though more elementary, questions 
 concerning the density of the ether in different media, and 
 the direction of the vibrations in polarized light. In fact, a 
 particular supposition respecting each of the latter questions 
 is included in my theory, the several principles of which, 
 making the single alteration that has been mentioned, I shall 
 here enumerate : 
 
 1. The density of the ether is the same in all media. 
 
 2. The vibrations of plane-polarized light are parallel to the 
 plane of polarization. 
 
 3. The vis viva is preserved. 
 
 4. The vibrations are equivalent at the common surface of 
 two media. 
 
 To these may be added the definition of the polarizing 
 angle of a crystal; namely, the angle of incidence at which 
 the plane of polarization of the reflected ray becomes indepen- 
 dent of the plane of polarization of the incident ray. At the 
 polarizing angle, the former plane does not, in general, coincide 
 with the plane of reflexion, but makes with it a small angle 
 which may be called the deviation. 
 
 It is curious that, about a year and a half ago, I employed 
 these four principles, precisely as I have now enumerated them, 
 in deducing Fresnel's well-known laws of reflexion for ordinary 
 media ; but I did not then apply the law of vis viva to crystals, 
 because my mind was preoccupied by the notion that there ex- 
 isted some relation among the pressures. This notion I had 
 
On the Laws of Crystalline Reflexion. 85 
 
 taken up from reading a little Paper, by M. Cauchy, in the 
 Bulletin des Sciences Mathematiques for July 1830 ; and by 
 combining such a relation with the three conditions afforded 
 by my own law of equivalent vibrations, I had actually obtained, 
 for the polarizing angles in different azimuths, a formula (that 
 marked (5) in my former Paper), which I found to agree very 
 well with Sir David Brewster's experiments, and which M. 
 Seebeck has found to agree still better with his own. 
 
 The formula for the polarizing angle is obtained by equat- 
 ing two values of the deviation ; and it is remarkable that the 
 very same formula comes out in my present theory, although 
 the values of the deviation are entirely different. Referring, 
 for brevity, to the notation of my former Paper, I find, for the 
 case of a uniaxal crystal, 
 
 tan |3 = cos (i + 0) tan 0, ...................... (a) 
 
 - * - - ** + , - * = 
 
 These equations (a) and (b) are to be substituted for equa- 
 tions (2) and (3),* which are the equations that M. Seebeck found 
 to be at variance with his experiments. 
 
 By means of formula (5), equation (a) becomes 
 
 from which the deviation in any azimuth may be readily calcu- 
 lated. The azimuth (as M. Seebeck reckons it) begins when 
 = 0, and p is then positive. This formula (c) perfectly represents 
 the experiments of M. Seebeck on Iceland spar. The corre- 
 sponding expressions for biaxal crystals may be easily deduced, 
 and will be given in a Paper which I am preparing to lay 
 before the Royal Irish Academy. 
 
 At the time of my last communication I was not aware that 
 the case in which the plane of incidence is a principal section of 
 the crystal (or the azimuth = 0) had been solved by M. Seebeck, 
 and that formula (7),f which I regarded as my own, had been 
 obtained by him long before. 
 
 * Supra, p. 79. t Supra, p. 80. 
 
86 On the Laws of Crystalline Reflexion. 
 
 It remains to say a word respecting the new principle of 
 equivalent vibrations, the most important, perhaps, of all, as it 
 is certainly the simplest that can be imagned. If we conceive 
 an ethereal molecule situated at the common surface of two 
 media, it would seem that its motion ought to be the same, 
 whether we regard the molecule as belonging to the first me- 
 dium or to the second. Now the incident and reflected vibra- 
 tions are superposed in the first medium, and the refracted 
 vibrations in the second; and therefore we may infer (when 
 the phase is not changed by reflexion or refraction), that if the 
 incident and reflected vibrations be compounded, like forces acting at 
 a point, their resultant will be the same, both in length and direc- 
 tion, as the resultant of the refracted vibrations similarly com- 
 pounded. This is the law of equivalent vibrations, and it 
 gives, at once, three equations. A fourth equation is afforded 
 by Fresnel's law of the vis viva; and thus we have the four 
 conditions necessary for a general solution of the problem. 
 From the principle of equivalent vibrations, as we have 
 stated it, it follows that the vibrations resolved parallel to the 
 separating surface are equivalent in the two media; and, in 
 fact, this part of the general principle was assumed by Fresnel ; 
 but the other part, namely, that the vibrations perpendicular 
 to the separating surface are equivalent, was not assumed by 
 him, nor is it by any means true in his theory. It appears 
 then that three conditions only are afforded by the hypotheses 
 which Fresnel successfully employed in solving the problem of 
 reflexion from ordinary media. These hypotheses, therefore, 
 are not sufficient when applied to crystals; except, indeed, in 
 the case before alluded to, where the azimuth = 0,. which has 
 been solved by M. Seebeck. It should be observed, that though 
 the reasons which I have assigned for the principle of equivalent 
 vibrations are extremely simple, yet it was not by such simple 
 reasoning that I was led to it originally. 
 
 TRINITY COLLEGE, DUBLIN, December 13, 1836. 
 
87 
 
 XI. ON THE LAWS OF CRYSTALLINE REFLEXION AND 
 REFRACTION. 
 
 [Transactions of the Royal Irish Academy, VOL, xvm. Read Jan. 9, 1837.] 
 
 WHEN a ray of light, which has been polarized in a given plane, 
 suffers reflexion and refraction at the surface of a transparent 
 medium, the rays into which it is divided are found to be po- 
 larized in certain other planes ; and it becomes a question to 
 determine the positions of these planes, as well as the relative 
 intensities of the different rays ; or, in theoretical language, to 
 find the direction and magnitude of the reflected and refracted 
 vibrations, supposing those of the incident vibration to be given. 
 The transparent medium may be either a singly-refracting sub- 
 stance, such as glass, or a doubly-refracting crystal, like Iceland 
 spar. When the medium is of the first kind, the problem is 
 comparatively simple, being, in fact, nothing more than a par- 
 ticular case of the problem which we have to consider when the 
 medium is supposed to be of the second kind. In the progress 
 of knowledge it was natural that the simpler question should be 
 first attended to ; and accordingly Fresnel, during his brief and 
 brilliant career, found time to solve it. But the general problem, 
 relative to doubly-refracting media, had not been attempted by 
 anyone, when, in the year 1834, my thoughts were turned to 
 the subject. I then recollected a conclusion to which I had 
 been led some years before, and which, on this occasion, proved 
 of essential service to me. Being fond of geometrical construe- 
 
88 On the Laws of Crystalline 
 
 tions, I amused myself, when I first became acquainted with 
 Fresnel's theories, by throwing his algebraical expressions, when- 
 ever I could, into a geometrical form ; and treating in this way 
 the well-known formulae in which he has embodied his solution 
 of the problem just alluded to, I obtained a remarkable result, 
 which gave me the first view of the principle that I have since 
 employed under the name of the principle of the equivalence of 
 vibrations. In order to state this result briefly, I will take leave 
 to introduce a new term for expressing a right line drawn 
 parallel to the plane of polarization of a ray, and perpendicular 
 to the direction of the ray itself. Calling such a right line the 
 transversal of the polarized ray, I found, from the formulae 
 of Fresnel, that when polarized light falls upon a singly- 
 refracting medium, the transversals of the incident, of the re- 
 flected, and of the refracted rays are all parallel to the same 
 plane, which is the plane of polarization of the refracted ray ; 
 and that the magnitudes of the vibrations, or the greatest ex- 
 cursions of the ethereal molecules, in the incident and the 
 reflected rays, are to each other inversely as the sines of the 
 angles which the respective transversals of those rays make with 
 the transversal of the refracted ray. I was struck by the strong 
 analogy which these relations among the transversals bore to 
 the composition of forces or of small vibrations in mechanics ; 
 but it happened unfortunately that, in the theory of Fresnel, 
 the vibrations of light were supposed to take place, not in the 
 direction of the transversals, but perpendicular to them, so that 
 there was no physical circumstance to support the analogy, there 
 being no motion in the direction of the transversals ; while, on 
 the other hand, no such analogy existed among the vibrations 
 themselves in the directions which Fresnel had assigned to them. 
 It was therefore with some interest that I afterwards learned, 
 upon the publication of the tenth volume of the Memoirs of the 
 Institute, that M. Cauchy* had actually inferred, from mecha- 
 nical principles, that the vibrations of polarized light are in the 
 
 * Memoir es de Plmtitut, tome x. p. 304. 
 
Reflexion and Refraction. 89 
 
 direction of the transversals ; but this inference was to be re- 
 ceived with caution, as being contrary to the hypothesis of 
 Fresnel ; and besides, I had in the meantime contrived a way 
 of adapting my analogy, in some degree, to that hypothesis, by 
 supposing areas to be compounded instead of vibrations ; so that 
 I hesitated which of the two opinions to prefer. Taking, how- 
 ever, the opinion of M. Cauchy as that which fell in more na- 
 turally with the aforesaid analogy, I was led to the conclusion, 
 that the vibration in the refracted ray is probably the resultant 
 of the incident and reflected vibrations ; and I saw that if this 
 principle were true for singly-refracting media, it should also, 
 from its very nature, be true, when properly generalized, for 
 doubly -refracting crystals ; so that in such crystals the resultant 
 of the two refracted vibrations would be the same, both in length 
 and direction, as the resultant of the incident and reflected 
 vibrations. 
 
 This was the principle of equivalent vibrations. But I had 
 no sooner begun to regard it as probable, than an objection 
 started up against it. In the case of a ray ordinarily refracted 
 out of a rarer into a denser medium, the magnitude of the 
 refracted vibration, as deduced from this principle, was greater 
 than that which came out from the theory of Fresnel, in the 
 proportion of the sine of the angle of incidence to the sine of 
 the angle of refraction. Consequently, assuming with Fresnel 
 that the ether is more dense in the denser medium, the law of 
 the preservation of vis viva was violated. 
 
 There was another embarrassment which I felt in my early 
 efforts to find out the laws of crystalline reflexion. Taking 
 for granted the hypothesis of Fresnel, that the density of the 
 ether in an ordinary medium is inversely as the square of its 
 refractive index, I was at a loss what hypothesis to make, in 
 this respect, for doubly-refracting crystals, wherein the refrac- 
 tive index changes with the direction of the ray. For the 
 density, being independent of direction, could not be con- 
 ceived to vary with the refractive index. About two years 
 ago I got over this difficulty, by supposing the density of the 
 
90 On the Laws of Crystalline 
 
 ether to be the same in all media.* At the same time I was 
 compelled to employ the principle of equivalent vibrations, in 
 order to have a sufficient number of conditions, though for a 
 while I overlooked the perfect agreement which now subsisted 
 between this principle and the law of vis viva : it happened, in 
 fact, that the new hypothesis of a constant density made the 
 vis viva of the refracted ray exactly the same as in the theory 
 of Fresnel.f 
 
 But to see why it was necessary to assume the principle of 
 equivalent vibrations, we must observe, that when a polarized 
 ray is incident on a crystal there are four things to be deter- 
 mined, namely, the direction and magnitude of the reflected 
 vibration, and the magnitudes of the two refracted vibrations. 
 Hence we must have four conditions, or we must have relations 
 affording so many equations. But the hypotheses of Fresnel, 
 by which he solved the problem of reflexion for ordinary media, 
 afford only three conditions. We will state his hypotheses at 
 length : 
 
 1st. The vibrations of polarized light are in the plane of the 
 wave, and perpendicular to the plane of polarization. 
 
 2nd. The density of the ether is inversely as the square of 
 the refractive index of the medium. 
 
 3rd. The vis viva is preserved. 
 
 4th. The vibrations parallel to the separating surface of two 
 media are equivalent; that is, the refracted vibration parallel to 
 the surface is the resultant of the incident and reflected vibra- 
 tions parallel to the same. 
 
 We see that the fourth hypothesis gives two conditions, and 
 the law of vis viva gives a third. 
 
 Let us now take the more general principle of equivalent 
 vibrations, in place of the fourth hypothesis of Fresnel, altering 
 
 * This hypothesis is maintained by Mr. Challis ; and certainly it falls in ex- 
 tremely well with the astronomical phenomenon of the aberration of light. See, 
 on this subject, Professor Lloyd's Report on Physical Optics, " Fourth Report of the 
 British Association for the Advancement of Science," pp. 311, 313. 
 
 t Supra, p. 100, note. 
 
Reflexion and Refraction. 9 1 
 
 the first hypothesis in the way that we have shown to be neces- 
 sary in order to suit that principle, and making the ethereal 
 density constant. Then, if we retain the law of vis viva, our 
 new hypotheses will be these : 
 
 1st. The vibrations of polarized light are in the plane of the 
 wave, and parallel to the plane of polarization ; which may be 
 expressed in a word, by saying that the vibrations are transver- 
 sal, according to the peculiar sense in which I use the term. 
 
 2nd. The density of the ether is the same in all bodies as in 
 vacuo. 
 
 3rd. The vis viva is preserved. 
 
 4th. The vibrations in two contiguous media are equivalent ; 
 that is, the resultant of the incident and reflected vibrations is 
 the same, both in length and direction, as the resultant of the 
 refracted vibrations. 
 
 It is evident that the last hypothesis affords three equations, 
 by resolving the vibrations parallel to three axes of co-ordinates ; 
 and the law of vis viva supplies a fourth equation. Thus we 
 have the requisite number of conditions. 
 
 The hypotheses that we have last enumerated are those 
 which will be employed in the present Paper. They have been 
 made to include the law of vis viva, because I lately found that 
 this law must necessarily accompany the rest ; but at first I 
 neglected it, and even made considerable progress without it; 
 for, by the help of another hypothesis, I obtained formulae 
 which represented such experiments as I was aware of at the 
 time. This other hypothesis I took up from reading an article 
 by M. Cauchy in the Bulletin des Sciences Mathematiques* in 
 which he arrives, by a peculiar process, at the formulae of 
 Fresnel for the case of ordinary reflexion. The hypotheses 
 
 * " Sur la Refraction et la Beflexion de la Lumiere," Bulletin des Sci. Math., 
 Juillet, 1830. In this Paper the vibrations of polarized light must be supposed 
 perpendicular to the plane of polarization, though the Paper was published im- 
 mediately after the author had promulged the contrary opinion. The latter 
 opinion, which I adopted from him because it harmonized with my analogy 
 before mentioned, he has formally renounced of late, and has returned to the 
 
92 On the Laws of Crystalline 
 
 which he chiefly employs are relations among certain quanti- 
 ties called pressures ; and it was such a relation that I adopted 
 instead of the law of vis viva. I supposed that, at the confines 
 of two media, the pressure on the separating surface, in a direc- 
 tion perpendicular to the plane of incidence, ought to be the 
 same, whether it he considered as resulting from the vibrations 
 in the first medium or in the second. This hypothesis I con- 
 ceived to be true in general, because I found it to be true for 
 ordinary media ; but I could never assign any better reason for 
 it. Combining it, however, with the principle of equivalent 
 vibrations, I deduced several expressions for uniaxal crystals, 
 and among others a formula for the polarizing angles in diffe- 
 rent azimuths of the plane of reflexion. When this formula was 
 compared with the experiments of Sir David Brewster* on the 
 polarizing angles of Iceland spar, the accordance was so satis- 
 factory as to leave no doubt upon my mind that I had arrived 
 at the true formula for these angles ; and though the truth of 
 the conclusion did not allow me to argue that the premises 
 
 hypothesis of Fresnel. M. Cauchy supposed too, in the above Paper, that the 
 ethereal density is the same in different media ; but he has found cause to abandon 
 this hypothesis also. See his notes addressed to M, Libri, in the Comptes rendus 
 des Seances de VAcademie des Sciences, Seance du 4 Avril, 1836, where he gives the 
 reasons for his present opinions. He says, "Ainsi Fresnel a eu raison de dire, 
 non-seulement que les vibrations des molecules etherees sont generalement com- 
 prises dans les plans des ondes, mais encore que les plans de polarisation sont 
 perpendiculaires aux directions des vitesses ou des deplacements moleculaires. 
 J'arrive au reste a cette derniere conclusion d'une autre maniere, en etablissant 
 les lois de la reflexion et de la refraction a 1'aide d'une nouvelle methode qui 
 
 sera developpee dans mon memorie [cette methode] ne m' oblige plus a 
 
 supposer, comme je 1'avais fait dans un article du Bulletin des Sciences, que la 
 densite de 1' ether est la meme dans tous les milieux. Mes nouvelles recherches 
 donnent lieu de croire que cette densite varie en general quand on passe d'un 
 milieu a un autre." More lately, in his Nouveaux Exercices de Mathematiques, 
 7 e Livraison, M. Cauchy states positively that his principles do not permit him 
 to adopt the hypothesis that the density of the ether is the same in all media. He 
 also gives the differential equations which, as he has found by his new method, 
 ought to siibsist at the separating surface of two media, and from which he has 
 obtained the formulae of Fresnel for ordinary reflexion. But these eqtiations do 
 not include the laws of crystalline reflexion. 
 * Phil. Trans., 1819, p. 150. 
 
Reflexion and Refraction. 93 
 
 were true, yet the presumption in their favour was very strong, 
 insomuch that, upon remarking, as I did soon after, that the 
 law of vis viva harmonized with my other hypotheses, I did not 
 think it worth while* to try what would be the consequence of 
 using this law, instead of the relation which I had put in its 
 place. In this state of my theory, I gave an account of it at 
 the meeting of the British Associationf in Dublin, in August, 
 1835 ; and the leading steps and results were afterwards pub- 
 lished in a letter to Sir David Brewster.J 
 
 Now we are to observe, that when common light is polarized 
 by reflexion at the surface of a doubly-refracting crystal, the 
 polarization does not, in general, coincide with the plane of 
 reflexion, as in the case of ordinary media, but is inclined 
 to it at a certain angle, which may be called the deviation; and 
 it was by equating two values of the deviation that I obtained 
 the formula above mentioned for the polarizing angle. This 
 formula, as we have seen, was correct ; but it happened, singu- 
 larly enough, that the expressions for the deviation, which were 
 
 * I had, besides, an objection to the law of vis viva, on the ground that it would 
 give an equation of the second degree ; and I wished to have all my equations 
 linear, lest, in the seemingly complicated question of crystalline reflexion, they 
 should give two answers when the nature of the question required but one. This 
 has actually happened, since the present Paper was read, in applying my hypothe- 
 ses to the case of internal reflexion at the second surface of a uniaxal crystal. 
 Supposing an ordinary ray to emerge after double reflexion, and putting for the 
 angle which the emergent transversal makes with the plane of incidence, I found, 
 for determining 0, an equation of the form 
 
 A + B tan + G tan 2 = 0, 
 
 wherein A is very small, but does not vanish ; so that the equation gives two roots, 
 one very small, the other about the proper value. It is clear, therefore, that there 
 is a want of adjustment somewhere : but I am now inclined to think that the fault 
 is not in the principle of vis viva. Possibly our laws of the propagation of light in 
 doubly refracting media are not quite accurate. "Whatever supplementary law 
 shall be found to remedy this untoward result will probably, at the same time, 
 account for the extraordinary phenomena observed by Brewster, in reflexion at 
 the first surface when the crystal is in contact with a medium of nearly equal 
 refractive power. 
 
 t London and Edinburgh Philosophical Magazine, Vol. vn. p. 295. 
 Ibid., Vol. vni. p. 103; February, 1836. (Supra, p. 75.) 
 
94 On the Laws of Crystalline 
 
 used in obtaining the formula, were erroneous. It is to M. Seebeck 
 that I am obliged for pointing out this curious circumstance. 
 In Poggendorff's Annals* he gave an abstract of my letter to 
 Sir David Brewster, and compared my results with his own 
 numerous and accurate experiments, both on the polarizing 
 angles of Iceland spar and on the angles of deviation. He 
 found that my formula represented the former class of ex- 
 periments as well as could be wished ; but the theoretical 
 values of the deviations did not at all agree with his experi- 
 mental measures. These measures of the deviation he pub- 
 lished on this occasion ; and, with their assistance, I traced 
 the error to its source, which was the relation among the press- 
 ures. The principle of vis vica was therefore introduced, instead 
 of that relation, and the theory became much simpler by the 
 change. I now obtained, for the deviation, a new expression, 
 which agreed with the experiments of M. Seebeck; but the 
 formula for the polarizing angle came out the very same as 
 before. This correction was made on the 6th of December, 
 and was published in the Philosophical Magazine-^ on the first 
 of the present month. 
 
 In the interval I have arrived at very elegant geometrical 
 laws, which can be easily remembered, and which embrace the 
 whole theory of crystalline reflexion. In enunciating these, it 
 will be convenient to draw our transversals always through the 
 same origin 0, which we shall suppose to be the point of inci- 
 dence, as this point is common to all the rays, whether incident, 
 reflected, or refracted ; and we may imagine wave planes to be 
 drawn through the origin, parallel to the plane of 'each wave, 
 so that every transversal will lie in its own wave plane. The 
 incident and reflected wave planes will be perpendicular to the 
 incident and reflected rays, but the two refracted wave planes 
 will in general be oblique to their respective rays. In the latter 
 case, a right line drawn through the origin perpendicular to 
 
 * Annalen der Physik und Chemie, Vol. xxxvur. p. 276. 
 
 1" London and Edinburgh Philosophical Magazine, Vol. x. p. 43. (Supra, p. 84.) 
 
Reflexion and Refraction. 95 
 
 the wave plane is called the wave normal. It is scarcely 
 necessary to remark, that all the four wave planes intersect 
 the surface of the crystal in the same right line which is per- 
 pendicular to the plane of incidence; and that the angles of 
 refraction are the angles which the refracted wave normals 
 make with a perpendicular to that surface. The index of 
 refraction is the ratio of the sine of the angle of incidence to 
 the sine of the angle of refraction, just as in ordinary media; 
 but here it is a variable ratio, and has different values for the 
 same angle of incidence. I have elsewhere* shown how to find 
 the refracted rays and waves when the incident ray is given. 
 
 As we suppose the ethereal molecules to vibrate parallel to 
 the transversals, we may take the lengths of the transversals 
 proportional to the magnitudes or amplitudes of the vibrations ; 
 these lengths being always measured from the common origin 
 0. Then, in virtue of our fourth hypothesis, the transversals 
 will be compounded and resolved exactly by the same rules as 
 if they were forces acting at the point 0. 
 
 "We must now conceive a wave surface of the crystal, with 
 its centre at 0, the point of incidence. As the veloci- / 
 
 ties of rays which traverse the crystal in directions n f. 
 parallel to the radii of its wave surface are repre- 
 sented by those radii, so let a concentric sphere be 
 described with a radius OS, which shall represent, 
 on the same scale, the constant velocity of light in 
 the medium external to the crystal. At any point 
 T on the wave surface apply a tangent plane, on 
 which let fall, from 0, a perpendicular OG, meeting 
 the plane in G. On this perpendicular take the length Fig. 17. 
 OP from towards G, so that OP shall be a third proportional 
 to OG and the constant line 08. Then, while the point T 
 describes the wave surface, the point P will describe another 
 surface reciprocal! to the wave surface. This other surface may 
 
 * Irish Academy Transactions, Vol. xvn. p. 252. 
 
 f For the general theory of reciprocal surfaces, see Irish Academy Transactions, 
 Vol. xvn. p. 241. 
 
g6 On the Laws of Crystalline 
 
 very properly be called the index surface* because its radius 
 OP is the refractive index of the ray whose velocity is OT, or 
 rather of the wave TG, which belongs to that ray; for if we 
 conceive an incident wave, touching the sphere, to be refracted 
 into the wave TG, touching the wave surface in T, the sine 
 of the angle of incidence will be to the sine of the angle of 
 refraction as OS to OG, or as OP to OS ; so that, taking the 
 constant OS for unity, the index of refraction will be repre- 
 sented by OP. The wave surface and the index surface will 
 thus be reciprocal to each other, every point T on the one 
 having a point P reciprocally corresponding to it on the other. 
 It is remarkable that the transversal of the ray OT is per- 
 pendicular to the plane OPT; for in the theory of Fresnel, as 
 I formerly proved, f the direction of the vibrations is the right 
 line TG', and as I suppose the transversal to be perpendicular 
 to the vibrations of that theory, and to be, at the same time, in 
 the wave plane, which is perpendicular to OP, it follows that 
 the transversal must be perpendicular to both the right lines 
 TG and OP, and therefore perpendicular to their plane OPT. 
 Therefore conceiving the transversal to be drawn through 
 at right angles to the plane OPT, the plane of polarization 
 of the ray OT must needs pass through it. But there is 
 nothing else to fix the position of this last plane. We may 
 make it pass through the ray itself OT, as an ordinary media, 
 or we may draw it through the wave normal OP with Fresnel. 
 Or, instead of drawing it through either of these two sides of 
 the triangle OPT, we may make it parallel to the third side 
 PT. The last is what I should prefer, because the plane so 
 determined possesses important properties. I shall call it, how- 
 ever, the polar plane, because the name, plane of polarization, is 
 a long one ; and the signification of the latter may, if any one 
 
 * This is the surface which I formerly called (Trans., p. 252) the surface of 
 refraction; a name not sufficiently descriptive. Sir W. Hamilton has called it 
 the surface of wave slowness, and sometimes the surface of components. But the 
 name index surface seems to recommend itself, as both short and expressive. 
 
 t Ibid. Vol. xvi. p. 76. (Supra, p. 12.) 
 
Reflexion and Refraction. 97 
 
 chooses, be kept distinct, though in an ordinary medium both 
 terms must mean the same thing. The polar plane then of the 
 ray OT is a plane passing through its transversal and parallel 
 to the right line PT\ so that if OK be drawn parallel to PT 9 
 the polar plane will pass through OK. In general, to find the 
 transversals and the polar plane of any ray, we take the point 
 where the ray meets its own nappe of the wave surface, and join 
 it with the corresponding point on the index surface, drawing a 
 plane through the origin and the joining line. Then a right 
 line perpendicular to this plane at the origin will be the trans- 
 versal, and a plane drawn through the transversal parallel to 
 the joining line will be the polar plane. 
 
 Now let a polarized ray be incident at upon the crystal. 
 It will in general be divided into two rays. But each of these 
 rays in turn may be made to disappear by polarizing the inci- 
 dent ray in a certain plane. Let us suppose then that there is 
 only one refracted ray OT. In what direction must the incident 
 ray be polarised, or, in other words, what must be the position 
 of its transversal, in order that this may be the case ? and what 
 will be the corresponding transversal of the reflected ray ? The 
 answer is simple both transversals mil lie in the polar plane of 
 the refracted ray. Let us pursue this remark a little. 
 
 The refracted ray OT being given, we can find its polar 
 plane, and thence the intersections of this plane with the inci- 
 dent and reflected wave planes. These intersections will be the 
 positions of the incident and reflected transversals when OT is 
 the sole refracted ray. The refracted transversal lies also in 
 the polar plane ; and this transversal is, by our fourth hypo- 
 thesis, the diagonal of a parallelogram, whose sides are the 
 other two transversals, which determines the relative lengths 
 of the three transversals, or the relative amplitudes of the 
 vibrations. The intensities of the reflected and incident rays 
 are, of course, proportional to the squares of their transversals. 
 When the ray OT dissappears, we must take the polar plane of 
 the other ray, and proceed as before. 
 
 Thus there are, in the incident wave plane, two transversal 
 
 H 
 
9 8 On tJie Laws of Crystalline 
 
 directions which give only a single refracted ray. These, as well 
 as the corresponding ones in the reflected wave plane, may be 
 called uniradial transversals. They are the intersections of the 
 two refracted polar planes with the incident and reflected wave 
 planes. 
 
 "When the incident transversal does not coincide with either 
 of the uniradial directions, it is to be resolved parallel to them, 
 and then each component transversal will supply a refracted 
 ray, according to the foregoing rules. The reflected transver- 
 sals, arising from the component incident ones, are to be found 
 separately by the same rules, and then to be compounded. 
 
 In ordinary reflexion, if the incident transversal be in the 
 plane of incidence, or perpendicular to it, the reflected trans- 
 versal will be so likewise. But this does not hold in crystalline 
 reflexion. The general method just given will, however, enable 
 us to determine the positions and magnitudes of the reflected 
 transversals in these two remarkable cases ; and then, if we 
 choose, we can reduce any other case to these two, by resolving 
 the incident transversal in directions parallel and perpendicular 
 to the plane of incidence. 
 
 If we conceive a pair of incident transversals, at right angles 
 to each other, to revolve about the origin, it is evident that there 
 will be a position in which the reflected transversals correspond- 
 ing to them will also be at right angles to each other. There 
 is no difficulty in finding this position, and there will be an 
 advantage in using it when common unpolarized light is in- 
 cident on the crystal. For, the incident transversals being 
 rectangular, we may suppose the light to be equally divided 
 between them, and then the intensities of the corresponding 
 reflected portions can be found by the preceding rules. As 
 the reflected transversals are also rectangular, the sum of these 
 intensities will be the whole intensity of the reflected light, and 
 their difference will be the intensity of the polarized part of 
 it. This part will be polarized in a plane passing through the 
 greater of the two reflected transversals. 
 
 Common light will be completely polarized by reflexion when 
 
Reflexion and Refraction. 99 
 
 the two uniradial directions in the reflected wave plane coincide 
 with each other ; that is, when this plane and the two refracted 
 polar planes have a common intersection. For then, if the inci- 
 dent light he polarized, it is manifest that the reflected transver- 
 sal will lie in that intersection, whatever be the position of the 
 incident transversal ; and therefore if common light be incident, 
 with its transversals in every possible direction, the reflected 
 transversals will have but one direction. Thus the reflected light 
 will be completely polarized in a plane passing through the above 
 intersection. 
 
 Hence, as the reflected ray is perpendicular to its wave plane, 
 it follows that, at the polarizing angle of a crystal, the reflected ray 
 is perpendicular to the intersection of the polar planes of the two re- 
 fracted rays. The reflected transversal, as we have seen, is this 
 very intersection. This transversal is inclined, in general, to the 
 plane of incidence, and we have had occassion to speak of its in- 
 clination under the name of the deviation. If we now suppose the 
 double refraction to diminish until it disappears, the intersection 
 of the polar planes will at last coincide* with the refracted ray. 
 There will then be no deviation, and the reflected and refracted 
 rays will be at right angles to each other, agreeably to the law 
 of Brewster, which prevails at the polarizing angle of an ordi- 
 nary medium. 
 
 There is a case in which the construction that we have given 
 for determining the polar plane of a ray becomes useless. It is 
 when the ray T is a normal to the wave surface ; for then OP 
 coincides with T, and we cannot fix the transversal by our con- 
 struction. But it is precisely in such a case that the polar plane 
 is most easily ascertained, for it is then nothing more than the 
 plane of polarization of the common theory. For example, if we 
 take the ordinary ray of a uniaxal crystal, its polar plane will 
 pass through the ray itself and the axis of the crystal. Of course 
 in an ordinary medium the polar plane and the plane of polari- 
 zation are synonymous. 
 
 * For the polar planes will become two planes of polarization at right angles to 
 each other. 
 
 H2 
 
ioo On the Laws of Crystalline 
 
 It may not be amiss to apply our general ' rules to the case 
 of ordinary reflexion and refraction. Suppose then a polarized 
 ray to fall on the surface of an ordinary medium. Draw a plane 
 through the incident transversal and the refracted ray ; this will 
 be the plane of polarization of the refracted ray, and it will 
 intersect the reflected wave plane in the reflected transversal. 
 The refracted transversal will be the diagonal of a parallelo- 
 gram, whose sides are the other two transversals; hence we 
 have the relative lengths of the transversals, and thus every- 
 thing is determined.* 
 
 * This construction was mentioned at the meeting of the British Association in 
 Dublin. See the Keports of the Association, or London and Edinburgh Phil. Mag. 
 vol. vii. p. 295. The following is an extract from the Paper which I read at that 
 meeting : 
 
 " The formulae given by Fresnel for the same purpose will be found to agree 
 exactly with this rule, in determining the positions of the planes of polarization ; 
 and his expression for the amplitude of the reflected vibration is also in accordance 
 with our construction. But the coincidence does not hold with regard to the am- 
 plitude of the refracted vibration, though the vis viva of the refracted ray is the 
 same in both theories. 
 
 " Now it is very remarkable that if we alter the hypotheses of Fresnel where 
 they are at variance with the preceding principles, we shall, from his own equa- 
 tions of condition, deduce formulae agreeing in every respect, even as to the ampli- 
 tude of the refracted wave, with the construction which we have accounted for in 
 a different way (i. e. by using the relation among the pressures instead of the law 
 of vis viva}. The requisite alterations are two in number. First, the vibrations 
 are to be supposed parallel to the plane of polarization, and not perpendicular to it, 
 as Fresnel conceived ; and secondly, the density of the ether is to be considered the 
 same in both media, from which it follows, that the corresponding ethereal masses, 
 imagined by Fresnel, are to each other as the sine of twice the angle of incidence 
 to the sine of twice the angle of refraction. Substituting in Fresnel's equations of 
 condition this value of the ratio of the masses, we obtain the formulas which I am 
 inclined to regard as correct." 
 
 The equations spoken of in this extract are those which arise from the prin- 
 ciple of vis viva, and from the equivalence of vibrations parallel to the separating 
 surface of the two media. But it is worth while to observe, that when the vibra- 
 tions are all in the same direction, that is, when the light is polarized perpendicular 
 to the plane of incidence, the very same formulae will come out from Young's re- 
 markable analogy of the two elastic balls, one of which impinges directly on the 
 other, supposed previously at rest, the masses of the balls being to each other in the 
 ratio of the ethereal masses mentioned above. And, perhaps, this consideration 
 affords the simplest possible explanation of Brewster's law relative to the pola- 
 
Reflexion and Refraction . I o I 
 
 The reason of this construction will be evident, if we con- 
 sider that, in an ordinary medium, the polar plane is the same 
 as the plane of polarization ; and that, when there is only one 
 refracted ray, the three transversals lie in the polar plane of 
 that ray, according to the general remark with which we set 
 out. We now proceed to show that the theorem asserted in 
 this remark is a consequence of our hypotheses, and we shaU 
 afterwards deduce a few results which may be readily compared 
 with experiments. 
 
 Let us suppose then that the direction of the incident trans- 
 
 rizing angle ; for, as there is no reflected motion when the balls are equal, the whole 
 velocity of impact being communicated to the ball that was at first quiescent, so 
 there is no reflected vibration when the ethereal masses are equal ; that is, when 
 the sine of twice the angle of incidence is equal to the sine of twice the angle of 
 refraction, or when the angles of incidence and refraction are together equal to a 
 right angle. The whole of the incident vibration then passes into the refracted 
 ray. In general, if t 1} t a denote the angles of incidence and refraction, the masses 
 of the imaginary balls will be as sin 2t 1? to sin 2t a ; and, if the velocity of the ori- 
 ginal impact be taken for unity, the common theory of the collision of elastic bodies 
 will give 
 
 6^2^-8^2*2 tan(i x -i 2 ) 
 
 sin 2t x + sin 2t 3 tan (^+0* 
 
 for the velocity retained by the impinging ball after the impact ; and 
 2 sin 2t sin 2t, 
 
 ^ . ^ 1 QJ. *-_ , /JT \ 
 
 sin 2t x + sin 2i a sin (i x + tj cos ( tl - t 2 ) ' 
 
 for the velocity communicated to the other ball. These expressions (i.) and (n.), 
 are the same as the values of T 3 and r 2 , which we should deduce from equations (1) 
 and (2), on the next page, by supposing T X to be unity, and the angles 0i, 2 , 03 to 
 be right angles. The general construction given in the text will lead to the same 
 results, if we find from it the limiting ratios of the transversals, on the supposition 
 that their directions approach each other indefinitely, and ultimately coincide in a. 
 right line perpendicular to the plane of incidence. 
 
 When the transversals are all in the plane of incidence, or when the light is 
 polarized in that plane, the incident, the reflected, and the refracted transversals 
 are to each other as sin (t^^ + tj, sin^-t,), and sin 2t 2 respectively; because each 
 transversal is proportional to the sine of the angle between the other two; and, in 
 the present case, the angle between any two transversals is equal to the angle 
 between the corresponding rays. Hence, taking the incident transversal for unity, 
 the reflected transversal is 
 
IO2 On the Laws of Crystalline 
 
 versal is such that there is only one refracted ray. It is evi- 
 dent that, in this case, the three transversals must lie in the 
 same plane, since, by the fourth hypothesis, the refracted vibra- 
 tion is the resultant of the other two vibrations ; and, therefore, 
 we have only to prove that the plane of the transversals is the 
 same as the polar plane of the refracted ray. Let TI, r 2 , r 3 be 
 the respective lengths of the incident, refracted, and reflected 
 transversals ; let ft, ft, ft be the angles which they make with 
 the plane of incidence, the angle ft being known from the 
 theory of Fresnel; put d, 2 , t 3 for the angles made by the 
 respective wave planes with the surface of the crystal, and 
 m^ m 2 , m z for the relative quantities of ether set in motion 
 by each wave. Then our hypotheses will give us the four 
 following equations : 
 
 m 3 r 3 2 , 
 
 TI sin ft + r 3 sin ft = r 2 sin ft, (2) 
 
 TI COS ft COS ti + T 3 COS ft COS 3 = T 2 COS 2 COS 2 , (3) 
 
 TI cos ft sin <i + r 3 cos ft sin * 3 = r 2 cos ft sin < 2 . (4) 
 
 The first equation is manifestly the translation of the law of 
 
 and the refracted transversal is 
 
 It has been already observed that our theory differs from that of Fresnel with 
 regard to the magnitude of the refracted transversals. The expressions (n.) and (iv.) 
 
 must, in fact, be multiplied each by - in order to produce the corresponding e x- 
 
 pressions which result from Fresnel' s hypotheses. But the two theories also differ 
 as to the relative directions of the incident and reflected transversals. For, suppos- 
 ing the light to fall upon the denser medium, or j x to be greater than t 2 , our con- 
 struction indicates that these transversals, when the angle of incidence is small, 
 point in the same direction ; whereas Fresnel concludes the contrary to be the case. 
 However, the disagreement in this respect ceases as we approach the limiting inci- 
 dence of 90 ; for then, according to both theories, the incident and reflected trans- 
 versals point in opposite directions. This last conclusion is conformable to the 
 inference which Professor Lloyd has drawn from his experiments on the interfer- 
 ence of direct light with light reflected at a very oblique incidence. See Irish 
 Acad. Transactions, VOL. xvn. p. 176. 
 
Reflexion and Refraction. 103 
 
 the preservation of vis viva ; the other three are obtained from 
 the principle of equivalent vibrations, by resolving the vibra- 
 tions, or transversals, in three rectangular directions. In the 
 second equation, the transversals are resolved perpendicular to 
 the plane of incidence ; in the fourth, perpendicular to the sur- 
 face of the crystal ; and in the third equation they are resolved 
 parallel to the intersection of these two planes. When the angles 
 0i, 02, 03 begin, the transversals are in the plane of incidence 
 in such a relative position, that, if they were turned round 
 together in that plane through a right angle, they would point 
 each in the direction of its own wave's progress. These angles 
 increase on the same side of the plane of incidence, and range 
 through the whole circumference. The angles ii, 2 , 3 are those 
 of incidence, refraction, and reflexion ; but, for the sake of sym- 
 metry, they are taken to be the angles which the wave normals, 
 drawn from the origin in the direction of each wave's motion, 
 make with the perpendicular to the surface, this perpendicular 
 being directed towards the interior of the crystal. Thus it 
 happens that t 3 is the supplement of *i. Attending to this 
 circumstance, equations (3) and (4) give us 
 
 T! COS 0i - r 3 COS 3 = r 2 COS 2 
 
 n a 
 
 TI COS 0i + T 3 COS (7 3 = T 2 COS 
 
 COS ti 
 
 sin e 2 
 
 sin <! 
 and, by adding and subtracting these, we find 
 
 cos 2 sin (/i + * 2 ) ->l 
 
 V (6) 
 
 mAoa-fti-n) I 
 
 which values if we substitute in equations (1) and (2), observing 
 that m 3 = m h as is evident, we shall get 
 
 sin 2 (<i + 2 ) sin 2 (i t - t a ) m z sin 2 2^ 
 
 cos 2 0! cos 2 3 mi cos 2 
 
 (7) 
 
IO4 On the Laws of Crystalline 
 
 sin (j + a ) tan 0! - sin (^ - * 2 ) tan 3 = sin 2*i tan 2 . (8 
 
 Subtracting from (7) the identity 
 
 sin 2 (i! + ia) - sin 2 (/ t - 1 2 ) = sin 2ii sin 2/ 2 , 
 there remains 
 
 sin 2 (i! + i a ) tan 2 0! - sin 2 (^ - i 8 ) tan 2 3 
 
 sin 2 2i! / w a sin 2i 2 - \ 
 = TTT - -^-x- cos 2 02 ; (9) 
 cos 2 02 \m l sin 2ix V 
 
 and this, by making 
 
 m 2 sin 2< 2 + 2/i sin 2 2 
 
 sinSti 
 
 (10 
 
 becomes 
 
 sin 2 (i! + 1 2 ) tan 2 0! - sin 2 (ti - 1 2 ) tan 2 3 = sin 2^ (sin 2* 2 + 2k) tan 2 2 , (11) 
 
 which is divisible by equation (8), the quotient being 
 
 sin (i! + 2 J tan 0i + sin (^ - i a ) tan 3 = (sin 2/ 2 + 2k) tan 2 . (12) 
 
 Then, by adding and subtracting equations (8) and (12) we 
 obtain 
 
 , /> A tan 2 ""i 
 
 tan 0! = cos (i!- ta) tan 2 + -, . , 
 
 + > (13) 
 
 tan 3 = - cos (i ! + i z ) tan 2 + -. -. 2 . 
 
 sm (/! - / 2 ) J 
 
 These equations give the positions of the incident and reflected 
 transversals when h is known. 
 
 Now let the directions in which the transversals have been 
 resolved in equations (2), (3), (4), be taken for the axes of ,#, y 
 respectively ; so that, the origin being at 0, fhe plane of xy may 
 be the plane of incidence, and the axis of x may lie in the sur- 
 face of the crystal. And, the reflected ray being conceived to lie 
 within the angle made by the positive directions of x and y, let 
 the initial condition that we have assumed for the angles 0i, 2 , 
 3 be satisfied by supposing that, when these angles begin, the 
 
Reflexion and Refraction. 105 
 
 transversals r ly r a lie between the negative directions of x and 
 ?/, and the transversal r 3 between the directions of + x and - y. 
 Then if ft, ft, ft be reckoned towards the positive axis of 2, so 
 that each angle may be 90 when the corresponding transversal 
 points in the direction of 2 positive, the equations of the trans- 
 versal TI will be 
 
 tan ft cos ti sin i! 
 and those of r 3 will be 
 
 _JL_ = __ = _-_, (15) 
 
 tan ft cos ii sin ii 
 
 Let 
 
 2 + Ax + By = (16) 
 
 be the equation of a plane passing through the directions of r 1? 
 r 2 and r 3 . To determine A and I?, let the variables be eliminated 
 from this equation by means of (14) and (15) successively, and 
 we shall get the two equations of condition, 
 
 tan ft - A cos i\ B sin ti = 0, 
 
 (17) 
 tan ft + A cos t - B sin 1 1 = ; 
 
 which, by addition and subtraction, give 
 
 _ tan ft + tan ft 
 
 2 sin tl 
 
 tan ft - tan ft < 
 2 cos i, ' 
 
 (18) 
 
 substituting, in these values, the expressions (13) for tan ft, 
 
 tan ft, we have 
 
 cos* 2 
 
 B = tan ft sint 2 + - 
 
 sm 2 *i - sm*i 2 , 
 
 (19) 
 A = tan ft I cos* 2 - .** 2 j; 
 
 whence, by making 
 
 tan K = . 8 A . 2 , (20) 
 
io6 On the Laws of Crystalline 
 
 we find 
 
 B tam 2 + tan K 
 
 - = tan-(c 8 + K). (21) 
 
 A 1-tan/otanfc 
 
 But if s = in (16), we have 
 
 for the equation of the right line in which the plane of the trans- 
 versals intersects the plane of incidence. This right line, lying, 
 like the refracted wave normal, between the directions of + x 
 and - ?/, makes with the direction of - y an angle v which ob- 
 
 Tt 
 
 viously has ^ for its tangent; and therefore, by (21), 
 
 v = < 2 + K ; (23) 
 
 which shows that the intersection of the two planes is inclined 
 to the refracted wave normal at an angle equal to K. 
 
 We must now find the value of /?, which depends on the rela- 
 tive ethereal masses put in motion by the incident and refracted 
 waves. Conceiving the incident and refracted rays to be cylin- 
 drical pencils, having of course a common section in the plane 
 of xz, which is the surface of the crystal, let each pencil be cut 
 by a pair of planes parallel to its wave plane, and distant a 
 wave's length from each other ; then the cylindrical volumes so 
 cut out will represent the corresponding masses, since, by our 
 second hypothesis, the densities are equal. These volumes are 
 to each other in the compound ratio of their altitudes, whieh are 
 the wave lengths, and of the areas of their bases. The altitudes 
 are evidently as sin *i to sin t a . The first base is a perpendicular 
 section of the incident pencil ; the second base an oblique section 
 of the refracted one, the obliquity being equal to the angle c at 
 which the wave normal is inclined to the ray. The perpendi- 
 cular sections are to each other as the cosines of the angles which 
 they make with the common section of the cylinders, or as cos i } 
 to cos t( 2 ); putting ( 2 ) for the angle which the refracted ray makes 
 with the negative direction of y. The second base is greater 
 than the perpendicular section of the refracted pencil in the pro- 
 
Reflexion and Refraction . 107 
 
 portion of unity to cos e. Therefore, compounding all these 
 ratios, we find 
 
 m<> sin t 2 cos IM 
 
 = -: LL -. (24) 
 
 nil sin ti cos ti cose 
 
 The same result may be otherwise obtained by observing that, 
 in a system of waves, the corresponding masses are proportional 
 to the ordinates y of the points where the rays meet their wave 
 surfaces. By a system of waves, I mean an incident wave with 
 all that are derived from it by reflexion or refraction at the same 
 surface of the crystal, or at parallel surfaces. If, at the point 
 where the incident ray intersects its spherical wave surface, we 
 apply a tangent plane intersecting the plane of xz in a right 
 line parallel to s, through which right line other planes are 
 drawn touching the wave surface of the crystal in four points, 
 these tangent planes will be the waves derived from the inci- 
 dent wave which touches the sphere ; and the points of contact, 
 including that on the sphere, will be the points where the rays 
 meet the wave surfaces. Then the corresponding masses will 
 be represented by prisms having a common rectangular base in 
 the plane a?s, one side of this rectangle being the distance, on 
 the axis of x, between the origin and the common intersec- 
 tion of the tangent planes; and the triangular face of each 
 prism having the same distance for one side, and a point of con- 
 tact for the opposite angle. These prisms, as they have a com- 
 mon base, will be proportional to their altitudes, which are the 
 ordinates y of the points of contact, The expression (24) may 
 be easily deduced from this relation. 
 
 Let OT, OP, and the negative direction of y meet the sur- 
 face of the wave sphere (described with the 
 radius 08) in the points T,, P,, Y, ; and let 
 the right line, in which the plane of the trans- 
 versals intersects the plane of incidence, meet 
 the sphere in L,. Then the points Y^ P^ L,, Fi g- 18 
 
 being all in the plane of incidence, will be on the same great 
 circle Yf^L/, and drawing the great circles TP /? Y,T /9 we 
 
io8 On the Laivs of Crystalline 
 
 shall have F / P / = <,, Y,T, = ( < 2) TP, = , T / Z / = v = t 2 + *, by 
 (23); whence P,,= K . 
 
 As the transversal r 2 is perpendicular to the plane OTP, or to 
 the plane of the great circle Tf^ the cosine of the spherical 
 angle Tf,Y f is the sine of 2 ; and therefore, from the triangle 
 Tf f Yfi we have 
 
 cos 1(2) = cos ( 2 cos + sin 2 sin sin 2? (25) 
 
 which being substituted in (24), gives 
 
 nh sin 2t 2 + 2 sin 2 < 2 sin 2 tan e 
 #h sin 2ti 
 
 and comparing this result with (10), we find 
 
 sin 2 * 2 tan 
 
 (26) 
 
 h = ^ ; (27) 
 
 sm0 2 
 
 whence, and from (20), it follows that 
 
 sin 2 < 2 tan e 
 
 tan K = J-T-Z - ~ 2 \ /i- 28 
 
 2 
 
 Draw the great circle Z^ff^ at right angles to Tf^ and meet- 
 ing it in K, ; then the plane of L,K, will be the plane of the 
 transversals, since the latter plane passes through L^ and is per- 
 pendicular to Tf r But the tangent of P,E, is equal to the 
 tangent of P^L, multiplied by the cosine of the angle P / or by 
 the sine of 2 ; therefore, denoting P,K, by 1, and recollecting 
 that P, = fc, we find, by (28), 
 
 tan sin 2 ii - sm 2 t 2 
 
 (29 ) 
 
 Now we have seen that the ratio of OP to OS, or OS to OG 
 (Fig. 17), is the index of refraction ; so that shrt, is to sin 2 t 2 as 
 OP to OG. Therefore, by (29), 
 
 tan fl OG OG 
 
 tan OP-OG 6P' 
 
 (-30) 
 
Reflexion and Refraction. 109 
 
 but OG is to GP as the tangent of the angle GPT is to the tan- 
 gent of the angle GOT', and since 6 is the angle GOT, it follows 
 that fi is equal to the angle GPT or KOP. Consequently, OK 
 will meet the surface of the sphere in the point J5T,. Thus we 
 have proved our assertion, that, when there is only one refracted 
 ray, the plane of the transversals is the polar plane of that ray. 
 
 The sign of the quantity h is always the same as that of the 
 cosine of the spherical angle T^P^Y^ But to remove all ambi- 
 guity respecting signs, we must make a few additional conven- 
 tions. Supposing, as we have hitherto done, that the refracted 
 light moves from to T, and conceiving a right line to be 
 drawn from the origin parallel to GT, and directed from G 
 towards T, let the angle 2 , which this right line makes with the 
 plane of incidence, be reckoned, like 0,, 2 , from an initial posi- 
 tion comprised between the negative directions of x and y ; and 
 let S 2 , like the angles Lt 2 , 3 , increase on the side of z positive, 
 and range from to 360. Then Sr z will always be equal either 
 to the angle P / of the spherical triangle Tf 4 F,, or to the re- 
 entrant angle, which is the difference between P 4 and 360. In 
 either case, the cosine of 3" 2 will be the same, both in magnitude 
 and sign, as the cosine of the angle Tf,Y,. Consequently, if, 
 instead of (25), we use the direct trigonometrical formula 
 
 cos t( 2 ) = cos 't z cos e + sin i 2 sin e cos 2 , (31) 
 
 we shall find 
 
 sin 2 i 2 tane 
 ~stfO~~ 
 
 showing that the sign h is always the same as the sign of cos 3v 
 Now as 2 differs from S 2 by a right angle, we will suppose 
 
 2 = S 2 + 90, (33) 
 
 and then we shall have sin 2 = cos S 2 , algebraically as well as 
 numerically. Thus we see that, by adopting these conventions, 
 the value of h in (27) will have the proper sign. Therefore, 
 substituting this value of h in formulae (13), we obtain 
 
i io On the Laws of Crystalline 
 
 sin 2 ( 2 tan 
 
 tan ft = cos (ii - 2 ) tan ft -r- j. ; , . , , 
 
 t (34) 
 a fl sin 2 ti tan E 
 
 tan ft = - cos (ii + i a ] tan ft + : , 
 
 cos ft sin (d- t 2 ) J 
 
 These formulae give the uniradial directions, or the positions 
 of the incident and reflected transversals, when the sole refracted 
 ray is that with which we have been occupied. The like direc- 
 tions, when the other ray exists alone, will be given by the 
 formulae 
 
 n, / , N , & sinY 2 tan * 
 
 tan i = cos Ui - 1 2 ) tan 6 2 + 7^7 --;, N ' 
 
 cos 2 sin (u + 1 2 ) 
 
 > (35) 
 
 * O / 1 / 
 
 BIT*** TQTI c 
 
 . frf s / \ J_ /!/ O1JJL t 2 tCtU. c 
 
 tan = - cos ftj + 1 2 ) tan 2 + f . 
 
 cos 2 sm (ti - t l ) J 
 
 where all the quantities, except i l9 which remains the same, are 
 marked with accents, to show that they belong to the second 
 refracted ray. 
 
 The uniradial directions having been found by these equa- 
 tions, the relative magnitudes of the uniradial transversals are 
 determined by equations (6). When the incident transversal 
 is not uniradial, it is evident, as we said before, that it may be 
 resolved* in the two uniradial directions ; that each component 
 
 * That, if an incident transversal be resolved in any two directions, the reflected 
 and refracted transversals arising from it will be the resultants of those which would 
 arise from each of its components separately, is a principle which appears very 
 evident, insomuch that we can hardly suppose it to be untrue, without doing 
 violence to our physical conceptions. Nevertheless, it is necessary to prove that 
 this principle is not contrary to the law of vis viva ; for though the vis viva may be 
 preserved by each set of components (as it is when these are uniradial), yet we 
 cannot therefore conclude that it will be preserved by their resultants. Here then 
 is a test of the consistency of our theory ; for we are bound to show that the law of 
 vis viva is not infringed by the adoption of the principle in question. Now it is 
 easy to see that, whatever be the two directions in which the incident transversal 
 is resolved, the final results will always be the same ; because, taking the compo- 
 nent in each of these directions separately, the reflected and refracted transversals 
 belonging to it must be obtained, in the first place, by the help of a resolution per- 
 
Reflexion and Refraction. 1 1 1 
 
 transversal, as if the other component did not exist, will furnish 
 a refracted ray and a partial reflected transversal uniradial in its 
 direction; and that the total (or actual) reflected transversal will 
 be the resultant of the two partial ones. 
 
 When B 3 = 0' 3 , the partial reflected transversals will coin- 
 cide, and their resultant will have a fixed direction, independent 
 of the direction of the incident transversal. The angle of inci- 
 dence at which this takes place is the polarizing angle, and the 
 common value of 3 and 0' 3 is the deviation. If, at the polarizing 
 angle, the partial reflected transversals be equal in magnitude, 
 and opposite in direction, their resultant will vanish, and the 
 reflected ray will disappear. This will happen when the inci- 
 dent transversal is in the plane of the two refracted transversals, 
 and therefore in the intersection of this plane with the incident 
 wave plane; for, when there is no reflected ray, the incident 
 transversal alone must be equivalent to the two refracted trans- 
 versals. 
 
 Since the reflected transversal can be made to vanish at the 
 polarizing angle, this angle might be found directly by putting 
 the vis viva of the incident ray equal to the sum of the vires vivce 
 of the two refracted rays, and by making the incident trans- 
 versal the resultant of the two refracted transversals. Eesolv- 
 ing the transversals parallel to the axes of co-ordinates, these 
 conditions would give four equations, from which we could 
 
 formed in the uniradial directions. "We need not, therefore, consider any case but 
 that in which the resolution is uniradial throughout. 
 
 The incident transversal being denoted by T\, let T 3 be the reflected transversal 
 determined by the rules given in the text ; and let the uniradial components of the 
 former be n, T'I, while those of the latter are TS, r'a. Then will 
 
 2V = Ti 2 + r'i 2 + 2-riT'i cos (0! - 0'i), 
 T 3 2 = T 3 2 -f r' 3 2 + 2T 3 T' 3 cos (03 - 0' 3 ) ; 
 
 where the signification of 0i, 0'i, 3 , 0' 3 is the same as in the text. The vis- viva of 
 one refracted ray is m\ (n 2 - r 3 2 ), and that of the other is m\ (TV - r' 3 2 ) ; there- 
 fore the vis viva of both refracted rays is 
 
1 1 2 On the Laws of Crystalline 
 
 eliminate the two ratios of the three transversals, together with 
 the angle at which the incident transversal is inclined to the 
 plane of incidence. In the equation produced by this elimina- 
 tion, the angle of incidence would be the polarizing angle, and 
 the other quantities would be known functions of that angle ; 
 whence the angle itself would be known. 
 
 a quantity which ought to be equal to 
 
 mi (2V - TV) ; 
 and consequently the equation 
 
 TIT'I cos (0i - 0'i) = TST'S cos (63 - 0' 3 ) (v.] 
 
 ought to be true, This equation, by help of the expressions (6) for n, ra, and the 
 like expressions for T'I, r'z, becomes 
 
 sin (t 1 + i a ) sin (t 1 + t' 2 ) (1 + tan 5 tan 0'J 
 
 = sin (t x - 1 2 ) sin (t x - t' 2 ) (1 + tan 3 tan 9' 3 ) ; (vi.) 
 
 which again, by substituting the values (13) and the other similar values, is 
 changed into 
 
 sin (i a + i' 2 ) {cos (t 2 - i' a ) + cotan 2 cotan 0' a } + A i h' = 0. (vn.) 
 
 where h 1 denotes for one refracted ray what h denotes for the other, the value of h 
 being given by formula (27), and that of h' by the same formula with accented 
 letters. The angle of incidence, we may observe, has disappeared from the 
 equation. 
 
 If, therefore, the laws of reflexion, which we have endeavoured to establish, are 
 consistent with each other, this last equation must be satisfied by means of the rela- 
 tions which the laws of propagation afford ; or rather, the equation must express a 
 property of the wave surface of the crystal, however strange it may be thought that 
 such a property should be derived from the laws of reflexion laws which would 
 seem, at first sight, to have no connexion at all with the form of the wave surface. 
 Now I have found that the equation (vn.) really does express a rigorous pro- 
 perty of the biaxal wave surface of Fresnel ; a very curious fact, which not only 
 shows that the laws of reflexion and the laws of propagation are perfectly adapted 
 to each other, but also indicates that both sets of laws have a common source in 
 other and more intimate laws not yet discovered. Indeed the laws of reflexion are 
 not independent even among themselves ; for the expressions (in.) and (iv.) in the 
 note on ordinary reflexion (page 101) have been deduced solely from the principle of 
 equivalent vibrations, and yet they satisfy the law of vis viva. Perhaps the next step 
 in physical optics will lead us to those higher and more elementary principles by 
 which the laws of reflexion and the laws of propagation are linked together as parts 
 of the same system. 
 
Reflexion and Refraction. 
 
 It deserves to be remarked, that, at any angle of incidence, 
 if the incident and reflected wave planes be intersected by a 
 plane drawn through the two refracted transversals, the inter- 
 sections will be corresponding transversal directions ; that is to 
 say, if the incident transversal coincide with one intersection, 
 the reflected transversal will coincide with the other. For it is 
 evident, from our fourth hypothesis, that if three of the trans- 
 versals be in one plane, the fourth transversal must be in the 
 same plane. 
 
 We come now to apply our theory to the case of uniaxal 
 crystals ; and, in doing so, we shall take the crystal to be of the 
 negative kind, like Iceland spar, so that the ordinary refraction 
 will be more powerful than the extraordinary. On the sphere 
 described with the centre and radius OS, let XY be a great 
 circle in the plane of incidence, the radii OX, OY being the po- 
 sitive directions of the co-ordinate axes of x and y. Suppose the 
 right lines iO and Oi', intersecting the sphere in i and /, to be 
 the incident and reflected rays; let the ordinary refracted ray and 
 the extraordinary wave normal be produced backwards from to 
 meet the sphere, at the side of the incident light, in the points o 
 and e respectively ; let the right line OA, cutting the sphere in 
 A, be the direction of the axis of the crystal ; and draw the great 
 circles Ao, Ae, A Y. The points i, e, o, i' are all on the circle 
 XY. The point E, where the extraordinary ray OE produced 
 backwards meets the sphere, will be on the circle Ae ; and if, as 
 in the figure, the arc Ae be less 
 
 than a quadrant, the point e will 
 lie between A and E. The polar 
 plane of the ordinary ray is ob- 
 viously the plane of the circle 
 Ao ; but the polar plane of the 
 other ray must be found by a 
 construction. On the arc AeE 
 take the portion ef, so that the point e may lie between the 
 points E and/, and so that the tangent of tf/may be to the tan- 
 gent of Ee as the square of the sine of the arc eYis to the dif- 
 
 i 
 
 Fig. 19. 
 
1 1 4 On the Laws of Crystalline 
 
 ference between the squares of the sines of ^Fand eY. Through 
 /draw the great circle ft perpendicular to the circle AcE\ and 
 it is manifest from (29) that the plane of ft is the polar plane of 
 the extraordinary ray. On each circumference Ao and ft, the 
 points which are distant 90 from i and i' 9 the distances being 
 measured by arcs of great circles, are the points where the unira- 
 dial transversals, prolonged from the centre, intersect the sphere. 
 Let Ao and ft intersect each other in t, and let ti f be an arc of a 
 great circle connecting the point t with the point i'. When the 
 connecting arc t% is a quadrant, the two uniradial transversals, 
 belonging to the reflected ray, coincide with each other and with 
 the right line Ot ; the angle of incidence is then the polarizing 
 angle ; the plane of ti' is the plane of polarization of the reflected 
 ray ; and the angle ti'Yis the deviation. 
 
 To find the equations appropriate to uniaxal crystals, we may 
 suppose formulae (34) to belong to the ordinary, and formulae 
 (35) to the extraordinary ray. Then will e = 0, and c' = the arc 
 Ee. Putting and 0' for the spherical angles Aoi and Aei, we 
 shall easily see that 2 = + 180, and 0' 2 = & + 90, if we con- 
 ceive the point A and the positive axis of z to be both on the 
 upper side of the plane XOY. And if a/ denote the arc Ae, 
 while b and a respectively express the reciprocals of the prin- 
 cipal indices, ordinary and extraordinary, the law of Huyghens, 
 for the double refraction of uniaxal crystals, will give us 
 
 tan '= - sin a/ cos a/, (36) 
 
 w. 
 
 here 
 
 sinV 2 
 sin t\ 
 
 Observing these relations, we have, from (34), 
 
 tan 0! = cos (t! - i 2 ) tan 0, ) 
 
 / (oo) 
 
 tan 3 = - cos (i l + i t ) tan 0, j 
 
 for the ordinary ray ; and from (35) we get 
 
Reflexion and Refraction. 115 
 
 f\f I r \ 4. & I z i2\ S i n w ' COS W ' Sm2 1 \ 
 
 tan V i = - cos Ui - / 2 ) cotan - (a z - cr) . -f^. - t J-T-, } 
 
 sm v sm (ii + LZ) / 
 
 > (39) 
 A/ / r\ /I//-. 7 9\ sin w' cos w' sin 2 *! ( 
 
 tan 9 3 = cos [4 + 1 2 ) cotan - ( 2 - 5 2 ) : -^. rr-, i 
 
 sin 6r sm (t| - 1 2 ) / 
 
 for the extraordinary ray. 
 
 The four preceding equations determine the uniradial direc- 
 tions ; and the following equation, 
 
 , / r\ n , 70N sn(icos(sn ,. M 
 
 cos (f i + 1 2 ) tan + cos (i l + 1 2 ) cotan -(a 2 - b z ) - ^= 0, (40) 
 
 sin v sin (<i t z) 
 
 obtained by putting tan 3 = tan r 3 , is that which determines the 
 polarizing angle. 
 
 In making use of this last equation to deduce the law of the 
 polarizing angles in various positions of the axis of the crystal, 
 we shall confine ourselves to the case in which the reflexion from 
 the crystal takes place in air, because the angle ii - t' 2 will then 
 be considerable, and the quantities cos (ii + i z ) and cos (ti + i z ) 
 will be small, so that it will be easy to arrive at approximate 
 results. For we shall have, in the first place, 
 
 cos (f t + 1' 2 ) = cos (ii + i a ) - (i t -( 2 ), (41) 
 
 nearly, since d + 2 will not differ much from a right angle ; and 
 because 
 
 sin t z = b sin 1 1, sin i z = s sin ii, (42) 
 
 we shall also have, rigorously, 
 
 sin 2 1', - sin 2 1 2 = (s 2 - b*) sin 2 l = (a* - b z ) sin 2 a/ sin 2 i l9 (43) 
 
 or 
 
 . /, x t , ,v sin 2 tu' sin 2 1! 
 sm( t .-.^(a'-y) 8in(tU(i) . (44) 
 
 which may be written 
 
 , / 9 TON sin 2 w"sin 2 ii 
 
 with sufiicient accuracy. This value of t z - iz having been sub- 
 stituted in (41), the resulting expression for cos (ti + i'z) must be 
 
 i2 
 
1 1 6 On the Laws of Crystalline 
 
 substituted in equation (40), which will then become 
 
 sin <> cotan 
 
 / QII 
 
 cos (/! + 2 ) (tan 9 + cotan 0) - ( 2 - & 2 ) sin 2 ^ sin wf - 
 
 sin 2/2 
 
 + ^2^--> ^ 
 
 if, denoting the arc Ao by w, we confound </ with w, 0' with 0, 
 and write cos 2/ 2 instead of sin (ii 1'). Multiplying all the 
 terms of (46) by sin cos 0, we find 
 
 ( 2 - tf) sin 2 t ! sin w cos 
 
 ~ 
 
 sm 2 it cos 2t 2 
 
 From A draw the arc AR meeting the arc iY at right angles 
 in the point R, and put RY=p, AR - q. Then by means of the 
 values 
 
 2 , 
 sin w cos 9 = cos # sin (p - 1%) , 
 
 afforded by the right-angled triangle ARo, the equation (47) 
 will take the form 
 
 / x i . , v / \ /^nx 
 
 cos (i + 2 ) = -^^ IT- cos 2 ^ sm (p - 1) sm (;j + 1), (49) 
 cos /c/2 
 
 or 
 
 cos (i 4 2 ) = Kvvtfq (sin 2 /? - sin 2 t 2 ), (50) 
 
 where 
 
 jr=(a> 2fta-V)- ; (51) 
 
 this value of JT being found by assuming tan 2 = cotan f! = b, 
 which is accurate enough for the purpose. 
 
 Thus we have obtained ii + i 2 , or the sum of the polarizing 
 angle and the angle of ordinary refraction. The former angle 
 itself may be inferred from formula (50) by help of the relation 
 sin i 2 = b sin i r . In this way, if we use zs l instead of ci to dis- 
 tinguish the polarizing angle from other angles of incidence, 
 and if we put 
 
 *-r^'w^j' ( 52 > 
 
Reflexion and Refraction. 117 
 
 we shall find 
 
 wi = -& - k cos 2 ^(sin 2 p - sin 2 2 ), (53) 
 
 in which us is the angle whose cotangent is equal to b ; in other 
 words, 73 is the polarizing angle of an ordinary medium whose 
 refractive index is equal to the ordinary index of the crystal. 
 
 This result accounts for a remarkable fact observed by Sir 
 David Brewster, who, in the year 1819, led the way in the ex- 
 perimental investigation of the laws of crystalline reflexion. He 
 found that the polarizing angle remains the same when the crystal 
 is turned round through 180, though one of the angles of refrac- 
 tion is changed, and though the situation of the refracted rays, 
 with respect to the axis of the crystal, becomes quite different 
 from what it was. This circumstance, which surprised me when 
 1 first met with it, is an immediate consequence of formula (53) ; 
 for the effect of a semi-revolution of the crystal is to change the 
 signs of p and q ; but the nature of the formula is such that 
 these changes of sign do not alter the value of CTL Neither is 
 that value altered by turning the crystal until the azimuth, as 
 the spherical angle A Yi is usually called, is changed into its 
 supplement ; for then the sign of p alone is affected. 
 
 Another remark, made by the same distinguished observer, 
 is also a consequence of formula (53). From his experiments 
 it appears that, on a given surface of the crystal, the polarizing 
 angle differs from a constant angle by a quantity proportional 
 to the square of the sine of the azimuth AYi. Now, calling 
 this azimuth a, and putting A for the acute angle at which the 
 axis of the crystal is inclined to its surface, so that A may be the 
 complement of the arc A Y, we have 
 
 sin q = cos A sin a, tan^? = cotan A cos a ; (54) 
 
 and by making these substitutions in formula (53), after having 
 changed sin < 2 into cos w, that formula becomes 
 
 CTI = 73 - k (sin 2 ?? - sin 2 A) + k sin 2 w cos 2 A sin 2 a, (55) 
 
 which agrees with the remark of Brewster. 
 
1 1 8 On the Laws of Crystalline 
 
 The deviation 3 or 0' 3 is found from the second of equations 
 
 (38), by putting . - r for tan 0, and by substituting for 
 
 sin {p tz) 
 
 cos (i + < 2 ) the value (49) or (50) which it has at the polarizing 
 angle. The result is 
 
 3 = O f z = -~ sin 2g sin (p + i a ), (56) 
 
 m 
 
 since the small arc 3 may be taken for its tangent. This result 
 is easily transformed into 
 
 3 = 0' 3 = - K sin q cos 0, (57) 
 
 where $ denotes the arc At, or the angle which the incident ray 
 makes with the axis of the crystal ; and this last expression is 
 equivalent to the following, 
 
 3 = 0' 3 = - jfiT'cos X sin a (sin X cos OT + cos X sin & cos a), (58) 
 
 which gives the deviation in terms of X and a. 
 
 As an example of the application of our formulae, we shall 
 make some computations relative to Iceland spar. According 
 to M. Rudberg, the ordinary index of that crystal, for a ray 
 situated in the brightest part of the spectrum, at the boundary 
 of the orange and yellow, is 1'66 ; and the least extraordinary 
 index for the same ray is 1*487. Dividing unity by each of 
 these numbers, we get a = *6725, b = "6024 ; whence OT = 58 56' ; 
 k = -1164 = 6 40' ; K= '1587 = 9 5'. Having thus determined 
 the constants, we can readily calculate the polarizing angle and 
 the deviation, for any given values of X and a. 
 
 First, let us see how the polarizing angle varies on different 
 faces of the crystal. 
 
 1. "When X = 90, the face of the crystal is perpendicular to 
 its axis, and OT X is independent of a. In this case the formula 
 (55) gives 
 
 which is the maximun value of the polarizing angle. 
 
Reflexion and Refraction . 119 
 
 2. When X = 0, the axis lies in the face of the crystal, and 
 formula (55) becomes 
 
 wj = ia - k sin 2 CT cos 2 a, 
 
 showing that w l = zr j when a is either 90 or 270. But when a 
 is or 180, we have 
 
 which is the minimum value of the polarizing angle. 
 
 3. For the natural fracture-faces of the crystal the value of 
 A is 45 23'. Hence, when a = or 180, 
 
 w^ty-k (sin 2 w - sin 2 A) = 57 26' ; 
 and when a = 90 or 270, 
 
 zs^w + k cos 2 -57 sin 2 A = 59 50'. 
 
 These values of the polarizing angles agree very well with the 
 experiments of Sir David Brewster, and still better with those 
 of M. Seebeck. 
 
 If we wish to know in what azimuths OTI is equal to w, on a 
 given surface of the crystal, it is obvious from (55) that we 
 must make 
 
 sin 2 sr - sin 2 A = sin 2 ^ cos 2 A sin 2 a, 
 
 whence we have, simply, 
 
 tan A 
 
 (59) 
 
 which shows that the thing is impossible when A is greater 
 than OT ; and that, when A is less than r, there are four such 
 azimuths ; as indeed there are, generally speaking, four values 
 of a corresponding to any other particular value of the polariz- 
 ing angle. If a' be the least of these azimuths, the others will 
 be 180- a', 180+ a', and 360- a'. On a natural face of the 
 crystal, the value of a, answering to the supposition ts l = w, is 
 found to be 52 22'. 
 
 Next, let us trace the changes which the deviation under- 
 goes in some remarkable cases. 
 
1 20 On the Laws of Crystalline 
 
 1. When the face of the crystal is perpendicular to its axis 
 there is evidently no deviation. 
 
 2. When the axis lies in the face of the crystal the devia- 
 tion vanishes in the azimuths 0, 90, 180, 270. In the inter- 
 mediate azimuths, differing 45 from each of these, the deviation 
 is a maximum ; for if we put X= in formula (55) the result will 
 be 
 
 3 = - ^r- sin ZT sin 2a ; 
 
 and this quantity (neglecting its sign) is a maximum when sin 
 2a = + 1. The coefficient of sin 2a is equal to 3 54', which is 
 consequently the greatest value of the deviation. According to 
 the experiments of M. Seebeck, the value is 3 57'. 
 
 3. On the fracture-faces of the crystal the deviation va- 
 nishes in the azimuths and 180, as also in two other azi- 
 muths for which 
 
 tanX 
 cos a = - , 
 
 and in which, therefore, ns l is equal to -&. In the azimuth 45 C 
 the deviation is -3 35'; in the azimuth 90 it is -2 32'; and 
 in the azimuth 127 38' it vanishes ; after which it attains a 
 small maximum with a positive sign, and vanishes again ir 
 azimuth 180. The calculated values of the deviation agree 
 pretty well with the values observed by M. Seebeck. , 
 
 The sign of the deviation shows at what side of the plane 
 of incidence the plane of polarization lies. But the position oJ 
 the latter plane is best indicated by that of the transversal oJ 
 the reflected ray. If this transversal and the axis of the crys- 
 tal be produced from the origin, towards the same side of the 
 plane of xz, until they intersect the sphere in the points t anc 
 A respectively, these points will be on the same side of the 
 great circle XY when the deviation and the sine of the azi- 
 muth have unlike algebraic signs ; and they will be on opposite 
 sides of that circle when those quantities have like signs. There- 
 fore if the crystal be supposed to revolve in its own plane, be- 
 
Reflexion and Refraction. 1 2 1 
 
 ginning at the azimuth 0, the points t and A will lie on the same 
 side of XY until A reaches the position A', where the angle 
 ATi is equal to 127 38' ; the point t will then pass over to the 
 side opposite A y at which side it will remain until A arrives at 
 the azimuth 232 22'. Thenceforward, to the end of the revo- 
 lution, both points will be found on the same side of the circle 
 XY. 
 
 We have seen that the deviation always vanishes when the 
 axis of the crystal lies in the plane of incidence. The reason is, 
 because the crystal is then symmetrical on opposite sides of that 
 plane. In this case the problem of reflexion offers peculiar faci- 
 lities for solution, since the uniradial directions are obviously 
 parallel and perpendicular to the plane of incidence. Let us, 
 therefore, consider the case at length. 
 
 1. In the first place, when the only refracted ray is the 
 ordinary one, the three transversals are in the plane of incidence, 
 and the transversal of each ray is proportional to the sine of the 
 angle between the other two rays. Hence the proportions are 
 
 Tl = Tz = T * _ (60) 
 
 sin (ii + 1 2 ) sin 2ii sin (^ - 1 2 )' 
 
 the same as in ordinary media. 
 
 2. In the second place, when the sole refracted ray is the 
 extraordinary one, the three transversals are perpendicular to 
 the plane of incidence ; and, if we use accents to mark the quan- 
 tities connected with this ray, we have the equations 
 
 T+miT, 
 
 which give the proportions 
 
 r / r 
 
 r * = _L 2 _ = Tz (62) 
 
 mi + m'z 2nii nh - m'J 
 wherein 
 
 m\ _ sin 2/2 2 sin Y 2 tan e' ( . 
 
 mi sin 2ii 
 
 by (26) ; the upper or lower sign being taken, in the numera- 
 
122 On the Laws of Crystalline 
 
 tor of (63), according as the refracted ray or its wave normal 
 makes the smaller angle with a perpendicular to the face of 
 the crystal. 
 
 To find the polarizing angle, we have only to make m\ = / 2 , 
 for then r x 3 will vanish by (62) ; and therefore, if common light 
 be incident, the whole reflected pencil will be polarized in the 
 plane of incidence. Supposing the crystal to be a negative one, 
 let us conceive the refracted ray to lie within the acute angle 
 made by the axis of the crystal with a perpendicular to its 
 surface. We shall then have to take the positive sign in the 
 numerator of (63), and the polarizing angle will be given by 
 the condition 
 
 sin 2h = sin 2/ 2 + 2 sinY 2 tan t'. (64) 
 
 But from (36) we have, in general, 
 
 sinY 2 tan / = ( 2 - b z ) sin a/ cos a/ sin 2 *!, (65) 
 
 and in the present instance it is evident that 
 (t/ = 90 A t' 2 , 
 
 where \ denotes, as before, the angle which the axis of the 
 crystal makes with its surface. Substituting these values in 
 (64), and multiplying all the terms by tan ' 2 , we get 
 
 sinY 2 = sin ^ cos /i tan i' 2 - (# 2 - b 2 ) sin (A + / 2 ) cos (X + 1 2 ) tan 
 
 Again, from (37) we have 
 
 sinY 2 = b* sin 2 *, + (a 2 - b z ) cos 2 (A + ' 8 ) sin 2 /! ; (66) 
 and by equating these two expressions for sinY 2 , we find 
 
 , a 2 cos 2 A + b 2 sin 2 A 
 
 tant 2 = - - 7-5 : T - r-. (o/j 
 
 cotan 1 1 + (a* - b~) sin A cos A 
 
 Then, if this value of tan t' 2 be substituted in equation (66), after 
 all its terms have been divided by cosY 2 , we shall obtain the 
 simple and rigorous formula 
 
 l- 2 cos 2 A- 2 sin 2 A 
 
 1 = " ~ 2 -- = sm " OT| ' (68) 
 
Reflexion and Refraction. 123 
 
 for determining the polarizing angle s^, when the axis of the 
 crystal lies in the plane of incidence. It is manifest, from the 
 nature of the formula, that this angle is the same, whether the 
 azimuth is or 180; that is, whether the light is incident at 
 the right or left side of the perpendicular to the surface of the 
 crystal. 
 
 This formula might be deduced more briefly by recollecting 
 what we have already proved, that the corresponding masses m^ 
 and m\ are proportional to the ordinates y of the points where 
 the incident ray and the extraordinary refracted ray meet their 
 respective wave surfaces ; whence it follows that these ordinates 
 must be equal at the polarizing angle ; and thus the question is 
 reduced at once to a geometrical problem. For as both rays are 
 in the plane of incidence, the axis of x will be intersected in one 
 and the same point by right lines touching the wave surfaces, 
 or their sections, at the extremities of the ordiuates. Now the 
 sections in the plane of xy are a circle and ellipse with their 
 common centre at the origin, the radius of the circle being unity, 
 and the semiaxes of the ellipse being a and 6, of which b is in- 
 clined at the angle A to the axis of #; and therefore it is re- 
 quired to draw, parallel to the axis of a?, a right line intersecting 
 the circle and ellipse, so that if tangents be applied to them at 
 two points of intersection which lie on the same side of the axis 
 of y, these tangents, when produced, may cut each other on the 
 axis of x. The angle which the tangent to the circle makes with 
 the axis of x is then the polarizing angle ^1 ; and the solution 
 of the problem just stated leads directly and easily to the for- 
 mula (68). From this way of viewing the matter we see the 
 reason why the polarizing angle is the same in the azimuths 
 and 180 ; for if tangents be applied at the two remaining 
 points where the parallel that we have spoken of intersects the 
 circle and ellipse, it is evident that these tangents also will cut 
 each other on the axis of x\ since tangents drawn at the ex- 
 tremities of any chord, either of a circle or an ellipse, intersect 
 the parallel diameter at equal distances from the centre. 
 
 Let the reflecting surface of the crystal be in contact with 
 
124 On the Laws of Crystalline 
 
 a fluid medium whose index of refraction out of vacuo is repre- 
 sented by N, and let B and A respectively denote the ordinary 
 and the principal extraordinary indices of refraction out of vacuo 
 
 N N 
 
 into the crystal. Then putting -j- for a, and for b, in the pre- 
 
 A Jj 
 
 ceding formula, and making 
 
 2 = ^ 2 sin 2 X 
 we readily deduce 
 
 
 Hence we perceive that if L~ = AB, that is, if 
 
 7) 
 
 tan A = / - 
 
 (in which case X will never be much above or below 45), the 
 value of t?! will be always possible ; for then we shall have 
 
 -;..; (70) 
 
 Bat if X be different from this, and of course L? not equal to 
 A B, the value of ^i may become impossible for certain values 
 of N. For it is clear that if N lie beticeen the limits L and 
 
 JT-, the numerator and denominator of the fraction (69) will 
 Jj 
 
 have unlike signs, and the tangent of CT! will be the square root 
 of a negative quantity. In this case, therefore, if common light 
 be incident, it will " refuse to be polarized," as Brewster ex- 
 presses it ; in other words, it will be impossible to find an angle 
 of incidence at which the reflected pencil will cease to contain 
 light polarized perpendicularly to the plane of incidence, or at 
 which the reflected transversal r' 3 will vanish. With all values 
 of N, except those which are included between the narrow limits 
 
 L and =r-, the polarizing angle is possible. It is zero at the 
 
 -L/ 
 
 latter limit, and 90 at the former. Outside these limits it 
 
Reflexion and Refraction. 125 
 
 changes rapidly at first, until N has passed either of them by 
 a quantity considerable in proportion to the interval between 
 them. 
 
 From (68) we find w^A, when = 1, or N-A\ and also 
 W! = 90-A, when 5 = 1, or N=B. In the latter case it is re- 
 markable that no light is reflected when common light is inci- 
 dent at the angle 90 -A. For then we have r' 3 =0; and because 
 *i = <2, we have likewise r 3 = 0. Therefore no light can enter the 
 reflected pencil. But this case deserves that we should consider 
 it more at large, without restricting ourselves to the supposition 
 that the axis of the crystal lies in the plane of incidence. 
 
 Assuming then that N=B, or that the refractive index of 
 the fluid, which covers the reflecting surface, is equal to the 
 ordinary index of the crystal itself, we may observe that, in 
 this case, every angle of incidence, in every azimuth, has a 
 right to be regarded as a polarizing angle. In fact, common 
 light cannot suffer reflexion at the separating surface of the 
 crystal and the fluid, without becoming completely polarized. 
 For if polarized light be incident, and if r 3 and r' 3 be the uni- 
 radial reflected transversals, respectively belonging to the ordi- 
 nary and to the extraordinary ray, the former transversal must 
 necessarily vanish, for the same reason that no reflexion can 
 take place at the separating surface of two ordinary media 
 whose refractive indices are equal; and thus the actual re- 
 flected transversal will always coincide in direction with / 3 , 
 whatever be the direction of the incident transversal. Conse- 
 quently, if common light be incident, the whole reflected pencil 
 will be polarized in a plane passing through r' 3 , and making 
 with the plane of incidence an angle 0' 3 determined by the 
 second of formulae (39). By putting t z = *i in that formula, 
 and employing the expression (44), we first obtain 
 
 n , cos (/i + ' 2 ) cos 0' tan a/ + sin (i t + t' 2 ) 
 
 tan V 3 = : TjTT- 7 
 
 sin u tan w 
 
 _ cos (ii + i' 2 ) tan (p - i' a ) + sin (^ + 1' 2 ) 
 sin 6T tan a/ 
 
126 On the Laws of Crystalline 
 
 and thence 
 
 sin (p + ii) sin (p + *i) cos a/ 
 
 x^ . ^ / v , \ > 
 
 sm tan w cos (p - 1 2 ) sm # cos (p - 1 2 ) 
 and finally, 
 
 = sin (p + i\) cotan g; (71) 
 
 a result which shows that the plane of polarization of the re- 
 flected ray is perpendicular to a plane drawn through the ray 
 itself and the axis of the crystal. 
 
 Moreover, we find, from the first of formulae (39), by 
 proceeding as above, 
 
 tan 0'i = - sin (p - ij cotan q = - cotan 9 ; (72) 
 
 and from (38) it is evident that Q l = 0. Therefore all that re- 
 lates to the case under our consideration may be summed up in 
 the following statement : 
 
 When N= B, and the incident light is polarized in a plane 
 passing through the axis, the course of the light is unaltered, 
 and there is neither reflexion nor refraction. When it is pola- 
 rized in the perpendicular plane, all the light which enters the 
 crystal undergoes extraordinary refraction. Whatever light is 
 reflected is always polarized in a plane at right angles to that 
 which passes through the reflected ray and the axis of the 
 crystal; and this is true, whether the incident light is pola- 
 rized or not. 
 
 Here, for the present, we must terminate our deductions 
 from the general theory propounded in this Paper. Several 
 other questions remain to be discussed, such as the reflexion of 
 common light* at the first surface, and the internal f reflexion 
 
 * The mode of treating the case in which common light is incident has heen 
 pointed out at the bottom of p. 100. 
 
 1 1 have since found that the problem of reflexion at the second surface may be 
 reduced to that of reflexion at the first surface by means of a very simple rule. 
 Let us suppose the two surfaces of the crystal to be parallel ; and let a ray JJi, 
 uniradially polarized, and incident on the first surface, give the ray J?s by reflexion, 
 and the single ray Hz by refraction. Let Hz be the ray which suffers internal reflex- 
 ion at the second surface, thereby giving the two reflected rays R tl , ' , and the 
 
Reflexion and Refraction. 127 
 
 at the second surface of a crystal ; but these must be reserved 
 for a future communication. It would be easy, indeed, to write 
 down the algebraical solutions resulting from our theory; but 
 this we are not content to do, because the expressions are rather 
 complicated, and, when rightly treated, will probably contract 
 themselves into a simpler form. It is the character of all true 
 
 single refracted ray E(\} emerging from the crystal in a direction parallel to R\. Put 
 TI, TS, Tg, and T,,, T',,, T(I) for the transversals of the rays in the order in which they 
 have been named. As the transversal TZ is supposed to be given4n magnitude, the 
 lengths as well as the directions of r\ and TS can be found by the, construction in 
 page 97. 
 
 Now, the direction of TS being changed, and its magnitude retained, let the ray 
 -Z?3 be turned directly back, so as to be incident again on the crystal, and to suffer 
 reflexion and refraction at the first surface. Then the two refracted rays which it 
 gives will be parallel to R tl , JR' ti , and their transversals will be equal and parallel 
 to T,,, T',,. The reflected ray which it gives will coincide with Hi ; and the re- 
 flected transversal, when compounded with TI, will furnish a resultant equal and 
 parallel to the emergent transversal T(i). 
 
 Thus the constructions, which have been given for the first surface, may be 
 made available for the second surface, and every question relative to crystalline 
 reflexion may be solved geometrically by means of the polar planes. 
 
 The foregoing rule was not, properly speaking, deduced from theory. I first 
 formed a clear conception of what the rule ought to be, and then verified it for the 
 simple case of singly-refracting media, and finally proved it for doubly-refracting 
 crystals. The truth of the rule, in crystals, depends upon the truth of the three 
 following equations : 
 
 sin (i tl -f t'J { cos (i,,-i',,) + cotan 0,, cotan 0',, } +h ll + h' it = Q, ^ 
 sin(i 3 - t j { cos (i' a ' -M J ~ cotan #2 cotan 0,, } + ^ 2 - A^O, (vin.) 
 
 sin (t 2 - i' J {cos (t a + /) - cotan 62 cotan 8' tt } + h z - h' = 0, 
 
 in which the notation is intelligible without any explanation. The first equation 
 is the same as equation (vn.) already noticed ; and the other two differ from it 
 only in appearance, the change in the signs being occasioned by a change in the 
 relative position of the rays. 
 
 "When the reflexion is total, I suppose we may follow the example which 
 Fresnel has set us in the case of ordinary media. The general algebraic expres- 
 sion for each reflected transversal will then become imaginary ; and by putting it 
 under the form 
 
 - 1 sin*), 
 
 we shall have T for the reflected transversal, and * for the change of phase. 
 
128 On the Laws of Crystalline 
 
 theories that the more they are studied, the more simple they 
 appear to be. And we may add, that a close examination of 
 such theories always meets with its reward, in the unexpected* 
 consequences which present themselves to view. Nothing can be 
 simpler than the laws of double refraction, as they were deli- 
 vered by Fresnel; yet the properties of his wave surface still 
 continue to furnish the geometer with beautiful and curious re- 
 lations. So we may hope that a little more time, devoted to 
 the laws of reflexion, will not be spent in vain. They promise 
 to supply many other theorems, not undeserving of attention, 
 though perhaps not as simple and comprehensive as those that 
 have already been made known. 
 
 From the nature of the rules which we have given for treating the question of 
 reflexion at either surface of the crystal, it follows that the final equation, for 
 determining the position of a transversal, is always linear, though the equation 
 of vis viva is of the second degree. This result very strongly confirms the theory ; 
 but it shows, at the same time, that the law of the preservation of vis viva is not to 
 be regarded as an ultimate principle, but rather as a consequence of some elemen- 
 tary law not yet discovered. 
 
 It now appears that the conjectures put forward in the note, p. 93, were hasty, 
 and that there was some mistake in the calculations which gave rise to them. It 
 js scarcely necessary to mention, that the sheet in which that note is found was 
 printed off before I had obtained the result announced in the subsequent note, 
 p. 111. Various delays occurred while my Paper was going through the press; 
 and I took advantage of them to increase its value, by appending notes on some 
 of the questions which I had overlooked or omitted in the first consideration of 
 the subject. 
 
 * As an instance of this, it may be mentioned, that the conclusion arrived at 
 in the note, p. Ill, was wholly unexpected. And in verifying the equation (viz.), 
 an unexpected and useful theorem was obtained ; for it became necessary to find 
 a manageable expression for the tangent of the angle c which the wave normal 
 makes with the ray. This expression is wanted in applying the formulae (34) and 
 (35) to biaxal crystals, and therefore I shall make no apology for introducing it 
 here. 
 
 Having described a sphere concentric with the wave surface, let the wave 
 normal OP and the two optic axes (which are the nodal diameters of the index 
 surface) be produced from the centre to meet the sphere in the points P t , A, A t , 
 respectively, thus marking out the angles of a spherical triangle PAA t . The same 
 wave normal may belong to two different rays ; and if we select one of these 
 rays, its transversal must lie in a plane drawn through the wave normal, and 
 bisecting either the internal angle APA t of the spherical triangle, or the ex- 
 
Reflexion and Refraction. 129 
 
 If we are asked what reasons can be assigned for the hypo- 
 theses on which the preceding theory is founded, we are far 
 from being able to give a satisfactory answer. We are obliged 
 to confess that, with the exception of the law of vis viva, the 
 hypotheses are nothing more than fortunate conjectures. These 
 conjectures are very probably right, since they have led to 
 elegant laws which are fully borne out by experiments ; but 
 this is all that we can assert respecting them. We cannot 
 attempt to deduce them from first principles; because, in the 
 theory of light, such principles are still to be sought for. It 
 
 ternal supplementary angle. By producing the optic axes in the proper direc- 
 tions, we may always make the above plane (which Fresnel calls the plane of 
 polarization) bisect the internal angle. Supposing this to have been done for the 
 ray which was selected, put w and <a t for the sides PA and PA, of the spherical 
 triangle, and J/ for the contained angle APA t , Let s be the length of the wave 
 normal from the centre to the point where it intersects the tangent plane applied 
 at the extremity of the ray, that is, applied at the point where the ray meets its 
 own nappe of the wave surface ; and let a and c be the greatest and least semiaxes 
 of the ellipsoid which generates the wave surface. Then we shall have 
 
 2 c 2 
 
 tan e = - sin (w - &>,) sin ty. (ix.) 
 
 . 2 
 
 And it is now manifest that if e, be the angle which the other ray makes with the 
 same wave normal, and *, the length of the wave normal intercepted between the 
 centre and the tangent plane at the extremity of this ray, we shall also have 
 
 a*-c z 
 tan e, = g- sin (o> + ,) cos i ^. (x.) 
 
 If a ray is given in direction it will have two wave normals ; and then the 
 angles e, e,, which it makes with each normal, may be found from the formulae 
 
 r 2 / 1 1 \ 
 
 tan e = - ( - - ^ 1 sin ( - w/ ) sin \ i|/, 
 
 where r and r t are the two radii of the wave surface which are in the direction of 
 the ray ; the spherical triangle P t AA 4 , of which the sides and contained angle are 
 expressed by the same letters as before, being now formed by producing the ray 
 and the two nodal diameters of the wave surface, until they intersect the sphere in 
 the points P t , A, A t . 
 
 K 
 
130 On tJie Laws of Crystalline 
 
 is certain, indeed, that light is produced by undulations, pro- 
 pagated, with transversal vibrations, through a highly elastic 
 ether; but the constitution of this ether, and the laws of its 
 connexion (if it has any connexion) with the particles of bodies, 
 are utterly unknown. The peculiar mechanism of light is a 
 secret that we have not yet been able to penetrate. As a proof 
 of this, we might observe, that some of the simplest and most 
 familar phenomena have never been explained. Not to mention 
 dispersion, about which so much has been fruitlessly written, 
 we may remark, that the very cause of ordinary refraction, or 
 of the retardation which light undergoes upon entering a trans- 
 parent medium, is not at all understood. Much less can it be 
 said that double refraction has been rigorously explained ; its 
 laws alone have been clearly developed by Fresnel. In short, 
 the whole amount of our knowledge, with regard to the propa- 
 gation of light, is confined to the laws of phenomena : scarcely 
 any approach has been made to a mechanical theory of those 
 laws. And if the case of uninterrupted propagation through 
 a continuous medium presents such difficulties, it would be use- 
 less to think of accounting for the laws which subsist at the 
 confines of two media, where the continuity is broken. 
 
 But perhaps something might be done by pursuing a con- 
 trary course ; by taking those laws for granted, and endeavour- 
 ing to proceed upwards from them to higher principles. In 
 this point of view, our second law, or hypothesis, is .extremely 
 remarkable ; for it seems to be opposed, in some degree, to* the 
 notion that the ethereal molecules are strongly attracted or 
 repelled by the particles of bodies. However that may be, it 
 would appear that a true theory must be in accordance with 
 this hypothesis ; and that any mechanical ideas, which would 
 make the mean density of the ether vary from one medium to 
 another,* cannot be admitted to represent the real state of 
 
 * Those who maintain that the density of the ether is different in different media, 
 ought to consider the following question : "What function of the three principal 
 indices of a doubly-refracting crystal represents the density of the ether within 
 the crystal ? 
 
Reflexion and Refraction. 131 
 
 things in nature. It is no objection to the hypothesis in 
 question, to say that it increases the difficulty of accounting 
 for refraction ; for, as there is positive evidence in favour of 
 the hypothesis, we ought rather to conclude that the common 
 opinion, which attributes refraction to a change of density in 
 the ether, is altogether erroneous. 
 
 In the next place we may remark, that our first hypothesis,* 
 concerning the direction of vibrations in polarized light, will be 
 useful in testing any proposed theory ; for as it now seems to be 
 certain that the vibrations are parallel to the plane of polariza- 
 tion, and not perpendicular to it, as Fresnel supposed, such a 
 direction of the vibrations ought to be a consequence of the 
 theory which we adopt. 
 
 The third hypothesis, or the principle of the preservation of 
 vis viva, is the most natural that can be imagined, inasmuch as 
 it implies only this, that the incident light is equal to the sum 
 of the reflected and refracted lights. Yet it is probable that 
 even this principle, like the law of vis viva in ordinary mechanics, 
 is a result of simpler laws, and will be shown to be so as soon 
 as the true mechanism of light shall be discovered. 
 
 The fourth hypothesis is a very important one, because the 
 whole theory turns upon it ; and therefore, in the beginning of 
 this Paper, a particular account has been given of the manner in 
 which it was originally suggested. If we wish to give a reason 
 for this hypothesis, we might say that the motion of a particle 
 of ether, at the common surface of two media, ought to be the 
 same, to whichsoever medium the particle is conceived to 
 belong ; and as the incident and reflected vibrations are super- 
 posed in one medium, and the refracted vibrations in the other, 
 we might infer that the resultant of the former vibrations ought 
 to be the same, both in length and direction, as the resultant of 
 the latter. At first sight this reasoning appears sufficiently 
 plausible; but it will not bear a close examination. For as 
 the argument is general, it would prove that the principle of 
 
 * This hypothesis properly belongs to the laws of propagation, as it relates only 
 to what passes within a given medium. 
 
 K2 
 
132 On tJie Laws of Crystalline 
 
 the equivalence of vibrations is true for metals,* as well as for 
 crystals, which it certainly is not. It is not easy to see why 
 the principle should hold in the one case and not in the other ; 
 but it is probably prevented from holding, in the case of metals, 
 
 * A few days after this Paper was read, I found reason to persuade myself 
 that in metals the vibrations parallel to the surface are equivalent, but not those 
 perpendicular to it ; and that in metals, as well as in crystals, the vis viva is 
 preserved. This persuasion was founded on a system of formulae -which I had 
 invented for expressing the laws of metallic reflexion and refraction ; and which 
 seem to represent very satisfactorily the experiments of Brewster, Phil. Trans. , 
 1830. As metallic and crystalline reflexion are kindred subjects, and will one 
 day be brought under the same theory, however distinct they may now appear, 
 it will not be out of place to insert the formulas for metals here. These formulae 
 are not proposed as true, but as likely to be true ; and they will be found to 
 express, at least with general correctness, all the circumstances that have hitherto 
 been regarded as anomalies in the action of metals upon light. 
 
 I suppose that for every metal there are two constants, M and x> of which 
 the first is a number greater than unity, and the second is an angle included 
 between and 90. The number M I call the modulus, and the angle x tne 
 characteristic of the metal. Both M and x varv witl1 tlie colour of the light, 
 
 and the ratio - is probably the index of refraction. From Brewster' s ex- 
 
 cosx 
 periments it appears that M diminishes from the red to the violet ; and therefore 
 
 I should suppose that cos x diminishes in a greater ratio, in order that the index 
 of refraction may increase as in transparent substances. 
 
 Put t x for the angle of incidence, and i 2 for the angle of refraction, so that 
 
 sin t. M 
 
 - i- = - ; (xn.) 
 
 sin t a cos x 
 
 and let p. be a variable determined by the condition 
 
 cost 
 
 fJL = - -. (XIII.) 
 
 COSi 2 
 
 These two relations combined will give 
 
 which shows that p. is equal to unity at a perpendicular incidence, and that it 
 vanishes at an incidence 90, decreasing always during the internal. 
 
 Now if plane polarized light be incident on the metal, we must distinguish two 
 principal cases, according as the light is polarized in the plane of incidence, or in 
 the perpendicular plane. In the first case, denoting the reflected and refracted 
 transversals by r 3 and r z respectively, let us put As for the change of phase in 
 
Reflexion and Refraction. 133 
 
 by the same cause, whatever it is, which produces a change of 
 phase in metallic reflexion. 
 
 It will be proper to conclude this Essay with a brief sketch 
 of the researches of Sir David Brewster and M. Seebeck, the 
 
 the reflected ray, and As for the change of phase in the refracted ray. Let the 
 same symbols, marked with accents, be used in the second case with similar signi- 
 fications. Then if the incident transversal be taken for unity, we shall have the 
 following formulae : 
 
 1. "When the incident transversal is in the plane of incidence, 
 
 , = M 2 + /t 2 - 2Jf /> cos X 1 
 
 ~ M 2 + /* 2 + 2M fji cos x 
 
 ^ = M* + ^ + 2#> coT? 
 2Jf u. sin Y 
 
 tanA 2 = 
 
 
 Mu ^-M Trr^ o > *** Ttir ' 
 
 M * p. A M. + p. COS X ^ 
 
 2. When the incident transversal is perpendicular to the plane of incidence, 
 
 1 + M 2 p? + 2M p cos x' 
 
 ~ 1 + Jf V a + 2Jf A* cos 
 
 2Jf u sin Y sin Y 
 
 tan A' 3 = W^ *> tan A'z = -^ - = - . 
 
 M 2 /i2 - 1 Jf^t + cos x J 
 
 "When x = 0, there is no change of phase, and the formulae become identical 
 with those given in the note, p. 101. When x = 90, there is total reflexion at 
 all incidences. The case of pure silver approximates to this. For good speculum 
 metal, x is about 70. The value of M ranges from 2 to 5 in different metals. 
 
 When the incident transversal is inclined to the plane of incidence, its compo- 
 nents, parallel and perpendicular to that plane, will give two reflected transversals 
 with a difference of phase equal to A'a - As. The reflected vibration will then be 
 performed in an ellipse ; and the position and magnitude of the axes of the ellipse 
 may be deduced from the preceding formulae. The consequences of these formulae 
 are very simple and elegant, but I cannot dwell upon them here. Suffice it to 
 observe, that every angle of incidence has another angle corresponding to it, which 
 I call its conjugate angle of incidence ; and that the value of A'a - AS at one of 
 these angles is the supplement of its value at the other, while the ratio 3 is the 
 same at both angles ; whence it follows that, ceteris paribus, the elliptic vibrations, 
 reflected at conjugate angles, are similar to each other, and have their homologous 
 
134 On the Laws of Crystalline 
 
 only other writers who have treated of the subject of crystalline 
 reflexion. 
 
 So early as the year 1819, Sir David Brewster published, 
 in the Philosophical Transactions, a Paper " On the Action of 
 Crystallized Surfaces upon Light." * In this Paper the Author 
 details a great variety of experiments on the polarizing effects 
 of Iceland spar. He gives the measures of the polarizing angles 
 in different azimuths, when the reflexion takes place in air; but 
 he does not notice the accompanying deviations, which were pro- 
 bably too small to attract his attention. In another instance, 
 however, he obtained very large deviations. He conceived the 
 idea of pushing his experiments into an extreme case, by mask- 
 ing, as it were, the ordinary reflecting action of the crystal, and 
 leaving the extraordinary energy at full liberty to display itself. 
 This was done by dropping on the reflecting surface a little oil 
 of cassia, a fluid whose refractive index is nearly equal to the 
 ordinary index of Iceland spar. When common light, incident 
 at 45, was reflected at the separating surface of the oil and the 
 spar, the reflected pencil was found to be partially, and some- 
 times completely, polarized in planes variously inclined to the 
 plane of incidence, the inclination going through all magnitudes 
 from to 180, as the crystal was turned round in azimuth. 
 This general result is no more than what theory would lead us 
 to expect, when the angle of incidence is nearly equal to one of 
 the angles of refraction ; but to institute a minute comparison 
 of theory with experiment would require troublesome calcula- 
 
 axes equally inclined to the plane of incidence, but on opposite sides of it. "When 
 A'a - AS = 90, the conjugate incidences are equal, the ratio is a minimum, and 
 
 T3 
 
 the axes of the elliptic vibration are parallel and perpendicular to the plane of 
 incidence. When A'a = 90, or Mp. = 1, the value of T'S is a minimum, and equal 
 to tan x- 
 
 The foregoing formulae differ slightly from those which I have given in No. I. 
 of the Proceedings of the Royal Irish Academy. The small quantity x', which 
 occurs in the latter, has been purposely neglected, as its presence interferes with 
 the simplicity of the expressions. 
 
 * Phil. Trans. 1819, p. 145. 
 
Reflexion an d Refraction . 135 
 
 tions, which I have not had time to make. With the view, 
 however, of showing clearly, from theory, that the range of 
 the deviation is unlimited, I have considered the simple case 
 in which N = B, or in which the refractive index of the fluid 
 is exactly equal to the ordinary index of the crystal. This case, 
 moreover, is remarkable on its own account ; and it might be 
 worth while to try whether it could not be verified by direct 
 experiment. If a fluid could be procured whose refractive 
 index, for some definite ray of the spectrum, should be equal 
 to the ordinary index of the crystal for the same ray, and if 
 common light, incident at any angle and in any azimuth, were 
 reflected at the confines of the fluid and the crystal, then, 
 supposing the theory to be exact, the definite ray aforesaid 
 would, as we have seen, be completely polarized by reflexion, 
 and the plane of polarization would always be perpendicular 
 to a plane drawn through the direction of the reflected ray 
 and the axis of the crystal. This experiment would be an 
 elegant test of the theory in its application to these extreme 
 and trying cases; and if it were successful, no doubt could 
 be entertained* as to the rigorous accuracy of the geome- 
 trical laws of reflexion. 
 
 * I was at this time in doubt whether the phenomena observed with oil of 
 cassia could bo reconciled to theory ; and when the note in page 93 was written, 
 I was almost certain that they could not. But I have since, I think, found out 
 the cause of this perplexity. Some of Brewster's experiments were made with 
 natural surfaces of Iceland spar; others with surfaces artificially polished. I 
 believe (though I have made very few calculations relative to the point) that the 
 former class of experiments will be perfectly explained by the theory ; the latter 
 I am certain cannot be so explained, nor ought we to expect that they should. 
 For the process of artificial polishing must necessarily occasion small inequalities, 
 by exposing little elementary rhombs with their faces inclined to the general 
 surface ; and the action of these faces may produce the unsymmetrical effects 
 which Brewster notices as so extraordinary (Sixth Eeport of the British Asso- 
 ciation, Transactions of the Sections, p. 16). If this will not account for such 
 effects, I do not know what will. From an old observation of Brewster (Phil. 
 Trans., 1819), it would appear that imperfect polish does actually produce a want 
 of symmetry in the phenomena; for when common light was reflected between 
 oil of cassia and a badly polished surface perpendicular to the axis, he found that 
 the reflected ray was polarized neither in the plane of incidence, nor perpendicular 
 
136 On the Laws of Crystalline 
 
 The experiments with oil of cassia must be very difficult 
 on account of the great feebleness of the reflected light. Sir 
 David Brewster, however, resumed them at different times; 
 and he laid an extensive series of his results before the Physi- 
 cal Section of the British Association at its late meeting in 
 Bristol. 
 
 It was not until the latter end of November, 1836, that I 
 became acquainted with the investigation of M. Seebeck, who 
 has contributed greatly to the advancement of the subject. He 
 made very acurate experiments on the light reflected in air from 
 Iceland spar. He detected the deviation, notwithstanding its 
 smallness, and measured it with great care. He also made the 
 first step in the theory of crystalline reflexion ; and the remark- 
 able formula (68), which gives the polarizing angle when the 
 axis lies in the plane of incidence, is due to him. The hypo- 
 theses which he employed were similar to those of Fresnel, 
 and they enabled him to solve the problem of reflexion in the 
 case just mentioned, but not to attempt it generally. The date 
 of his first Papers* is the year 1831 ; but he did not publish his 
 experiments on the deviation until a 'recent occasion, when he 
 was led to compare them f with the theory which I had origi- 
 nally given in my letter to Sir David Brewster. I have already 
 stated the correction which the theory underwent in conse- 
 
 to it, but 75 out of it. The same surface, when the light was reflected in air, 
 gave the polarizing angle more than two degrees below its proper value. 
 
 To show that, in other respects, the general character of the phenomena is in 
 accordance with theory, we may observe that when N= B, and A = or 90, if 
 common light be incident at 45 in the plane of the principal section of the crystal, 
 the whole of the reflected light will be polarized perpendicularly to that plane ; and 
 therefore if N be nearly equal to B, while every thing else remains the same, the 
 reflected pencil will contain some unpolarized light, and will be only partially pola- 
 rized in a plane perpendicular to the plane of incidence ; so that (as Brewster has 
 found by experiment) the crystal will then produce by reflexion the same effect 
 which is produced by ordinary refraction. This (as he also found) will not happen 
 when A. and the angle of incidence are each equal to 45, because the light is then 
 incident at the polarizing angle. 
 
 * Poggendorff s Annals, Vol. xxi. p. 290 ; Vol. xxn. p. 126. 
 <, Vol. xxxvin. p. 280. 
 
Reflexion and Refraction. 137 
 
 quence of those experiments, and by which it was brought to 
 its present simple form.* 
 
 * Two or three months after this correction had been published in the Philoso- 
 phical Magazine, a notice of it was inserted in Poggendorff's Annals, vol. xl. 
 p. 462. Up to that time, I believe, nothing had been published in Germany on 
 the general theory of crystalline reflexion ; at least the writer of the notice (whom 
 I take to be M. Seebeck) does not seem to have heard of any other theory, or any 
 other principles than mine. But in the next number of Poggendorff, vol. xl. 
 p. 497, there appeared a letter from M. Neumann, in which the writer speaks of 
 a theory of his own, founded on principles exactly the same as those which I had 
 already announced, and refers to a Paper which he had communicated on the 
 subject to the Academy of Berlin. The Paper has been printed in the Transactions 
 of that Academy for the year 1835 ; and through the kindness of the author I have 
 received a copy of it, just in time to acknowledge it here. On casting my eye over 
 it, I recognize several equations which are familiar to me in particular, the equa- 
 tions (vii.), (viu.), (ix.), (x.), which I discovered independently in November last. 
 M. Neumann's Paper is very elaborate, and supersedes, in a great measure, the 
 design which I had formed of treating the subject more fully at my leisure ; nor 
 can I do better than recommend it to those who wish to pursue the investigations 
 through all their details. 
 
 TRINITY COLLEGE, DUBLIN, March, 1838. 
 
 NIVERSITY 
 
( 138 ) 
 
 XII. ON A NEW OPTICAL INSTRUMENT, INTENDED 
 CHIEFLY FOR THE PURPOSE OF MAKING EXPERI- 
 MENTS ON THE LIGHT REFLECTED FROM METALS. 
 
 [Proceedings of the Royal Irish Academy, April 9, 1838.] 
 
 THE instrument consists of two hollow arms or tubes, moveable 
 about the centre, and in the plane, of a large divided circle, each 
 arm being provided with a Nicol's eye-piece, or some equivalent 
 contrivance for polarizing light in a single plane ; while in one 
 arm, which is of course crooked, a Fresnel's rhomb is interposed 
 between the eye-piece and the centre of the circle. At this centre 
 is placed a stage for carrying the reflector, with its plane perpen- 
 dicular to the plane of the circle, and having a motion to and fro 
 for adjustment. Each eye-piece, as well as the Fresnel's rhomb, 
 turns freely about the axis of the arm to which it belongs, and 
 is provided with a small circle for measuring its angle of rota- 
 tion. When the two arms are set at equal angles with the re- 
 flector, and the observer looks through the crooked arm, he will 
 see a light admitted through the straight one; and then, by 
 turning the Fresnel's rhomb, and the eye-piece next his eye, he 
 will be able, by means of their combined movements, to find a 
 position in which the light will entirely disappear. An obser- 
 vation will then have been made ; for the light, before its inci- 
 dence on the metal, is polarized in a given plane by the first 
 eye-piece; but after reflexion from the metal (as we know from 
 Sir David Brewster's experiments) it is elliptically polarized ; 
 and our object is to determine the position and species of the 
 
On a New Optical Instrument, &c. 139 
 
 little ellipse in which the reflected vibration is supposed to be 
 performed. Now, the axes of this ellipse are parallel and per- 
 pendicular to the principal plane of the rhomb, when it is in 
 the situation above described, where the light completely dis- 
 appears ; and the ratio of the axes is the tangent of the angle 
 which that plane makes with the principal section of the eye- 
 piece next the eye. The angles are read off from the divided 
 circles ; and thus, for any angle of incidence, and any plane 
 of primitive polarization, we can at once ascertain the nature 
 of the reflected elliptic vibration. Professor Mac Cullagh men- 
 tioned that the instrument was made last year with the view 
 of testing certain formulae which he has proposed for the case 
 of metallic reflexion, and which have been printed in Yol. xvm. 
 pp. 70, t 71, of the Transactions of the Academy*; but that he 
 had not yet found leisure to make the various adjustments 
 which are necessary in order to obtain satisfactory results with 
 it. The instrument is beautifully executed by Mr. Grubb, who 
 himself contrived the subordinate mechanism, by which the 
 requisite movements are effected with perfect ease to the ob- 
 server. 
 
 * Supra, pp. 132, 133. 
 
XIII. LAWS OF CRYSTALLINE REFLEXION. QUESTION 
 OF PRIORITY. 
 
 [Proceedings of the Royal Irish Academy, Nov. 30, 1838.] 
 
 THE President, Sir William E. Hamilton, read the following 
 letter which had been addressed to him by M. Neumann of 
 Konigsberg, on some points connected with the history of the 
 Laws of Crystalline Reflexion : 
 
 MONSIEUR, 
 
 Le haut prix que j' attache a votre suffrage et a celui de Pillustre Academic, 
 a laquelle vous presidez, et 1'honorable mention, que vous avez voulu faire de mon 
 memoire sur la theorie de la lumiere dans la seance de cette Academie du 25 Juin, 
 m'engagent a vous dresser la lettre suivante. Vous avez donne dans cette seance 
 un jugement dans la question de priorite, qui pouvait s' clever entre Mr. Mac 
 Cullagh et moi par rapport a la decouverte des lois suivant lesquelles la lumiere 
 est reflechie et refractee par des milieux crystallins j'ai 1'honneur de vous com- 
 muniquer dans ce qui suit quelques faits et quelques reflexions fondees sur ces 
 faits, et qui auraient ete peut-etre de quelque influence sur ce jugement. 
 
 Au commencement de 1'annee 1833 j'ai communique a M. Seebeck de Berlin 
 non seulement I'ensemble des principes de ma theorie tels qu'ils se trouvent 
 imprimes dans le 2 de mon memoire, mais j'avais illustre encore ces principes 
 par leur application aux milieux non crystallins. En meme terns j'ai annonce a 
 M. Seebeck, que les resultats tires de ces principes par rapport aux milieux 
 crystallins etaient parfaitement d' accord avec ses observations sur Tangle de 
 polarisation du kalkspath, et je lui fis part de la formule meme, qui exprime 
 1'inclinaison du plan de polarisation du rayon polarise par reflexion vers le plan 
 de reflexion. Sous la date du 11 Mai, 1833, M. Seebeck m'ecrivit, que cette 
 formule aussi s'accordait parfaitement avec ses observations, qu' il n'avait pas 
 encore publiees et qu' il avait la complaisance de me communiquer en manuscrit. 
 Dans le printems de 1834 le manuscrit de mon memoire tel qu'il a paru depuis 
 allait etre acheve ; mais un voyage que je fis dans ce terns et qui m'eloigna 
 
Question of Priority. 141 
 
 assez long-terns de Konigsberg, m'empecha de la publier incessamment. Cepen- 
 dant j'avais pris soin d'en faire un abrege dans lequel je developpai completemen^ 
 les principes de ma tneorie et les resultats auxquels elle m'avait conduit par 
 rapport aux crystaux a un axe. 
 
 J'envoyai cet extrait en Mai ou Juin, 1834, par la librairie de M. Schropp 
 de Berlin a M. Arago, en le priant de le faire imprimer dans les Annales de 
 Chimie et de Physique, ce savant ayant dans une note publiee dans ce terns 
 marque un grand interet pour 1' investigation des lois des intensites du rayon 
 ordinaire et extraordinaire, lois qui se trouvaient parmi les resultats mentionnes. 
 II n'y a pas de doute que cet extrait ne soit parvenu dans les mains de M. Arago, 
 entre lesquelles il doit se trouver encore a present. Du reste, M. Jacobi en avait 
 pris une connaissance detaillee, et a Berlin il a ete entre les mains de MM. Weiss 
 et Poggendorf . 
 
 En passant par Vienne dans Pete de 1834, j'avais le plaisir d'entretenir de mes 
 resultats et de ma methode M. Ettinghausen, savant tres distingue et tres verse 
 dans les parties les plus epineuses de 1'optique. Anterieurement j'avais enseigne 
 mes doctrines a M. Senff, maintenant professeur a 1'Universite de Dorpat, pendant 
 le sejour que fit a Koenigsberg ce jeune et habile physicien, qui vient de- publier 
 un excellent travail sur les proprietes optiques et crystallographiques du fer sulfate. 
 
 II suit de tout ce qui precede, que deja en 1834, mes resultats trouves par 
 rapport aux lois de reflexion et de refraction des crystaux n'etaient guere 
 inconnus aux physiciens de 1'Allemagne, qui s'occupent de 1'optique, et si des lors 
 ils n'ont pas re9u une plus grande publicite, vous voyez, Monsieur, cela tenait 
 aux Annales de Chimie. La publication de mon memoire a ete retardee par 
 1'espoir que j'avais con9u de pouvoir lui a j outer une partie experimentale. 
 Mais 1' execution des appareils me faisant attendre trop long terns, j'ai presente 
 vers la fin de 1835 a P Academie de Berlin mon ouvrage tel qu'il a ete imprime 
 depuis parmi les memoires de cette Academie. La partie experimentale a et^ 
 publiee en 1837 dans le volume 42 des Annales de M. Poggendorf. 
 
 Je vois du discours que vous avez tenu, Monsieur, dans la Seance de votre 
 Academie du 25 Juin passe, et qui vient de m'etre communique, que c'est deja 
 en Aout, 1835, que Mr. Mac Cullagh a fait a 1'Association Britannique une 
 communication sur les lois de reflexion et refraction par les crystaux, et qui a 
 ete imprimee dans le Lond. and Edinb. Phil. Mag., Fevrier, 1836. Je crois tres 
 volontiers, que Mr. Mac Cullagh est parvenu aux resultats qui se trouvent dans 
 cette publication, par ses propres efforts et sans avoir eu connaisance de mes 
 travaux sur ce meme sujet. Toutefois ce ne sont pas ces resultats qui.pourraient 
 etre 1'objet d'une question de priorite. En effet dans une note publiee dans les 
 Annales de M. Poggendorf (vol. xxxviii. 1836), M. Seebeck a montre que les 
 formules auxquelles est parvenu Mr. Mac Cullagh ne sont pas justes, et qu'elles 
 ne representent pas les lois de reflexion et de refraction par les crystaux. Dans 
 la meme note M. Seebeck a expose', comment les lois de reflexion et de refraction 
 des milieux non crystallins conformes a cette definition du plan de polarisation, a 
 laquelle on est conduit dans la theorie de la double refraction, peuvent etre deduites 
 des suppositions faites par Fresnel, avec la seule modification de 1'homogeneite 
 
142 L aws of Crystalline Reflexion . 
 
 de 1' ether dans tous les milieux. Mais les suppositions de Fresnel ainsi modifiees 
 forment la base principale de ma methode, dont j'avais deja fait part a M. Seebeck 
 depuis plusieurs annees. II est vrai, que dans les deux milieux Fresnel ne suppose 
 que 1'egalite de deux composantes paralleles au plan de separation, mais 1'egalite 
 de la troisieme n' est qu'une simple consequence de celle des deux autres et des 
 autres suppositions. Ce sont les suppositions de Fresnel modifiees de la dite 
 maniere, qu'a adoptees Mr. Mac Cullagh, apres s'etre convaincu par la note de 
 M. Seebeck de la faussete des resultats qu'il avait jusque-la obtenus, conviction 
 qui 1'engagea a rejeter tout ce qui n'etait pas confovme a ces suppositions, et des 
 lors seulement en 1837, dans le Lond. and Edinb. Phil. Mag., Mr. Mac Cullagh 
 est parvenu aux memes lois de reflexion et de refraction que j'avais eues 1'honneur 
 de presenter a 1' Academie des Sciences de Berlin en 1835. 
 
 Vous voyez par tout ceci, Monsieur, que des 1833 j'ai etc en pleine possession 
 de la methode, et que des le commencement de 1834 j'ai ete en pleine possession 
 des resultats qu'elle fournit, que dans ce meme terns j'ai envoye un abrege con- 
 tenant ces resultats et lu en manuscrit par plusieurs savans bien connus a M. 
 le redacteur des Annales de Physique et Chimie pour le publier dans ce recueil, 
 et qu'a la fin de 1835, j'ai presente 1'ouvrage complet a present imprime a 
 1'Academie de Berlin ; vous voyez en meme terns, que Mr. Mac Cullagh ayant 
 communique' a 1'Association Britanique en 1835 des lois de reflexion et de refrac- 
 tion crystallin, ces lois ont ete demontrees etre fautives par M. Seebeck in 1836, 
 et que Mr. Mac Cullagh n'est parvenu en 1837 aux vraies lois qu' apres avoir 
 pris connaissance du fondement de ma methode, et s'en etre servi. 
 
 De tout cela resulte, Monsieur, que la priorite de la decouverte des lois de 
 reflexion et refraction par des crystaux n'est pas douteuse, et" qu'il n'y a pas de 
 simultaneite entre mes travaux et ceux de Mr. Mac Cullagh, dont du reste 
 personue ne peut estimer plus que moi le talent distingue. 
 
 Daignez, Monsieur, agreer les assurances de la plus haute consideration avec 
 
 laquelle je suis, &c., 
 
 F. E. NEUMANN. 
 
 KONIGSBEBG, 5 Octobre, 1838. 
 
 When this letter was read, Professor Mac Cullagh requested 
 permission to make a few remarks. After expressing much 
 regret that his researches in the theory of light should have 
 clashed with those of any other person (though in the present 
 state of science such collisions were perhaps inevitable), he 
 proceeded to say, that he did not think it necessary to detain 
 the Academy with a formal reply to the communication which 
 had just been read ; it would be sufficient for him to observe, 
 in general, that the facts brought forward by the writer, with 
 reference to the history of his own investigations, were all, 
 without exception, of a private nature, not one of them being 
 
Question of Priority. 143 
 
 taken from any published document ; that the first document 
 of the kind, which professed to give any account of M. Neu- 
 mann's "method," or any statement of the principles employed 
 in it, appeared in the Annals of Poggendorf (Yol. XL. p. 497), 
 some months after Mr. Mac Cullagh had published his last 
 Paper on the subject in the Philosophical Magazine (Yol. x. 
 p. 43), and even after that Paper had been noticed in the 
 aforesaid Annals (Yol. XL. p. 462) ; that M. Neumann's Memoir 
 in the Berlin Transactions was not published until a later 
 period ; that, therefore, there could be no question about prio- 
 rity of publication ; and that, consequently, if it were to be 
 imagined for a moment that either author had borrowed from 
 the other, the presumption must necessarily be against M. Neu- 
 mann. With respect to M. Seebeck's note, it would be enough 
 to state, that M. Neumann is not mentioned there at all ; that 
 the principles there given by M. Seebeck are not adequate to 
 the general solution of the problem ; and that such of them 
 as differ from those of Fresnel had been previously published 
 by Mr. Mac Cullagh. It was clear, therefore, that Mr. Mac 
 Cullagh owed nothing on the score of theory to anyone but 
 Fresnel. He had, indeed, made one alteration in his theory 
 as it originally stood; for he had at first rejected Fresnel's law 
 of the vis viva, and had been obliged to restore it afterwards, 
 in order to account for certain experiments of M. Seebeck, 
 which M. Seebeck himself, from want of sufficient principles, 
 had not attempted to account for; but the real service which 
 M. Seebeck had rendered him, and for which he had frequently 
 acknowledged his obligations, was the communication of these 
 experiments, and not any suggestion of the law of vis viva, 
 which he knew well enough before. In all this, however, it 
 was plain that M. Neumann had no concern, unless he chose 
 to say that he had appropriated to himself Fresnel's law of 
 the vis viva, that he had determined to regard it as the foun- 
 dation of his method (le fondement de sa methode), and that 
 thenceforward no one else (however ignorant of such appro- 
 priation) could have any right to use it. 
 
144 Laws of Crystalline Reflexion. 
 
 Having thus endeavoured to prove his claim to priority of 
 publication, and to establish the independence of his own re- 
 searches, which was all that was necessary for self defence, 
 Mr. Mac Cullagh concluded by saying, that he would there 
 drop the argument, without discussing his claim to priority in 
 the abstract, as he had an objection to disputes of such a kind, 
 and did not wish to pursue them any farther than he was 
 compelled to do. But if anyone thought it worth while to 
 examine the merits of this second question, he would find 
 the circumstances relating to it very fully and clearly stated in 
 the last number of the Proceedings of the Academy,* and 
 would thence be enabled to form a judgment for himself. 
 
 * Vol. i. p. 217. 
 
XIV. AN ESSAY TOWARDS A DYNAMICAL THEORY OF 
 CRYSTALLINE REFLEXION AND REFRACTION. 
 
 [Transactions of the Royal Irish Academy, VOL. xxi. Read December 9, 1839.] 
 
 SECT. I. INTRODUCTORY OBSERVATIONS. EQUATION or MOTION. 
 
 NEARLY three years ago I communicated to this Academy* the 
 laws by which the vibrations of light appear to be governed in 
 their reflexion and refraction at the surfaces of crystals. These 
 laws remarkable for their simplicity and elegance, as well as 
 for their agreement with exact experiments I obtained from a 
 system of hypotheses which were opposed, in some respects, to 
 notions previously received, and were not bound together by any 
 known principles of mechanics, the only evidence of their truth 
 being the truth of the results to which they led. On that occa- 
 sion, however, I observed that the hypotheses were not indepen- 
 dent of each other ; and soon afterwards I proved that the laws 
 of reflexion at the surface of a crystal are connected, in a very 
 singular way, with the laws of double refraction, or of propaga- 
 tion in its interior ; from which I was led to infer that "all these 
 laws and hypotheses have a common source in other and more 
 intimate laws which remain to be discovered ;" and that " the 
 next step in physical optics would probably lead to those higher 
 and more elementary principles by which the laws of reflexion 
 
 * In a Paper "On the Laws of Crystalline Eeflexion and Refraction." Trans- 
 actions of the Royal Irish Academy, VOL. xvm. p. 31. (Supra, p. 87.) 
 
146 On a Dynamical Theory of 
 
 and the laws of propagation are linked together as parts of the 
 same system."* This step has since been made, and these anti- 
 cipations have been realised. In the present Paper I propose to 
 supply the link between the two sets of laws by means of a very 
 simple theory, depending on certain special assumptions, and 
 employing the usual methods of analytical dynamics. 
 
 In this theory, the two kinds of laws, being traced from a 
 common origin, are at once connected with each other and 
 severally explained ; and it may be observed, that the explana- 
 tion of each, as well as the source of their connexion, is now 
 made known for the first time. For though the laws of crys- 
 talline propagation have attracted much attention during the 
 period which has elapsed since they were discovered by Fresnel,f 
 they have hitherto resisted every attempt that has been made to 
 account for them by dynamical reasonings ; and the laws of re- 
 flexion, when recently discovered, were apparently still more 
 difficult to reach by such considerations. Nothing can be easier, 
 however, than the process by which both systems of laws are 
 now deduced from the same principles. 
 
 The assumptions on which the theory rests are these : First ^ 
 that the density of the luminiferous ether is a constant quantity ; 
 in which it is implied that this density is unchanged either by 
 the motions which produce light or by the presence of material 
 particles, so that it is the same within all bodies as in free 
 space, and remains the same during the most intense- vibrations. 
 Second, that the vibrations in a plane-wave are rectilinear, and 
 that, while the plane of the wave moves parallel to itself, the 
 vibrations continue parallel to a fixed right line, the direction of 
 this right line and the direction of a normal to the wave being 
 functions of each other. This supposition holds in all known 
 crystals, except quartz, in which the vibrations are elliptical. 
 
 Concerning the peculiar constitution of the ether we know 
 
 * Ibid, p. 53, note. (Supra, p. 112.) The note here referred to was added 
 some time after the Paper itself was read. 
 
 f These laws were published in his Memoir on Double Refraction Memoir es 
 de V Institiit, torn. vii. p. 45. 
 
Crystalline Reflexion and Refraction. 147 
 
 nothing, and shall suppose nothing, except what is involved in 
 the foregoing assumptions. But with respect to its physical 
 condition generally, we shall admit, as is most natural, that a 
 vast number of ethereal particles are contained in the differen- 
 tial element of volume ; and, for the present, we shall consider 
 the mutual action of these particles to be sensible only at dis- 
 tances which are insensible when compared with the length of a 
 wave. 
 
 By putting together the assumptions we have made, it will 
 appear that when a system of plane waves disturbs the ether, the 
 vibrations are transversal, or parallel to the plane of the waves. 
 For all the particles situated in a plane parallel to the waves are 
 displaced, from their positions of rest, through equal spaces in 
 parallel directions ; and therefore if we conceive a closed surface 
 of any form, including any volume great or small, to be de- 
 scribed in the quiescent ether, and then all its points to partake 
 of the motion imparted by the waves, any slice cut out of that 
 volume, by a pair of planes parallel to the wave-plane and inde- 
 finitely near each other, can have nothing but its thickness 
 altered by the displacements ; and since the assumed preserva- 
 tion of density requires that the volume of the slice should not 
 be altered, nor consequently its thickness, it follows that the 
 displacements must be in the plane of the slice, that is to say, 
 they must be parallel to the wave-plane. And conversely, when 
 this condition is fulfilled, it is obvious that the entire volume, 
 bounded by the arbitrary surface above described, will remain 
 constant during the motion, while the surface itself will always 
 contain within it the very same ethereal particles which it en- 
 closed in the state of rest ; and all this will be accurately true, 
 no matter how great may be the magnitude of the displace- 
 ments. 
 
 Let x, y, z be the rectangular co-ordinates of a particle before 
 it is disturbed, and x + 5, y + rj, z + its co-ordinates at the time t, 
 the displacements , TJ, being functions of #, y, * and t. Let 
 the ethereal density, which is the same in all media, be regarded 
 as unity, so that dxdydz may, at any instant, represent indif- 
 
148 On a Dynamical Theory of 
 
 ferently either the element of volume or of mass. Then the 
 equation of motion will be of the form 
 
 JJJ dxdydz ( S5 + -^ Srj + ? d% } = JjJ dxdydzSV, (1) 
 
 where Y is some function depending on the mutual actions of 
 the particles. The integrals are to be extended over the whole 
 volume of the vibrating medium, or over all the media, if there 
 be more than one. 
 
 Setting out from this equation, which is the general formula 
 of dynamics applied to the case that we are considering, we per- 
 ceive that our chief difficulty will consist in the right determina- 
 tion of the function Y ; for if that function were known, little 
 more would be necessary, in order to arrive at all the laws which 
 we are in search of, than to follow the rules of analytical me- 
 chanics, as they have been given by Lagrange. The determina- 
 tion of Y will, of course, depend on the assumptions above stated 
 respecting the nature of the ethereal vibrations ; but, before we 
 proceed further, it seems advisable to introduce certain lemmas, 
 for the purpose of abridging this and the subsequent investiga- 
 tions. 
 
 SECT. II. LEMMAS. 
 
 Lemma I. Let a right line making with three' rectangular 
 axes the angles a, /3, 7, be perpendicular to two other right lines 
 which make with the same axes the angles a', /3', 7' and a", /3", 7" 
 respectively, and which are inclined to each other at an angle 
 denoted by ; then it is easy to prove that 
 
 sin cos a = cos ft' cos 7" - cos ft" cos 7', 
 
 sin 6 cos j3 = cos 7' cos a" - cos 7" cos a', (A) 
 
 sin cos 7 = cos a' cos j3" - cos a" cos ft' ; 
 
 supposing the first right line to be prolonged in the proper direc- 
 tion from the origin, in order that the opposite members of any 
 
Crystalline Reflexion and Refraction. 149 
 
 one of these equations may have the same sign, as well as the 
 same magnitude. 
 
 If the last two right lines be perpendicular to each other, 
 we have sin = 1, and the formulae become 
 
 cos a =. cos |3' cos 7" - cos ft" cos 7', 
 
 cos /3 = cos 7' cos a" - cos y" cos a', (B) 
 
 cos 7 = cos a' cos |3 r/ - cos a" cos )3' ; 
 
 but in this case the three right lines are perpendicular to each 
 other, and therefore we have, in like manner, 
 
 cos a' = cos |3" cos 7 - cos j3 cos 7", 
 
 cos ]3' = cos 7" cos a - cos 7 cos a", (B') 
 
 cos 7' = cos a" cos/3 - cos a cosjS"; 
 and also 
 
 cos a" = cos /3 cos 7' - cos j3' cos 7, 
 
 cos ]3" = cos 7 cos a' - cos 7' cos a, (B") 
 
 cos 7" = cos a cos ]3' - cos a' cos /3. 
 
 The last three groups of formulas will still be true, if we 
 suppose the first right line to make with the axes the angles 
 a, a', a", the second the angles ]3, /3', /3", and the third the 
 angles 7, 7', 7" 
 
 Lemma II. Let , 17, denote, as before, the displacements 
 of a particle whose initial co-ordinates are #, y, 2 ; and after 
 putting 
 
 rfi, 4i M 
 
 ~te~dy> -Tx~te> ~ dy dx* 
 
 suppose the axes of co-ordinates, still remaining rectangular, to 
 have their directions changed in space, whereby the quantities 
 X, F, Z will be changed into X', Y', Z' ', answering to the new 
 co-ordinates x' 9 /, a', and to the new displacements ', ?', 2' ; then 
 will the quantities X', Y', Z' be connected with X, F, Z by 
 
150 On a Dynamical Theory of 
 
 the very same relations which connect the co-ordinates x ', ?/', z 
 with z, y, z, or the displacements ', r/, ' with , r?, 2. 
 
 That is to say, if the axes of #, ?/, s make with the axis of x 
 the angles a, /3, 7, with the axis of y the angles a', |3', 7', and 
 with the axis of s' the angles a", j3'', 7" respectively, we shall 
 
 have 
 
 X = X' COS a + T' COS a' + Z' cos a", 
 
 Y= X' cos /3 + F' cos /3' + Z' cos /3", (D) 
 
 Z' = J' cos 7 + F' cos 7' + Z' cos 7", 
 and 
 
 X' = X cos a + Y cos )3 + Z cos 7, 
 
 F' = X cos a' + F cos j3 r + Z cos 7', (D') 
 
 just as we have, for example, 
 
 % = % COS a + i/ COS a' + ?' COS a", 
 
 n = ' cos )3 + / cos j3' + T cos )3", (d) 
 
 = ' cos 7 + r! cos 7' + y cos 7", 
 
 and 
 
 #' = x cos a + y cos ]3 + 2 cos 7, 
 
 ?/ = # cos a' + ?/ cos j3' + s cos 7', (d') 
 
 2' = # cos a" + y cos /3" + s cos 7". 
 
 For, the change of the independent variables #, y, 2 into 
 of, y', z' gives us the equations 
 
 di\ dr\ dx drt dy drj dz' 
 
 = __ _ 
 
 dy ~ fa' ~dy dy' dy dz' dy 9 
 
 in the right-hand members of which we have to substitute the 
 
Crystalline Reflexion and Refraction. 151 
 
 values of the differential coefficients obtained from (d) and (). 
 Thus we get 
 
 dn (dg dj ., d? D ,A 
 
 ~ = , cos /3 + -j-, cos j3 + ~ cos /3 cos 7 
 
 dz \dx dx dx J 
 
 cos j3 + -~ cos 8' + - cos i3" ) cos ?' 
 
 ,, 
 
 = , COS 7 + -7-7 COS 7 + -7-7 COS 7 COS 3 
 
 dy \dx dx dx 
 
 + -j-7 COS 7 + -T-? COS 7' + , COS 7" COS 8' 
 
 y dy dy 
 
 "- cos 7 + -^7 cos 7' + -3-7 cos 7" ] cos 3" ; 
 
 3 <& fl&^ / 
 
 and when we subtract these equations, attending to the formulae 
 in Lemma I., we find 
 
 = I -7-7 - -7-7 ) cos a + I -7-7 - -r-7 ) cos a' 
 
 3 dy \dz dy J \dx dz J 
 
 'd% dn \ 
 
 ~T7 T~7 cos a > 
 
 ,C?^ ^/ 
 
 or simply, 
 
 X = X' COS a + y COS a' + /T COS a", 
 
 which is the first of formulae (D) . And in like manner the others 
 may be proved. 
 
 The same things will obviously hold with respect to quanti- 
 ties derived from X, F, Z in the same way that these are derived 
 from > 17, 2- That is, if we put 
 
 ____ --~ -- 
 
 ' ~ Hz " ~dy' ' " dx ~ ~dz* ' ~ dy dx' 
 
 and then suppose the axes of co-ordinates to be changed, the 
 
152 On a Dynamical Theory of 
 
 formulae for the transformation of the quantities X# F y , Z^ will 
 be similar to those for the transformation of the co-ordinates 
 themselves. The like will be true of the quantities X //5 F^, ^ /y , 
 if we put 
 
 _ V = - -' 7 = .< - 
 
 dz dy' " dx dz' " dy dx 
 
 and so on successively. 
 
 It is to be observed that, in this Lemma, the displacement 
 is not limited by any restriction whatever. Each of its com- 
 ponents may be any function of the co-ordinates. But the 
 displacements produced by a system of plane waves are re- 
 stricted by our definition of such waves ; they must be the 
 same for all particles situated in the same wave plane. If 
 the waves be parallel, for instance, to the plane of x ', ^', the 
 quantities ', i/, % will be independent of the co-ordinates x ', y ', 
 and will be functions of z' only. This consideration reduces 
 formulae (D) to the following : 
 
 dri d% 
 
 X = -7-7 COS a - , COS a , 
 dz dz 
 
 F-J'eosp-gooBp', :() 
 
 dr,' 
 
 in which it is remarkable that the normal displacement ?' does 
 not appear. If ' = 0, these formulae become 
 
 -r- dr{ ~ dr\ , . 
 
 or if r\ = 0, then we have 
 
 (F') 
 
 Lemma III. If, in an ellipsoid whose semiaxes are equal 
 to a, by c, there be two rectangular diameters, one making with 
 
Crystalline Reflexion and Refraction. 153 
 
 the semiaxes the angles a, |3, y, and the other the angles 
 a', /3', y', such as to satisfy the condition 
 
 cos a cos a cos cos 
 
 these diameters will be the axes of the ellipse in which their 
 plane intersects the ellipsoid. 
 
 For, the above condition expresses that either diameter is 
 parallel to the tangent plane at the extremity of the other ; 
 they are therefore conjugate diameters of the elliptic section ; 
 and hence, as they are at right angles to each other, they must 
 be its axes. 
 
 If the semiaxes of the ellipsoid be represented by -, =-, -, 
 
 cioc 
 
 the equation of condition will become 
 
 a? cos a cos a' + b* cos j3 cos ]3' + c 2 cos y cos y' = 0. (G') 
 
 Lemma TV. Let s, *' be the lengths of perpendiculars let 
 fall from the centre of an ellipsoid upon any two tangent 
 planes, and r, / the lengths of radii drawn to the respective 
 points of contact. Then putting w for the angle between the 
 directions of r and s', and a/ for the angle between the directions 
 of r and s, we shall have 
 
 rs cos a) = rs' cos a/. 
 
 For if the semiaxes of the ellipsoid, having their lengths 
 denoted by a, b, c, make with the direction of s the angles 
 a, /3, 7, and with that of s' the angles a', j3', y'; with the 
 direction of r the angles a , j3 , yo, and with that of / the 
 angles ai, ]3i, yi, there will exist the relations 
 
 a 1 ' cos o = rs cos a , b z cos /3 = rs cos /3 , c* cos y = rs cos y , 
 a 2 cos a' = rV cos ai, b z cos j3 r = rV cos )3i, c 2 cos y' = rV cos yi, 
 by one set of which the quantity 
 
 tf COR a cos a' + b 2 cos ]3 cos /3' + c z cos y cos y' 
 
154 On a Dynamical Theory of 
 
 will be converted into 
 
 rs (cos a c cos a' + cos /3 cos |3' + cos 70 cos 7') = rs cos w, 
 and by the other set into 
 
 r' s' (cos ai cos a + cos /3i cos j3 + cos ji cos 7) = r f s' cos to' ; 
 so that we shall get 
 
 rs cos co = r' s' cos to' = a 2 cos a cos a' + b z cos /3 cos /3' 
 
 + c 2 cos 7 cos 7'. (H ) 
 Corollary. When the condition 
 
 a 2 cos a cos a' 4- b 2 cos ]3 cos j3' + c 2 cos 7 cos 7' = (i) 
 
 is satisfied, each of the angles <o, to' is a right angle. Let us 
 suppose, at the same time, that the direction of s is perpendi- 
 cular to that of s'. Then will the directions of s and r' coin- 
 cide with the axes of the ellipse in which their plane intersects 
 the ellipsoid ; for s is perpendicular to r' and parallel to the 
 tangent plane at its extremity. The directions of s' and r, in 
 the same manner, will coincide with the axes of another elliptic 
 section. 
 
 SECT. III. DETERMINATION or THE FUNCTION ON WHICH THE 
 MOTION DEPENDS. PRINCIPAL AXES OF A CRYSTAL. 
 
 We come now to investigate the particular form which 
 must be assigned to the function Y, in order that the formula 
 (1) may represent the motions of the ethereal medium. For 
 this purpose conceive the plane of x f y' to be parallel to a 
 system of plane waves whose vibrations are entirely transversal 
 and parallel to the axis of y', so that ' = 0, ' = 0. Imagine 
 an elementary parallelepiped dd dtf dz' ^ having its edges parallel 
 to the axes of of, y, s', to be described in the ether when at rest, 
 and then all its points to move according to the same law as the 
 ethereal particles which compose it. The faces of the parallele- 
 piped which are perpendicular to the edge dz' will be shifted, 
 each in its own plane, in a direction parallel to the axis of y f ; 
 but their displacements will be unequal, and will differ by V, 
 
Crystalline Reflexion and Refraction. 1 55 
 
 so that the edges connecting their corresponding angles will no 
 longer be parallel to the axis of 2', but will be inclined to it at 
 
 7 / 
 
 an angle K whose tangent is -, . 
 
 Now the function Y can only depend upon the directions 
 of the axes of #', y' ', z f with respect to fixed lines in the crystal, 
 and upon the angle K, which measures the change of form pro- 
 duced in the parallelepiped by vibration. This is the most 
 general supposition which can be made concerning it. Since, 
 however, by our second assumption, any one of these directions, 
 suppose that of #', determines the other two, we may regard Y 
 as depending on the angle K and on the direction of the axis of 
 x alone. But from the equations (r)*it is manifest that the 
 angle K and the angles which the axis of x f makes with the 
 fixed axes of #, /, 2 are all known when the quantities X, F, Z 
 are known. Consequently Y is a function of JT, F, Z. 
 
 Supposing the angle K to be very small, the quantities 
 X, F, Z will also be very small; and if Y be expanded 
 according to the powers of these quantities, we shall have 
 
 Y = K + AX + BY + CZ+ DX* + EY* + FZ* 
 
 + GYZ + HXZ + IXY + &o., 
 
 the quantities K, A, B, C, D, &o., being constant. But in the 
 state of equilibrium the value of 8 Y ought to be nothing, in 
 whatever way the position of the system be varied ; that is to 
 say, when the displacements , ij, , and consequently the 
 quantities X, F, Z 9 are supposed to vanish, the quantity 
 
 SY = ASX + J93F + CSZ + 2DXSX + &c., 
 
 ought also to vanish independently of the variations 85, Srj, 8, 
 or, which comes to the same thing, independently of 8JT, 8F, 
 8Z. Hence* we must have ^4 = 0,5 = 0, (7=0; and therefore, 
 if we neglect terms of the third and higher dimensions, we may 
 conclude that the variable part of Y is a homogeneous function 
 
 * See the reasoning of Lagrange in an analogous case, Mecanique Analytique, 
 torn. i. p. 68. 
 
156 On a Dynamical Theory of 
 
 of the second degree, containing, in its general form, the squares 
 and products of X, F, Z, with six constant coefficients. 
 
 Of these coefficients, the three which multiply the products 
 of the variables may always be made to vanish by changing the 
 directions of the axes of #, y, z. For this is a known property 
 of functions of the second degree, when the co-ordinates are the 
 variables ; and we have shown, in Lemma II., that the quan- 
 tities X, Y, Z are transformed by the very same relations as 
 the co-ordinates themselves. Therefore, in every crystal there 
 exist three rectangular axes, with respect to which the function 
 Y contains only the squares of X, Y 9 Z ; and as it will presently 
 appear that the coefficients of the squares must all be negative, 
 in order that the velocity of propagation may never become ima- 
 ginary, we may consequently write, with reference to these axes, 
 Y = -i(fl 2 X 2 + & 2 r 2 + c 2 ^ 2 ), (2) 
 
 omitting the constant K as having no effect upon the motion. 
 
 The axes of co-ordinates, in this position, are the principal axes 
 of the crystal, and are commonly known by the name of axes of 
 elasticity. Thus the existence of these axes is proved without 
 any hypothesis respecting the arrangement of the particles of 
 the medium. The constants #, b, c are the three principal velo- 
 cities of propagation, as we shall see in the next section. 
 
 Having arrived at the value of Y, we may now take it for 
 the starting point of our theory, and dismiss the assumptions by 
 which we were conducted to it. Supposing, therefore, in the 
 first place, that a plane wave passes through a crystal, we shall 
 seek the laws of its motion from equations (1) and (2), which con- 
 tain everything that is necessary for the solution of the problem. 
 The laws of propagation, as they are called, will in this way be 
 deduced, and they will be found to agree exactly, so far as mag- 
 nitudes are concerned, with those discovered by Fresnel ; but the 
 direction of the vibrations in a polarized., ray will be different 
 from that assigned by him. In the second place, we shall in- 
 vestigate the conditions which are fulfilled when light passes 
 out of one medium into another, and we shall thus obtain the 
 laws of reflexion and refraction at the surface of a crystal. 
 
Crystalline Reflexion and Refraction. 157 
 
 SECT. IV. PROPAGATION OF LIGHT IN A CRYSTALLIZED ME- 
 DIUM. LAWS OF FRESNEL. ALTERATION REQUIRED TO BE 
 MADE IN THEM. WAVE-SURFACE, INDEX-SURFACE, AND 
 THEIR PROPERTIES. 
 
 The principal axes of the crystal being the axes of #, y, s, 
 we have, by equation (2), 
 
 + b z PS Y + c*Z3Z ; (3) 
 
 or, by taking the variations from formulae (c), and interchanging 
 the characteristics d and S, 
 
 dz J \dy dx 
 
 and if we substitute this value in equation (1), and then inte- 
 grate by parts the right-hand member, in order to get rid of the 
 differential coefficients of the variations, we shall obtain 
 
 JJ dydz (c*Z8n - b*Y$Z) + JJ dxdz (a z 
 
 (4) 
 
 j -- u ~J~ *** "^ I a ~j c j 
 
 dy dz J \ dz dx) 
 
 :r - - 
 d# dy 
 
 But as the variations 85, 817, S? are arbitrary and independent, 
 this equation cannot hold unless the double integrals, which re- 
 late to the limits of the system, reduce themselves to zero, leav- 
 ing the equality to subsist, independently of the variations, 
 between the triple integrals alone. Equating, therefore, the 
 
158 On a Dynamical Theory of 
 
 coefficients of the corresponding variations in the triple integrals, 
 we get 
 
 dy 
 
 -- 
 dt z dx dy' 
 
 which are the equations of propagation, giving the expression 
 for the accelerating force parallel to each axis of co-ordinates. 
 
 When there is a single medium extending indefinitely on all 
 sides, the conditions relative to the limits are of no importance, 
 and we have only to consider the equations (5), from which we 
 shall now deduce the laws by which a system of plane waves is 
 propagated. 
 
 Supposing the waves to be parallel to the plane of x /, the 
 displacements will be functions of z f only ; and if *// be any 
 function of the displacements, we shall have, by formulae (d'), 
 
 
 
 T~ = ~T~' ~T~ = T~' cos a > T~ = ~J~f cos P 9 ~T = T> cos 7 9 
 dx dz dx dz dy dz dz dz 
 
 so that the equations (5) may be written 
 
 
 _ _ -- / 
 
 = a- 
 
 d& dz' dz' 
 
 and when we combine these with the following, 
 
Crystalline Reflexion and Refraction . 159 
 
 attending to the relations (B), (B'), we find 
 dX , . dY 
 
 _ = _ ^ r) 
 
 ** 0- 
 
 rf? = ' 
 
 from which it appears that there is no accelerating force in the 
 direction of a normal* to the wave, and consequently no vibra- 
 tion in that direction. Introducing now the values of Jf, F, Z 
 from formulae (E), the first two of these equations become 
 
 - (a 2 cos a cos a' + b z cos )3 cos )3' + <? cos 7 cos y') -~J 
 
 j^- = (a 2 cos 2 a + b* cos 2 j3 + c 2 cos 2 7) -^ 
 
 - (a 2 cos a cos a' + > 2 cos /3 cos /3' + c 2 cos 7 cos 7') -r-n. 
 
 fife 
 
 But as the axes of /, y' are arbitrarily taken in the plane of x' y' 
 we may subject their directions to the condition 
 
 a 2 cos a cos a + b* cos |3 cos j3' + c 2 cos 7 cos 7' = ; (7) 
 
 * In the ingenious, but altogether unsatisfactory theory, by which Fresnel has 
 endeavoured to account for his beautiful laws, the direction of the elastic fojrce 
 brought into play by the displacement of the ethereal molecules is, in general, in- 
 clined to the plane of the wave. He supposes, however, that the force normal to 
 that plane does not produce any appreciable effect, by reason of the great resistance 
 which the ether offers to compression. Memoires de V Institut, torn. vii. p. 78. 
 
160 On a Dynamical Theory of 
 
 and then, if we put 
 
 s 2 = a 2 cos 2 a + >' cos 2 13 + c z cos 2 y, 
 
 (8) 
 s' 2 = a* cos 2 a' + # cos 8 j3> c 2 cos 2 y', 
 
 the equations (6) will be reduced to the well-known form 
 
 This result shows that, when the directions of x f and y' fulfil 
 the condition (7), the vibrations ' and r[ are propagated inde- 
 pendently of each other, the former with the velocity of s', the 
 latter with the velocity s. The vibrations must therefore be 
 parallel exclusively to one or other of these directions, else the 
 system of waves will split into two systems, one vibrating pa- 
 rallel to #', the other parallel to ?/. 
 
 "When the plane of the wave is parallel to one of the prin- 
 cipal axes, it is easy to infer that the vibrations must be either 
 parallel or perpendicular to that axis ; and that, in the latter 
 case, the velocity of propagation is constant, being equal to 
 a, b, or c, according as the wave is parallel to the axis of #, y, 
 or z. These constants are therefore called the principal velo- 
 cities of propagation ; and we now perceive the reason of the 
 negative sign in equation (2) ; for if any of the terms in the 
 right-hand member of that equation were positive, the corre- 
 sponding velocity would be imaginary. 
 
 According to Fresnel, the wave which is propagated with 
 the velocity a has its vibrations not perpendicular to the axis 
 of x, but parallel to it ; and it is to be observed that a difference 
 of the same character distinguishes his views, throughout, from 
 the results of the present theory. It will appear in fact, by 
 what immediately follows, that the equations (7), (8), (9), ex- 
 press exactly the laws of Fresnel, provided the quantities ' and 
 ij', in the equations (9), be interchanged. To make these laws 
 agree with our theory, it is therefore necessary to alter them in 
 one particular, and in one only ; it is necessary to suppose that 
 
Crystalline Reflexion and Refraction. 161 
 
 the direction of the vibrations is always perpendicular to that 
 assigned by Fresnel. And since, in order to make his views 
 agree with the phenomena, Fresnel was obliged to say that, in 
 an ordinary medium, the vibrations of a ray polarized in a cer- 
 tain plane are perpendicular to that plane, it is clear that, on the 
 present principles, we must come to a different conclusion, and 
 say that the vibrations of a polarized ray are parallel to its plane 
 of polarization. 
 
 Conceive an ellipsoid with its centre at 0, the common origin 
 of the co-ordinates a?, y, z, #', /, z' ; and let its semiaxes be pa- 
 rallel to a, y, z, their lengths being equal M 
 
 to -, T , - respectively. From the iden- 
 a b c 
 
 tity of the condition (7) with that 
 marked (G') in Lemma III., it is evi- 
 dent that the directions of x f and y f , when 
 they are the two directions of vibration, 
 coincide with the axes of the ellipse in 
 which the plane of x'y' intersects the el- 
 lipsoid ; and if the right line OR, meet- 
 
 Q 
 
 Fig. 20. 
 
 ing the ellipsoid in R, be the direction of x, we have 
 1 
 
 or, by (8), 
 
 (OR)'' 
 
 a? cos 2 a + b z cos 2 /3 + c z eos 2 y, 
 
 1 
 OR~ S ' 
 
 so that OR is the reciprocal of the velocity with which the vi- 
 brations parallel to y' are propagated. Thus we see that the 
 vibrations parallel to either semiaxis of the elliptic section are 
 propagated with a velocity which is measured by the reciprocal 
 of the other semiaxis. 
 
 Again, conceive an ellipsoid with its centre at 0, and its 
 semiaxes parallel to #, y, z, as before, but equal to a, b, c re- 
 spectively. Let this ellipsoid be touched in the point Q by a 
 plane which cuts OR perpendicularly in P, and draw the right 
 lines OP, PQ. Then as the condition (7) is identical with that 
 
 ivr 
 
1 62 On a Dynamical Theory of 
 
 marked (i) in the corollary to Lemma IV., it follows that Oy' 
 (if we so call the direction of y'} is perpendicular to OQ, and 
 also that Oy' and OQ coincide with the axes of the elliptic sec- 
 tion made in this ellipsoid by the plane Q,0y', just as Oy' and 
 OR coincide with the axes of the section ROy' in the first ellip- 
 soid. The plane QOR is therefore perpendicular to Oy and to 
 the plane of the wave. Moreover, we have 
 
 (OP) 2 = a? cos 2 a + b 2 cos 2 /3 + c* cos 2 ? = s 2 , 
 
 so that OP is the reciprocal of OR, and is equal to the velocity 
 s with which the wave is propagated when its vibrations are 
 parallel to Oy' . 
 
 Now let the figure TOSH be equal in all respects to QOPR, 
 but in a position perpendicular to it, so that if QOPR were 
 turned round in its own plane through a right angle, the point 
 being fixed, the points Q, P, R would fall upon T, 8, M re- 
 spectively ; and supposing the wave-plane ROy' to take various 
 positions passing through 0, imagine a construction similar to 
 the preceding one to be always made by means of the two ellip- 
 soids. Then while the points R and Q describe the ellipsoids, 
 the points M and T describe two biaxal* surfaces reciprocal to 
 each other, the latter surface being touchedf in the point T by 
 a plane which cuts OM perpendicularly in 8. But this plane 
 is parallel to the central wave-plane ROy' , and distant from it 
 by an interval OS (= OP) which represents the velocity of the 
 wave ; and as the surface whose tangent planes possess this 
 property is, by definition, the wave-surface of the crystal, it is 
 obvious that the point T describes the wave-surface. The radius 
 OT, drawn to the point of contact, is then, by the theory of 
 waves, the direction of the ray which belongs to the wave ROy', 
 and the length OT represents the velocity of light along the 
 ray. As to the surface described by the point M 9 it is that 
 
 * See Transactions of the Royal Irish Academy, VOL. xvii. p. 244 (supra, p. 24). 
 f Ibid. VOL. xvi. p. 6 (supra, p. 4). 
 
Crystalline Reflexion and Refraction. 163 
 
 which I have called the surface of indices, or the index-surface* 
 because its radius ON, being the reciprocal of OS, represents 
 the index of refraction, or the ratio of the sine of the angle of 
 incidence to the sine of the angle of refraction, when the wave 
 ROy', to which OM is perpendicular, is supposed to have passed 
 into the crystal out of an ordinary medium in which the velocity 
 of propagation is unity. The angles of incidence and refrac- 
 tion are understood to be the angles which the incident and 
 refracted waves respectively make with the refracting surface 
 of the crystal. 
 
 The wave-surface and the index-surface have the same geo- 
 metrical properties since they are both biaxal surfaces. Let us 
 consider the former, which is generated by the ellipsoid whose 
 semiaxes are a, b, c ; and let us conceive this ellipsoid to be in- 
 tersected by a concentric sphere of which the radius is r. Then 
 the equations of the ellipsoid and the sphere being respectively 
 
 x z + y z + z z 
 L ' ~~ 
 
 we get, by subtracting the one from the other, 
 
 for the equation of the cone A which has its vertex at 0, and 
 passes through the curve of intersection. If OQ be equal to r, 
 it will be a side of this cone ; and a plane touching the cone 
 along OQ will make in the ellipsoid a section of which OQ will 
 be a semiaxis ; so that OT will be perpendicular to that plane, 
 and equal in length to r. Therefore, as OQ describes the cone 
 A, the right line OT describes another cone B reciprocal to A, 
 and the point T describes the curve in which the wave-surface is 
 intersected by the sphere above mentioned ; this curve being a 
 spherical ellipse, reciprocal to that which the point Q describes on 
 
 * See Transactions of the Royal Irish Academy, VOL. xvui. p. 38 (supra, p. 96). 
 I had previously called it the surface of refraction, YOL. xvu. p. 252 (supra, p. 36). 
 
164 On a Dynamical Theory of 
 
 the surface of the ellipsoid. The equation of the cone B is 
 found from that of A, by changing the coefficients of the squares 
 of the variables into their reciprocals, and is therefore 
 
 which, of course, is also the equation of the wave-surface, if r be 
 supposed to be the radius drawn from to the point whose 
 co-ordinates are #, y, z. Combining this equation with that of 
 the sphere, we have 
 
 /v* 0y* #2 
 
 2_^2 + IT~T2 + 12 J = 1> (12) 
 
 which represents a hyperboloid passing through the common 
 intersection of the sphere, the cone J5, and the wave-surface. 
 
 Since the differences between the coefficients of the squares 
 of the variables in the equation (10) are the same as the corre- 
 sponding differences in the equation of the ellipsoid, the cone A 
 has its planes of circular section coincident with those of the 
 ellipsoid. The cone .#, being reciprocal to A, has therefore its 
 focal lines perpendicular to the circular sections of the ellipsoid. 
 These focal lines are consequently the nodal diameters* of the 
 wave-surface, that is, the diameters which pass through the 
 points where the two sheets of that surface intersect each 
 other. 
 
 If the direction of OT cut the other sheet of the wave- 
 surface in T', and if two radii of constant lengths, equal to OT 
 and OT' respectively, revolve within the surface, the cones B 
 and B' described by these radii will intersect each other at right 
 angles, since they have the same focal lines. And supposing 
 the axis of y to be the mean axis of the ellipsoid, so that the 
 nodal diameters lie in the plane of a?s, the axis of x will lie 
 within one of the cones, as B, and the axis of z within the other 
 cone B'. Now the angle contained by the two sides of either 
 
 * See Transactions of the Royal Irish Academy, VOL. xvn. p. 247 (supra, p. 29). 
 
Crystalline Reflexion and Refraction. 165 
 
 cone, which lie in the plane of #s, is given by the angles 6 and 
 6' which the direction of the right line OTT' makes with the 
 nodal diameters ; because the angles which any side of a cone 
 makes with its focal lines have a constant sum, or a constant 
 difference, according to the way in which they are reckoned. 
 But if the angles and 6' be reckoned (as they may be) so that 
 their sum shall be equal to the angle contained by the two sides 
 of the cone B which are in the plane of #s, their difference will 
 be equal to the angle contained by the two sides of the cone B' 
 which are in the same plane ; the contained angle, in each.case, 
 being that which is bisected by the axis of x. Therefore, the 
 lengths OTand OT', which we denote by r and /, are equal to 
 two radii of the ellipse whose equation is 
 
 - + -T = 1, 
 
 a 2 c 2 
 
 these radii making with the axis of z the angles i (0 + 0') and 
 % (9 - 6') respectively. Hence 
 
 !_ = sin 2 j- (0 + 0') cos 2 j- (0 + 0') _ n 1 
 = ~~ ~~ ~ 
 
 (13) 
 
 _! _ sin 2 j- (0-0') cos 2 i(0-0') _ 
 r'*~ a* c* * 
 
 These formulae give the two velocities of propagation along a 
 ray which makes the angles 0, / with the nodal diameters. 
 Subtracting them, we have 
 
 - = - 8in e sin ff - 
 
 All the preceding equations, relative to the wave-surface, 
 
1 66 On a Dynamical Theory of 
 
 may be transferred to the index-surface, by changing the quan- 
 tities a, b, c into their reciprocals. For example, if the normal 
 to a wave make the angles , 0i with the nodal diameters of 
 the index-surface, the formulae (13) give 
 
 s 2 = I- (a 2 + c 2 ) - i (a* - c 2 ) cos (0o + ft), 
 
 (15) 
 <*) - ( - c 2 ) cos (0 - ft) ; 
 
 observing that s and /, the two normal velocities of propaga- 
 tion^ are the reciprocals of the radii of this surface which coin- 
 cide with the wave-normal. Subtracting these expressions, we 
 
 get 
 
 s 2 - s ' 2 = (a* - c 2 ) sin sin ft. (16) 
 
 As the position of the tangent plane, at any point T of a 
 biaxal surface, depends on the position of the axes of the section 
 QOy' made in the generating ellipsoid by a plane perpendicular 
 to OT, it is obvious that when this section is a circle, that is, 
 when the point Tis a node of the surface, the position of the 
 tangent plane is indeterminate, like that of the axes of the 
 section ; and it is easy to show that the cone which that plane 
 touches in all its positions is of the second order. Again, when 
 the section ROy' of the reciprocal ellipsoid is a circle, the right 
 line OS is given both in position and length ; and the tangent 
 plane, which cuts OS in S, is fixed ; but the point of contact T 
 is not fixed, since the semiaxis OR, to which the right line ST 
 is parallel, may be any radius of the circle ROtf. In this case, 
 the point T describes a curve e in the tangent plane, and this 
 curve is found to be a circle. But both these cases have been 
 fully discussed elsewhere.* 
 
 * See Transactions of the Royal Irish Academy, VOL. xvn. pp. 245, 260 (supra, 
 pp. 25-7, 49-51). 
 
Crystalline Reflexion and Refraction. 167 
 
 SECT. V. CONDITIONS TO BE SATISFIED WHEN LIGHT PASSES 
 OUT OF ONE MEDIUM INTO ANOTHER. EEMARKABLE CIRCUM- 
 STANCES CONNECTED WITH THEM. RELATIONS AMONG THE 
 
 TRANSVERSALS OF THE INCIDENT, REFLECTED, AND RE- 
 FRACTED RAYS. 
 
 Now let light pass out of one medium into another sup- 
 pose out of an ordinary into a doubly-refracting medium ; and 
 taking the origin of rectangular co-ordinates # fl , y n , z at a point 
 on the surface which separates the two media, let this surface 
 be the plane of # y*> Then if the components of the displace- 
 ment of a particle whose initial co-ordinates are # , ^o, %o be 
 denoted by ' , TJ' O , ?'o when the particle is in the first medium, 
 and by ", rj " 5 o" when it is in the second, the equation (1), 
 adapted to the present case, will be 
 
 ?' /7 2 */ AW 
 
 * Z' , a ^ $ ' , a * s> 
 
 S + - 8.1. + - 8 
 
 = Iff dx Q dy dz Q 8V + Iff dx, dy, dz SV" ; (17) 
 
 wherein 8V and 8 V are the respective values of SV for the two 
 media, which are conceived to extend indefinitely on each side 
 of the plane of X Q y Q ; that plane being an upper limit of the 
 integrations relative to one medium, and a lower limit of the 
 integrations relative to the other. Each medium is conceived 
 to be occupied by systems of plane waves the first by incident 
 and reflected waves, the second by refracted waves ; and, except 
 where they are bounded by the plane of X Q y , these waves are 
 regarded as unlimited in extent. 
 
 For the ordinary medium, if we put 
 
 dz dy^ dx Q dz^ r/// dx Q J 
 
1 68 On a Dynamical Theory of 
 
 and suppose the velocity of propagation to be unity, we have 
 
 z dy Q ) \ dx Q dz 
 
 For the crystallized medium, if its principal axes be those of 
 a?, y, a, the value of 8V" will be the same as that of 8V in for- 
 mula (3) ; but instead of the variations of ?, rj, , we must use 
 those of " , ij" , " . Denoting the cosines of the angles which 
 the principal axes respectively make with the axis of X Q by /, m, n ; 
 with the axis of y by f , m', ri ; with the axis of Z Q by T, m", n" ; 
 and putting 
 
 Yf, _ dvi"o d%" _ dZ" d" _ d" Q dr{' 9 
 
 "**- ~ ~~j i > * o~ ; ; > " o j j > 
 
 azo dy Q dx Q az dy dx Q 
 
 we have 
 
 = 
 
 These expressions for SX, 8 F, SZ" having been written in 
 formula (3), the resulting value of 8 V", as well as the above value 
 of 8V', is to be substituted in the equation (17), and then the 
 right-hand member of that equation is to be integrated by parts, 
 in order to get rid of the differential coefficients of the varia- 
 tions. When this operation is performed, the triple integrals 
 on one side of the equation will be equal to those on the other ; 
 and by equating the coefficients of the corresponding variations 
 
 * It is assumed here, and in what follows, that when there are two or more 
 coexisting waves in a given medium, the form of the function V is the same as for 
 a single wave, provided the displacements which enter into the function be the re- 
 sultants of the displacements due to each wave separately. This, however, ought 
 evidently to be the case, in order that the principle of the superposition of vibrations 
 may hold good. 
 
Crystalline Reflexion and Refraction. 169 
 
 in each medium, we should get the laws of propagation in each. 
 But we are not now considering these laws, and we need only 
 attend to the double integrals produced by the operation afore- 
 said. The double integrals are together equal to zero ; but we 
 are concerned only with that part of them which relates to the 
 common limit of the media, the plane of x Q y^\ and this part 
 must be separately equal to zero, since the conditions to be ful- 
 filled at the plane of X Q y Q are independent of anything that 
 might take place at other limiting surfaces, if such were sup- 
 posed to exist. Collecting therefore the terms produced by in- 
 tegrating with respect to s , and observing that a negative sign 
 must be interposed between those which belong to different 
 media, we get 
 
 y Q (F' S' - *' &i'o) - If dx, dy Q (Q$?' - P^o) = 0, (18) 
 where 
 
 P = a 2 lX + WmY + c z nZ, Q = a*?X + b*m'Y + c*n'Z. (19) 
 
 In each of these equations it is understood that s = 0. But 
 when z = 0, we have obviously 
 
 r, - r, * - ," 
 
 and therefore 
 
 Sc, Q = o? o) 017 = Sj o ; 
 
 so that the equation (18) becomes 
 
 which, as the variations rf' and dif Q are arbitrary and indepen- 
 dent, is equivalent to the two equations 
 
 X\ = P, Y\ = Q. (21) 
 
 Thus, to find the relations which subsist among the vibra- 
 tions' incident, reflected, and refracted, at the common surface 
 of two media, we have four conditions, expressed by the equa- 
 tions (20) and (21) ; and these conditions are sufficient to deter- 
 mine the reflected and refracted vibrations, when the incident 
 
170 On a Dynamical Theory of 
 
 vibration is given. But though, by the nature of the question, 
 four conditions only are required for its solution, there remains 
 another condition which ought to be satisfied; for we ought 
 evidently to have 
 
 = 0. (22) 
 
 This condition is apparently independent of the rest ; but it 
 cannot really be so, if the preceding theory is consistent with 
 itself. We shall accordingly see, in what follows, that the last 
 condition is included in the other four ; which is a remarkable 
 circumstance, and a singular confirmation of the theory.* 
 
 As the incident and reflected waves coexist in the first me- 
 dium, and two sets of refracted waves in the second, the resolved 
 displacements, and all the quantities which depend upon them, 
 are composed of two parts, due to the coexisting waves. Let 
 the point be, for each set of waves, the origin of a system of 
 rectangular co-ordinates, which we shall call x^ y^ z l for the in- 
 cident, and o/i, /'i, s'i for the reflected wave, the axes of z l and z\ 
 being perpendicular to the respective waves, and their positive 
 directions being those of propagation. Let the displacements in 
 these waves be parallel to y\, y\, and be denoted by iji, 1/1, re- 
 spectively. Then if the axes of # , / , z make with the axis of 
 #! the angles ai, |3i, 71, and with the axis of x\ the angles 
 a'i, /3'i, y'i, we have, by the formulae (F), 
 
 J'. = P cos , + C ^ cos ' r o = pi cos ft + J < /3',. (23) 
 azi az i uZi az i 
 
 Again, let the co-ordinates #2,2/2, s 2 have reference to one set of 
 refracted waves, and #' 2 , y' 2 , s' 2 to the other, the axes of z 2 
 and z' z being perpendicular to the respective waves, and their 
 positive directions being those in which the waves are propa- 
 
 * In considering the question of reflexion at the common surface of two ordi- 
 nary media (Memoires de VInstitut, torn. ix. p. 396), Fresnel assumes the conditions 
 (20) ; but his other suppositions violate the condition (22). In fact, this last con- 
 dition is inconsistent with the supposition that, in a polarized ray, the direction of 
 the vibrations is perpendicular to the plane of polarization. See the Transactions of 
 the Eoyal Irish Academy, VOL. xvui. p. 32 (supra, p. 88). 
 
Crystalline Reflexion and Refraction. 1 7 1 
 
 gated. Suppose the displacements to be parallel to y^ y'^ and 
 to be denoted by 172, *A respectively. Then if the axes of x y y, z 
 make with the axis of x z the angles ct( 2 ), j3( 2 ), 7(2), and with the 
 axis of x\ the angles a'( 2 ), j3'( 2 ), 7(2), we have, by the formulae^), 
 
 driz din?, 
 
 ^=-cos 7(2) + 5? - 
 and thence, by the relations (19), 
 
 P = -^ (a 2 1 cos a (2) + b*m cos/3( 2 ) + c*n 0087(2)) 
 
 6f 2 
 
 7 / 
 
 + -^- (al z cos a' ( 2 ) + b*m cos j3( 2 ) + c 2 n cos y'w), 
 az 2 
 
 Q = ~ (a?l f cos a ( 2 ) + b*m' cos )3( 2 ) + cV cos 7(2)) 
 uZ'z 
 
 + -^rlW' cos o'(2) + b z m' cos j3 r ( 2 ) + cV cos y'(a)). 
 
 2 2 
 
 Suppose the ellipsoid which generates the wave-surface of 
 the second medium to have its centre at 0, and to be touched in 
 the points Q and Q' by two planes which cut the axes of a? 2 and 
 x\ perpendicularly in the points P and P f ; the lengths OP and 
 OP' being expressed by s, s', and the lengths OQ and OQ' by 
 r, /. Let the axes of a? , / , s make with the direction of OQ 
 the angles a 2 , /3 2 , 72, and with the direction of OQ! the angles 
 a' 2 , j3' 2 , 7^2- Then, from the equation (H) in Lemma IV., it is 
 manifest that 
 
 a 2 1 cos a ( 2 ) + b z m cos/3(2) + c z n cos 7(2) = rs cosa 2 , 
 a z l cos a' ( 2 ) + b z m cos j3'( 2 ) + c z n cos 7%) = rV cos a^, 
 cfl' cos a ( 2 ) + ^ 2 ^ cos j3( 2 ) 4 cV cos 7(2) = rs cos j3 2 , 
 a 2 / r cos a' (2) + b~m cos /3'( 2 ) + c 2 ^' cos 7^2) = rs cos jS^. 
 
172 On a Dynamical Theory of 
 
 Hence, 
 
 n i / / / 
 
 P = rs cos a 2 + r s cos a 2 , 
 az 2 dz z 
 
 (24) 
 
 The quantities s, s' are, as appears by the last section, the 
 normal velocities with which the two sets of refracted waves are 
 propagated. The velocity with which the incident and reflected 
 waves are propagated is taken as unity. Therefore, if TI, T\ be 
 the transversals, or amplitudes of vibration in the incident and 
 reflected waves, and r 2 , r' 2 the transversals of the refracted waves, 
 the lengths of the latter waves being denoted by A 2 , A' 2) and the 
 length of an incident or reflected wave by AI, and if we put 
 
 fr=??L (t-z l + Vl ) ^ l= ,^L( t - z \ + v \\ 
 
 A 2 A 2 
 
 where vi, 1/1, v 2 , v' 2 are constants, and IT is the ratio of the cir- 
 cumference to the diameter of a circle, we may write 
 
 rii = TI COS 0i, t/i = T'I COS 0'i, IJ 2 = T 2 COS 2 , i/ 2 = / 2 COS 0' 2 . (26) 
 
 By means of these values the formulse (23) and (24) become 
 X' = Y~ (TI cos ai sin 0i + T'I cos a\ sin 0'J, 
 
 F 7 ^,:: (TI cos/3i sin 0i + /i cos|3'i sin^'i), 
 
 P = 2ir (r- TZ co&az sin 2 + -p- r\ cos a\ sin r 2 ), 
 \A 2 A 2 J 
 
 Q = 2ir ( ^ T 2 cos /3 3 sin 2 + TT- r' a cos^ r 2 sin 0' 2 ]. 
 \A 2 A 2 / 
 
 The angles 0i, 0'i, 2 , 0' 2 are the phases of vibration in the 
 
 (27) 
 
Crystalline Reflexion and Refraction. 173 
 
 different waves at the time t. To see how they depend on the 
 co-ordinates x w y m 2 , conceive the axis of z to be directed from 
 towards the interior of the second medium, and the axis of X Q 
 to lie in the plane of incidence, so that the positive directions of 
 21, 2 2 > s'a may lie within the angle made by the positive direc- 
 tions of X Q and 2 , while the positive direction of z\ lies within 
 the angle made by the positive direction of X Q and the negative 
 direction of s . Let ^ be the angle of incidence, and i z , i\ the 
 angles of refraction ; then 
 
 21 = x<> sin n + s cos ii t z\ = # sin h - Z Q cos i it 
 
 (28) 
 z z = X Q sin iz + s cos i z , z 2 = X Q sin i 2 + z cos i . 
 
 These values are to be written in the expressions (25). They 
 show that the phases, and therefore the displacements, are in- 
 dependent of y Q . 
 
 Since the conditions relative to the plane of # ?/o must hold 
 at. every instant of time, and for every point of that plane, the 
 co-efficients of t, as well as those of a? > in the values of the 
 different phases, must be identical; so that we must have 
 
 1 _ s s' sin ii _ sin i t _ sin i' z 
 
 AI A2 A 2 AI \z A 2 
 
 Therefore, when Z Q = 0, the supposition 
 
 Vi = l/i = V Z = l/ 2 (30) 
 
 renders the phases identical, independently of t and X Q . And, 
 from the form of the equations of condition, it is easy to see 
 that this supposition is necessary; because the equations (20), 
 when the values (26) are substituted in them, contain only the 
 cosines of the phases ; and the equations (21), when the values 
 (27) are substituted in them, contain only the sines of the 
 
174 On a Dynamical Theory of 
 
 phases. Making the latter substitution, and attending to the 
 relations just mentioned, we find 
 
 TI COS QI + T\ COS a'i = rr 2 COS a 2 + /r' 2 COS a' 2 , 
 
 (31) 
 TI cos ]3i + T\ cos )3'i = rr 2 cos j3 2 + // 2 cos /3' 2 . 
 
 In these equations, the angles by whose cosines each trans- 
 versal is multiplied are the angles which a plane, passing 
 through the directions of that transversal and of the corre- 
 sponding ray, makes with the planes of y Q z and X Q z . This 
 is evident with regard to the incident and reflected rays. And 
 if we refer to the diagram in the preceding section, it will also 
 be evident with regard to the refracted rays ; for OQ is perpen- 
 dicular to the transversal r 2 , and to the right line OI 7 , which is 
 the direction of the corresponding ray. 
 
 Taking for the point of incidence, let right lines proceed- 
 ing from it represent the different rays ; and let the length of 
 each ray, measured from in the direction of propagation, be 
 assumed proportional to the velocity with which the light is 
 propagated along it. Through the extremity of each ray con- 
 ceive its transversal to be drawn, and let the transversals so 
 drawn have their moments taken, with respect to the point 0, 
 as if they represented forces applied to a rigid body. The 
 length of the incident or reflected ray being considered as 
 unity, the lengths of the refracted rays (as appears by the 
 last Section) are r and r f respectively. Hence, as each trans- 
 versal is perpendicular to its ray, the moments of the incident 
 and reflected transversals are proportional to TI, r'i, and the 
 moments of the refracted transversals to rr 2 , /r' 2 respectively. 
 The equations (31) therefore signify that when the moments 
 are projected, either upon the plane of y Q s , or upon the plane 
 of #o So, the total projected moments are the same for the two 
 media ; or that, if the transversals themselves be projected on 
 either of these planes, the moments of the projections of the 
 incident and reflected transversals are together equal to the 
 moments of the projections of the refracted transversals. 
 
Crystalline Reflexion and Refraction. 175 
 
 But the second of the equations (31) has another signifi- 
 cation. For if the transversals applied at the extremities of 
 the refracted rays be projected on the plane of # s< which is 
 the plane of incidence, and contains the axes of z z and s' 2 , the 
 projections will be perpendicular to these axes, since the trans- 
 versals themselves are perpendicular to them ; and the distances 
 of the projections from the point will be proportional to 8 
 and s', or, by the relations (29), to sin i 2 and sin / 2 ; so that 
 if 2 and 0' 2 be the angles which the transversals make with 
 the plane of incidence, the moments of the projections will 
 be represented by r 2 cos 3 sin 4 and / 2 cos 0' 2 sin / 2 . At the 
 same time, if 0i and 9\ be the angles which the incident and re- 
 flected transversals make with the plane of incidence, the mo- 
 ments of the corresponding projections of these transversals will 
 evidently be represented by TI cos 0i sin i l9 and - T\ cos 0\ sin ii ; 
 the latter quantity being taken with a negative sign, because 
 the extremity of the reflected ray, where the transversal T\ 
 is applied, lies in the first medium, while the extremities of 
 the incident and refracted rays lie in the second, and it is 
 supposed that when any of the angles O l9 0\, 2 , 0' 2 is zero, 
 the direction of the corresponding transversal makes an acute 
 angle with the axis of o? . Hence we have 
 
 TI cos 0i sin h - T\ cos 9\ sin ^ = r 2 cos 2 sin i z + r' 2 cos 0' 2 sin i\ ; 
 
 an equation which expresses that if each transversal be pro- 
 jected upon the axis of s 0) the sum of the projections of the 
 incident and reflected transversals will be equal to the sum 
 of the projections of the refracted transversals. Therefore, 
 since the phases of the different vibrations are identical when 
 Z Q = 0, the condition (22) is fulfilled, as it ought to be. 
 
 On account of this identity of phases, it follows from the 
 conditions (20) and (22), that if the transversals be drawn 
 through the point 0, and those which belong to each medium 
 be compounded like forces acting at a point, their resultants 
 will be the same ; that is, the resultant of the incident and 
 
176 On a Dynamical Theory of 
 
 reflected transversals will be the same as the resultant of the 
 refracted transversals. 
 
 Hence, recollecting what has heen proved respecting the 
 moments of the transversals applied at the extremities of the 
 rays, we have the following theorem : 
 
 Supposing the length of each ray, measured from the point 
 of incidence and in the direction of propagation, to be taken 
 proportional to the velocity with which the light is propagated 
 along it, and its transversal to be drawn through the extremity 
 of this length, the incident and reflected transversals having 
 their proper directions, but the refracted transversals having 
 their directions reversed ; if all the transversals so drawn be 
 compounded like forces applied to a rigid body, their resultant 
 will be a couple, lying in a plane parallel to the plane which 
 separates the two media. 
 
 This theorem affords a complete solution of the question 
 of reflexion and refraction.* Expressed analytically it gives 
 five equations, of which four are independent. 
 
 To apply the preceding results to a simple case, suppose 
 the second medium, as well as the first, to be an ordinary one. 
 We have then only one refracted ray, and one refracted trans- 
 versal T Z . 
 
 1. When the incident ray is polarized in the plane of inci- 
 dence, the transversals are all in that plane ; and as they are 
 perpendicular to the rays, and the refracted transversal is the 
 resultant of the other two, we have evidently 
 
 , sin fe - 2 ) sin 
 
 r 2 = 
 
 i I - j-. , 2 I -p . 
 
 sin (^ + * a ; sin (f| + H) 
 
 2. When the incident ray is polarized perpendicularly to 
 
 * The same theorem applies to the other case of reflexion and refraction, when 
 a ray which has entered the crystal emerges from it into an ordinary medium, 
 undergoing double reflexion at the surface where it emerges. In fact, the con- 
 ditions (20) and (21) hold good whether the ordinary medium is the first or the 
 second; and in the latter" case, as well as in the former, it may be shown that 
 the condition (22) is fulfilled, and that the theorem above mentioned is true. 
 
Crystalline Reflexion and Refraction. 177 
 
 the plane of incidence, the transversals are all perpendicular 
 to that plane. Taking 2 sin *\ to represent the length of the 
 incident or reflected ray, the proportional length of the re- 
 fracted ray is 2 sin fl' 2 , and the projections of these lengths on 
 the plane of y So are 2 sin e\ cos ^ and 2 sin i* cos * 2 , or sin 2^ 
 and sin 2 i 2 . The transversals applied at the extremities of the 
 rays are not altered by being projected on the plane of y s ; 
 therefore the moments of the incident, reflected, and refracted 
 transversals, projected on this plane, are represented by the 
 quantities TI sin 2^ - T\ sin 2&i, and r z sin 2i 2 respectively. 
 Equating the last moment to the sum of the other two, and 
 the refracted transversal to the sum of the other two trans- 
 versals, we get 
 
 (TI - T'I) sin 2ii = T Z sin 2/ 2 , TI + r\ = r 2 ; 
 
 and thence 
 
 tan (! - 4) sin 2\ 
 
 i I . -/-, . Z i . . 
 
 tan fyi + 4) sm (*i + h) cos (^ - 4) 
 
 This case has been considered by JPresnel. The relative 
 magnitudes of the incident and reflected transversals, as given 
 by him, are in accordance* with the formulae (32) and (33) ; 
 but with respect to the refracted transversals, his results do 
 not agree with the formulae. 
 
 SECT. VI. PRESERVATION OF Vis VIVA THEOREM OF THE 
 POLAR PLANE CONCLUSION. 
 
 Eeturning to the general question, if we resolve the trans- 
 versals parallel to the axes of # , 2/ 0> *o, and equate the sums 
 of the parallel components in one medium to the corresponding 
 
 * There is, however, a difference as to the relative directions of the incident 
 and reflected transversals. "When the second medium is the denser, and the inci- 
 dence is perpendicular, these transversals, according to the present theory, have 
 the same direction, but according to Fresnel they have opposite directions. 
 
 TV 
 
1 78 On a Dynamical Theory of 
 
 sums in the other, we get the three conditions 
 
 (T! COS 0! + r'i COS S\) COS \ = T 2 COS 2 COS l a + r' 2 COS 0' 2 COS / 2 , 
 
 TI sin 0! + T'I sin / 1 = r 2 sin 2 + r' 2 sin 0' 2 , (34) 
 
 (TI cos 0, - r'i cos / i) sin e\ =r 2 cos 2 sin ^ + r' 2 cos 0' 2 sin i\. 
 
 A fourth condition is supplied by the first of the equations (31), 
 in which equation we have to write 
 
 cos H! = sin 0! cos i, cos a\ - - sin 9\ cos i i9 
 
 and to substitute similar expressions for cos a 2 , cos a' 2 . 
 
 The right line OQ is perpendicular to the transversal r 2 and 
 to the ray OT. The cosines of the angles a 2 , ]3 2 , y 2 may therefore 
 be found by means of the cosines of the angles which the trans- 
 versal and the ray make with the axes of a? > y^ *o 
 
 The cosines of the angles which the transversal r 2 makes 
 with these axes are respectively 
 
 cos 2 cos iz, sin 2 , - cos 2 sin i 2 . 
 
 As the plane which passes through the ray and the wave- 
 normal OS is perpendicular to the transversal T 2 , this plane 
 makes with the plane of incidence an angle equal to 90 + 2 or 
 90 - 2 . Let a sphere, having its centre at 0, be intersected in 
 the points $ , T Q by the right lines OS, OT, and in the points 
 X 09 Fo, ZQ by the axes of ar , ^o, s ; and conceive the points T Q 
 and YQ to be at the same side of the plane # 0} * > ^ ne spherical 
 angle T S X being 90 + 2 , and the spherical angle T S Z 
 being 90 - 2 . Let e be the angle which the ray makes with 
 the wave-normal. Then, the angles which the ray makes with 
 the axes of co-ordinates being measured by the arcs T X Q , T Y , 
 T Z , the cosines of these angles respectively are 
 
 sin it cos c - sin 2 cos 4 sin e, cos 2 sin c, 
 
 cos iz cos c + sin 2 sin i z sin t. 
 Hence, as the transversal is at right angles to the ray, we 
 
Crystalline Reflexion and Refraction. 179 
 
 have, by Lemma I., 
 
 cos a 2 = sin & 2 sin + sin 2 cos 2 cos c, cos j3 2 = - cos 2 cos c, 
 
 cos y 2 = cos 2 sin * - sin 2 sin i z cos f. (35) 
 
 In like manner, putting c' for the angle which the other re- 
 fracted ray makes with its wave-normal, we have 
 
 cos a' 2 = sin *" 2 sin c' + sin 0' 2 cos i\ cos c', cos /3' 2 = - cos 0' 2 cos E ', 
 cos 7' 2 = cos i\ sin t' sin 0' 2 sin ' 2 cos c'. (36) 
 
 If we substitute, in the first of the equations (31), the values 
 just given for cos a 2 , cos a' 2 , along with the above values of 
 cos ai, cos o'i, and attend to the relations 
 
 T COS = S. / COS e = /, 
 
 (37) 
 sin 4 = * sin e'i, sin i\ = s' sin i iy 
 
 we find, after multiplying all the terms of the equation by 
 sin t\, 
 
 (TI sin 0i - r'i sin 0'i) sin t'i cos e\ = r 2 (sin 2 sin ea cos 2 
 
 (38) 
 + sin^ 2 tan a) + r' 2 (sin ^ 2 sin i' t cos / 2 + sin 2 / 2 tan e). 
 
 Joining this equation to the equations (34) we have all the con- 
 ditions that are necessary for the solution of the question. 
 
 Multiplying the first of the equations (34) by the third, also 
 the second of these equations by the equation (38) , adding the 
 products together, and then dividing by sin i^ we obtain 
 
 ^ (n 2 - r?) = JI,T,' + /,rV + Mr z r^ (39) 
 
 where we have put 
 
 /ui = cos e\, fji z = s (cos 4 + sin 2 sin h tan ), 
 
 ju r 2 = s' (cos e' z + sin 0' 2 sin ^ 2 tan '), 
 
 M sin i\ = sin (i z + ?*) {cos 2 cos 0' 2 + sin 2 sin 0' 2 cos ( 2 - i\) } 
 + sin ^jj sin 2 i t tan e + sin 2 sin 2 i\ tan s'. 
 
1 80 On a Dynamical Theory of 
 
 The last expression may be put under the form 
 
 M sin fc\ sin (i z - i') 
 = sin 2 2 {cos 2 cos 0' 2 + sin 2 sin 0' 2 cos (i z - ') (40) 
 
 + sin 0' 2 sin ( 2 - ' 2 ) tan e) 
 - sinVajoos 2 cos 0' 2 + sin 2 sin 0' 2 cos (e a - / 2 ) 
 
 - sin 2 sin (i z - ^2) tan c'} . 
 
 Let the axes of x m y , s make with the direction of OP the 
 angles a y/ , /3 //? 7,,, and with the direction of OP' the angles 
 a '//> /3',/j 7'//- The cosines of these angles may be found from 
 the expressions (35) and (36) by supposing e and e' to vanish. 
 Therefore 
 
 cos er /y = sin 2 cos i z , cos /3 7/ = - cos 2 , cos y f/ = - sin 3 sin 2 , 
 
 (41) 
 cos a' y/ = sin 0' 2 cos ^2 / , cos jS'^ = - cos r 2 , cos 7'^ = - sin 0' 2 sin i\. 
 
 If w be the angle which OQ makes with OP', and w' the 
 angle which OQ' makes with OP, so that 
 
 cos u> = cos a 2 cos a' /y + cos j3 2 cos j3' /y + cos 72 cos 7' /y , 
 
 COS O)' = COS a' 2 COS a /7 + COS )3' 2 COS /3 /y + COS y' 9 COS 7 /x , 
 
 we find, by the formulae (35), (36), (41), 
 
 cos w = cos efcos 2 cos 0' 2 + sin 2 sin 0' 2 cos (i z - i' z ) 
 
 + sin 0' 2 sin (i 2 - i' z ] tan cj, 
 
 cos w' = cos s { cos 2 cos 0' 2 + sin 2 sin 0' 2 cos (i z - i' z ) 
 
 - sin 2 sin (i 2 - i' z ) tan c'J. 
 
 Hence, observing the relations (37), we see that the right-hand 
 member of the equation (40) is equal to the quantity 
 
 sin 2 ^ (rs cos w - r's' cos a/). 
 
Crystalline Reflexion and Refraction. 1 8 1 
 
 But, by the property of the ellipsoid (Lemma IV.), this quantity 
 is zero ;* therefore M = 0, and the equation (39) becomes 
 
 /Ui n 2 = m rV + ,W + /x'.rV. (42) 
 
 On each ray let a length, representing the velocity with 
 which the light is propagated along it, be measured, as before, 
 from the point 0. The distances of the plane of # 2A> from the 
 extremities of these lengths will be proportional to the coeffi- 
 cients of the squares of the transversals in the preceding equa- 
 tion. For if we take, on the incident or reflected ray, a length 
 equal to unity, its projection on the axis of s will be cos i { or p\\ 
 and if, on the refracted ray OT, we take a length equal to r, its 
 projection on the same axis will be 
 
 r (cos i-i cos e + sin 2 sin ? 2 sin e), 
 
 which is equal to ju 2 . Similarly, the length /, assumed on the 
 other refracted ray, will have its projection equal to juV The 
 quantities by which the squares of the transversals are multi- 
 plied, in the equation (42), are therefore the corresponding ethe- 
 real volumesf which we may conceive to be put in motion by the 
 different waves ; and as we suppose the density of the ether to 
 be the same in both media, the equation expresses a principle 
 analogous to that of the preservation of vis viva.% 
 
 By giving a certain direction to the incident transversal, 
 that is, by polarizing the incident ray in a certain plane, we may 
 make one of the refracted rays disappear. If OT be the ray 
 which remains, we have r' 2 = 0, and the equations (34) and (38) 
 become 
 
 * The equation M = is the same as the equation (vn.) in my former Paper. 
 Transactions, R.I. A., VOL. xvm. p. 52 (supra, p. 112). 
 
 t Ibid., p. 48 (supra, p. 106). 
 
 J A similar equation of v is viva holds when the light passes out of a crystal into 
 an ordinary medium. 
 
1 82 On a Dynamical Theory of 
 
 (ri cos 0i + r'i cos 0^) cos e'i ~ r 2 cos 2 cos 4, 
 TI sin 0! + r'i sin 0\ = r 2 sin 2 , 
 
 (TI cos 0i - r'i cos 0^) sin e\ = r 2 cos 2 sin e,, (43) 
 
 (ri sin 0i - T'I sin 0\) sin ^ cos ^ 
 
 = r 2 (sin 2 sin 2 cos e a + sin 2 ' 2 tan e). 
 
 In this case, the three transversals are in the same plane, the 
 refracted transversal being the resultant of the other two. 
 Therefore if we find this plane, everything will be determined. 
 
 The axes of # , y w Z Q make, with the incident transversal, 
 angles whose cosines are 
 
 cos 0i cos f'i, sin X , - cos 0! sin *!, 
 and, with the reflected transversal, angles whose cosines are 
 cos 0^ cos i, sin 0\ 9 cos 9\ sin ^ ; 
 
 therefore, by Lemma I., the cosines of the angles which these 
 axes make with a right line perpendicular to the plane of the 
 transversals are proportional to the quantities 
 
 sin (0i + 0'i) sin \, - cos : cos 9\ sin 2*i, sin (B\ - 0i) cos i^ 
 
 Now from the product of the first and second of the equations 
 (43), combined with the product of the third and fourth, we 
 find, by the help of the relations (37), 
 
 2ri/i sin (0! + 0^1) sin e\ = r 2 2 tan \ cos 2 {sin 2 cos i z 
 
 - s 2 (sin 2 cos iz + sin 2 tan e) } . 
 
 From the squares of the first and third of those equations we 
 find 
 
 - Sri/i cos 0i cos 0' t sin 2&\ = r 2 2 tan ^ cos 2 2 (s 2 - 1), 
 
 and from the product of the first and fourth, combined with the 
 product of the second and third, 
 
 2ri r'i sin (0'i - : ) cos \ = T 2 2 tan ^ cos 3 {- sin 2 sin i z 
 
 + s* (sin 2 sin i 2 - cos / 2 tan c ) J . 
 
Crystalline Reflexion and Refraction. 183 
 
 In the three equations just found, the left-hand members are 
 proportional, as we have seen, to the cosines of the angles which 
 a right line perpendicular to the plane of the transversals makes 
 with the axes of co-ordinates ; and the right-hand members, as 
 appears by the formulae (35) and (41), are proportional to the 
 quantities 
 
 cos a,, cos ft cos -v . 
 
 - r cos a 2 , - r COS p, - r cos y 2 , 
 
 s s s 
 
 which are obviously the differences between the corresponding 
 co-ordinates of the points R and Q. The plane of the transver- 
 sals is therefore perpendicular to the right line QR, which joins 
 those points. 
 
 A plane parallel to the right line TM, and passing through 
 the transversal of the ray OT, is that which I have called the 
 polar plane of the ray;* and this plane is perpendicular to QH. 
 Therefore, when there is only one refracted ray, the incident 
 and reflected transversals lie in the polar plane of that ray ; and 
 their directions being thus determined, the relative magnitudes 
 of the three transversals are known. In this case the incident 
 and reflected transversals are called uniradial ; and as each re- 
 fracted ray in turn may be made to disappear, there are two 
 uniradial directions in the plane of the incident wave, and two 
 in that of the reflected wave. 
 
 When the incident transversal is not uniradial, it may be 
 considered as the resultant of two uniradial transversals, each of 
 which will supply a refracted ray, and will produce a uniradial 
 component of the reflected transversal. 
 
 It is needless to extend these deductions further. They 
 have been carried far enough to show that the results of the 
 foregoing theory are in perfect accordance with the laws estab- 
 lished in my former Paper on the subject of crystalline reflexion. 
 The theory itself suggests much matter for consideration ; but 
 at present we shall confine ourselves to one remark, which may 
 
 * Transactions, Koyal Irish Academy, VOL. xvm. p. 39 (supra, p. 96). 
 
1 84 On Crystalline Reflexion and Refraction. 
 
 be necessary to prevent any misconception as to the nature of 
 the foundation on which it stands. In this theory, everything 
 depends on the form of the function Y ; and we have seen that, 
 when that form is properly assigned, the laws by which crystals 
 act upon light are included in the general equation of dynamics. 
 This fact is fully proved by the preceding investigations. But 
 the reasoning which has been used to account for the form of 
 the function* is indirect, and cannot be regarded as sufficient, in 
 a mechanical point of view. It is, however, the only kind of 
 reasoning which we are able to employ, as the constitution of 
 the luminiferous medium is entirely unknown. 
 
 * Since this Paper was read to the Academy, I have found that the form of the 
 function V is more general than it would seem to be from the mode in which it is 
 here deduced ; and I have obtained from it a theory of the Total Reflexion of Light. 
 For a sketch of this theory, see the Proceedings of the Royal Irish Academy, 
 VOL. n. p. 96 (supra, p. 187). 
 
XY. ON THE OPTICAL LAWS OP BOCK- CRYSTALS. 
 
 [Proceedings of the Royallrish Academy, VOL. i. p. 385. Read Jan. 13, 1840.] 
 
 PROFESSOR MAC CULLAGH made a communication respecting 
 the optical Laws of Bock-crystal (Quartz). 
 
 In a Paper read to the Academy in February 1836, and 
 published in the Transactions (YoL. xvn. p. 461),* he had shown 
 how the peculiar properties of that crystal might be explained, 
 by adding to the usual equations of vibratory motion certain 
 terms depending on differential co-efficients of the third order, 
 and containing only one new constant C. This hypothesis, 
 which was very simple in itself, not only involved as conse- 
 quences all the laws that were previously known, but led to 
 the discovery of a new one the law, namely, by which the 
 ellipticity of the vibrations depends on the direction of the ray 
 within the crystal. He was not able, however, to account for 
 his hypothesis, nor has it since been accounted for by anyone. 
 
 But the theory developed in the Paper which he read at 
 the last meeting of the Academy now enables him to assign, 
 with a high degree of probability, the origin of the additional 
 terms above mentioned, and, if not to account for them mecha- 
 nically, at least to advance a step higher in the inquiry. In 
 that theory it was supposed (and the supposition holds good 
 in all known crystals, except quartz), that the molecules of the 
 ether vibrate in right lines, the displacements remaining always 
 parallel to each other as the wave is propagated; and it was 
 shown that the function Y, by which the motion is determined, 
 
 * Supra, p. 63. 
 
1 86 On the Optical Laws of Rock- Crystals. 
 
 then depends only on the relative displacements of the molecules. 
 But when this is not the case when, as in quartz, each mole- 
 cule is supposed to vihrate in a curve then it is natural to con- 
 ceive that the function Y may depend, not only on the relative 
 displacements, but also on the relative areas which each mole- 
 cule describes about every other more or less advanced in its 
 vibration. This idea, analytically expressed, introduces a new 
 term v into the value of the function 2V ; and, if the plane of 
 the wave be taken for the plane of xy 9 it is easy to show that 
 
 v = T~ T~2 ~ T~ -> 2 / 
 
 V <f2 G?Z 2 ^S rfs 3 / 
 
 Now if we integrate by parts the expression 
 
 so as to get rid of the variations of differential co-efficients, the 
 reduced form of the triple integral will be 
 
 from which it appears that the quantities 
 
 are to be added to the usual expressions for the force in the 
 directions of x and y respectively. These are the very terms 
 in the addition of which the hypothesis before alluded to 
 consists. 
 
XYI. ON A DYNAMICAL THEORY OF CRYSTALLINE 
 REFLEXION AND REFRACTION. 
 
 [Proceedings of the Royal Irish Academy, VOL. n. p. 96. Read May 24, 1841.] 
 
 PROFESSOR MAC CULLAGH read a supplement to his Paper 
 " On a Dynamical Theory of Crystalline Eeflexion and Befrac- 
 tion." 
 
 In his former Paper on that subject,* the author had given the 
 general principles for solving all questions relative to the propa- 
 gation of light in a given medium, or its reflexion and refrac- 
 tion at the separating surface of two media ; but he had applied 
 them only to the common case of waves, which suffer no dimi- 
 nution of intensity in their progress, and in which the vibration 
 may be represented by the sine or cosine of an arc multiplied 
 by a constant quantity. Some months after that Paper was read, 
 it occurred to him that he might obtain new and important 
 results by substituting in his differential equations of motion a 
 more general expression for the integral, that is (as usual in 
 such problems), by making the displacements proportional to 
 the sine or cosine of an arc, multiplied by a negative exponen- 
 tial, of which the exponent should be a linear function of the 
 co-ordinates. Such vibrations would become very rapidly insen- 
 sible, and would, therefore, be fitted to represent the disturbance 
 which, in the case of total reflexion, takes place immediately 
 behind the reflecting surface ; and, the laws of this disturbance 
 being thus discovered, the laws of polarization in the totally 
 
 * See Proceedings, 9th December, 1839 (supra, p. 146). 
 
1 88 On a Dynamical Theory of 
 
 reflected light would also become known, by means of the 
 general formulas which the author had established for all cases 
 of reflexion at the common surface of two media. 
 
 The present supplement is the fruit of these considerations. 
 It contains the complete theory of the new kind of vibrations, 
 not only in ordinary media, but in doubly-refracting crystals ; 
 and also the complete discussion of the laws of total reflexion at 
 the first or second surface of a crystal, including, as a particular 
 case, the well-known empirical formulae of Fresnel for total 
 reflexion at the surface of an ordinary medium. 
 
 The existence of vibrations represented by an expression con- 
 taining a negative exponential as a factor had been recognised 
 by other writers, and was indeed sufficiently indicated by the 
 phenomenon of total reflexion ; but it was impossible to obtain 
 the laws of such vibrations, so long as the general equations for 
 the propagation of light were unknown. 
 
 The method of deducing these equations was given in the 
 abstract of the author's former Paper ;* but as they were not there 
 stated, it may be well to transcribe them here. If then we put 
 
 2C = -7" ~T~ -t = -7 7-. Zi = -7 . 
 
 cfe dy dx dz dy dx 
 
 and suppose the axes of co-ordinates to be the principal axes of 
 the crystal, the equations in question may be thus written : 
 
 d?Z_ z dZ_dY 
 dt* dy dz 9 
 
 _ _ 
 
 ~ l dz ? dx' 
 
 (2) 
 
 df- dx dy ' 
 and if we further put 
 
 S = ~~7 ~~7 > 1) == ~~J 7~~ 9 ^> = ~~7 ~~ 7 > \ / 
 
 dz dy dx dz dy dx 
 
 * See Proceedings, 9th December, 1839 (supra, p. 157). 
 
Crystalline Reflexion and Refraction. 1 89 
 
 they will take the following simple form : 
 
 in which it is remarkable that the auxiliary quantities i> rji, &, 
 are exactly, for an ordinary medium, the components of the dis- 
 placement in the theory of Fresnel. In a doubly-refracting 
 crystal, the resultant of &, j!, ?i is perpendicular to the ray, and 
 comprised in a plane passing through the ray and the wave-nor- 
 mal. Its amplitude, or greatest magnitude, is proportional to 
 the amplitude of the vibration itself, multiplied by the velocity 
 of the ray. 
 
 The conditions to be fulfilled at the separating surface of two 
 media were given in the abstract already referred to. From 
 these it follows, that the resultant of the quantities &, 171, , 
 projected on that surface, is the same in both media ; but the 
 part perpendicular to the surface is not the same ; whereas the 
 quantities , TJ, ?, are identical in both. These assertions, analy- 
 tically expressed, would give five equations, though four are 
 sufficient ; but it can be shown that any one of the equations is 
 implied in the other four, not only in the case of common, but 
 of total reflexion ; which is a very remarkable circumstance, 
 and a very strong confirmation of the theory. 
 
 The laws of double refraction, discovered by Fresnel, but not 
 legitimately deduced from a consistent hypothesis, either by him- 
 self or any intermediate writer, may be very easily obtained, as 
 the author has already shown, from equations (2), by assuming 
 
 =p cos a sin 0, rj = p cos ]8 sin 0, =p cos 7 sin 0, (5) 
 
 where 27r, 7 N 
 
 = -^- (Ix + my + nz - sf) ; 
 A 
 
 but the new laws, which are the object of the present supple- 
 ment, are to be obtained from the same equations by making 
 
 g = [p cos a sin + q cos a' cos 0), 
 
 ij = e (p cos ]3 sin + q cos ]3' cos 0), ^ (6) 
 
 ^ = c (p cos y sin + q cos A' cos 0), 
 
190 On a Dynamical Theory of 
 
 where <j> has the same signification as before, and 
 
 e = e 
 
 the vibrations being now elliptical, whereas in the former case 
 they were rectilinear. In these elliptic vibrations the motion 
 depends not only on the distance of the vibrating particle from 
 the plane whose equation is 
 
 Ix + my + nz = 0, (7) 
 
 but also on its distance from the plane expressed by the equation 
 
 and if the constants in the equation of each plane denote the 
 cosines of the angles which it makes with the co-ordinate planes, 
 we shall have A for the length of the wave, and s for the velocity 
 of propagation ; while the rapidity with which the motion is ex- 
 tinguished, in receding from the second plane, will depend upon 
 the constant r. The constants p and q may be any two conju- 
 gate semi- diameters of the ellipse in which the vibration is per- 
 formed ; the former making, with the axes of co-ordinates, the 
 angles a, /3, 7, the latter the angles a', j3', 7'. 
 
 As vibrations of this kind cannot exist in any medium, unless 
 they are maintained by total reflexion at its surface, we shall 
 suppose, in order to contemplate their laws in their, utmost 
 generality, that a crystal is in contact with a fluid of greater re- 
 fractive power than itself, and that a ray is incident at their 
 common surface, at such an angle as to produce total reflexion. 
 The question then is, the angle of incidence being given, to de- 
 termine the laws of the disturbance within the crystal. 
 
 The author finds that the refraction is still double, and that 
 two distinct and separable systems of vibration are transmitted 
 into the crystal. He shows that the surface of the crystal itself 
 (the origin of co-ordinates being upon it at the point of inci- 
 dence) must coincide with the plane expressed by equation (8), 
 a circumstance which determines the three constants /, g, h. 
 The plane expressed by (7) is parallel to the plane of the re- 
 
Crystalline Reflexion and Refraction. 191 
 
 fracted wave ; and a normal, drawn to it through the origin, 
 lies in the plane of incidence, making with a perpendicular to 
 the face of the crystal an angle o>, which may be called the angle 
 of refraction ; so that, if i be the angle of incidence, we have 
 
 sin o> = s sin i, 
 
 the velocity of propagation in the fluid being regarded as unity. 
 To each refracted wave, or system of vibration, corresponds 
 a particular system of values for r, s, o>. These the author shows 
 how to determine by means of the index-surface (the reciprocal of 
 Fresnel's wave-surface), which he has employed on other occa- 
 sions,* and the rule which he gives for this purpose affords a 
 remarkable example of the use of the imaginary roots of equa- 
 tions, without the theory of which, indeed, it would have been 
 difficult to prove, in the present instance, that there are two, and 
 only two, refracted waves. Taking a new system of co-ordi- 
 nates #', i/ ', s', of which z' is perpendicular to the surface of the 
 crystal, and y' to the plane of incidence, while of lies in the in- 
 tersection of these two planes ; put tf = in the equation of the 
 index-surface referred to those co-ordinates, the origin being at 
 its centre ; we shall then have an equation of the fourth degree 
 between x' and z', which will be the equation of the section made 
 in the index-surface by the plane of incidence. In this equation 
 put x f = sin i, and then solve it for z'. "When i exceeds a certain 
 angle /, the four values of z' will be imaginary ; and if they be 
 denoted by 
 
 u v <, u f 
 
 each pair will correspond to a refracted system, and we shall 
 have, for the first, 
 
 sin i sin w 
 
 tan o> = - , s = r, r = sv ; (9) 
 
 u smz 
 
 and for the second, 
 
 , sinz sin a/ , , , 
 
 tan u/ = -, s' = r , r = s'v . (10) 
 
 u sin i 
 
 * Transactions of the Academy, VOLS. xvn. and XYIII. (supra, pp. 36, 96). 
 
On a Dynamical Theory of 
 
 When i lies between i and a certain smaller angle i"> two of the 
 roots will be real, and two imaginary. The real roots correspond 
 to waves which follow the law of Fresnel ; the imaginary roots 
 give a single wave, following the other laws just mentioned. 
 
 Lastly, when i is less than i" 9 all the roots are real, the re- 
 fraction is entirely regulated by Fresnel's law, and the reflexion 
 by the laws already discovered and published by the author. 
 
 If the crystal be uniaxal, and all the values of z imaginary, 
 the ordinary wave-normal will coincide with the axis of x ; 
 whilst the extraordinary wave-normal and the axis of z' will be 
 conjugate diameters of the ellipse in which the index surface is 
 cut by the plane of incidence. 
 
 When a = b = c, the crystal becomes an ordinary medium ; 
 there is then only single refraction, and the refracted wave is 
 always perpendicular to the axis of of. 
 
 With regard to the ellipse in which the vibrations are per- 
 formed, it may be worth while to observe, that if it be projected 
 perpendicularly on the plane of incidence, the projected diameters 
 which are parallel to the surface of the crystal and to the wave- 
 plane will, in all cases, be conjugate to each other, and their re- 
 spective lengths will be in the proportion of r to unity. The 
 vibrations, it is obvious, are not performed in the plane of the 
 wave, though they take place without changing the density of 
 the ether. 
 
 The new laws here announced are, properly spealdng, laws 
 of double refraction, and are necessary to complete our know- 
 ledge of that subject. Between them and the laws of Fresnel 
 a curious analogy exists, founded on the change of real into 
 imaginary constants. 
 
 The laws of the total reflexion, which accompanies the new 
 kind of refraction, need not be dwelt upon in this abstract, as 
 nothing is now more easy than to form the equations which con- 
 tain them. In fact, the difficulties which formerly surrounded 
 the problem of reflexion, even in the simplest cases, have com- 
 pletely disappeared, since the author made known the conditions 
 which must be fulfilled at the separating surface of two media. 
 
Crystalline Reflexion and Refraction. 1 93 
 
 In what precedes, it has been supposed that the reflexion and 
 refraction take place at the first surface of the crystal, because 
 this is the more difficult and complicated of the two cases into 
 which the question resolves itself. But it will usually happen 
 in practice that a ray which has entered the crystal will suffer 
 total reflexion at the second surface, while the new kind of vibra- 
 tion is propagated into the air without. The refracted wave will 
 then be always perpendicular to the axis of x' ; the two reflected 
 rays, within the crystal, will be plane-polarized, according to the 
 common law, but they will each undergo a change of phase ; 
 and the vis viva of the two rays together will be equal to that of 
 the incident ray, the vis viva being measured by the square of 
 the amplitude multiplied by the proportional mass. 
 
 In conclusion, the author states a mathematical hypothesis 
 by which both the laws of dispersion, and those of the elliptic 
 polarization of rock crystal, may be connected with the laws 
 already developed. 
 
( 194 ) 
 
 XVIL NOTES ON SOME POINTS IN THE THEORY OF 
 
 LIGHT. 
 
 [Proceedings of the Royal Irish Academy, VOL. n. p. 139. Bead Nov. 8, 1841.] 
 
 I. 
 
 On a Mechanical Theory which has been proposed for the Explana- 
 tion of the Phenomena of Circular Polarization in Liquids, and 
 of Circular and Elliptic Polarization in Quartz or Rock-crystal; 
 with Remarks on the corresponding Theory of Rectilinear Pola- 
 rization. 
 
 THE theory of elliptic polarization, which I feel myself 
 called upon to notice, was first stated by M. Cauchy, and has 
 been made the subject of elaborate investigation by other 
 writers. That celebrated analyst, conceiving (though without 
 sufficient reason, as will presently appear) that he had fully 
 explained the known laws of the propagation of rectilinear 
 vibrations by the hypothesis that the luminiferous ether, in 
 media transmitting such vibrations, consists of separate mole- 
 cules symmetrically arranged with respect to each of three 
 rectangular planes, and acting on each other by forces which 
 are some function of the distance, was led very naturally to 
 imagine that he would find the laws of circular and elliptic 
 vibrations, in other media, to be included in the more general 
 hypothesis of an unsymmetrical arrangement. Accordingly, 
 in a letter read to the French Academy on the 22nd of 
 February, 1836 a letter to which he attached so much im- 
 portance that he desired it might not only be published in 
 
Notes on some Points in the Theory of Light. 195 
 
 the Proceedings, but also " deposited in the Archives " of that 
 body* he gave a precise statement of his more extended views, 
 informing the Academy that he had submitted his new theory 
 to calculation, and that, among other remarkable results, he had 
 obtained (with a slight variation or correction) the laws of cir- 
 cular polarization, discovered by Arago, Biot, and Fresnel. Re- 
 ferring to his Memoir on Dispersion, published at Prague, under 
 the title of Nouveaux Exercices de Mathematiques, he observes, that 
 the results therein contained may be generalized, by " ceasing to 
 neglect " in the equations of motion [the equations marked (24) 
 in 2 of that memoir] certain terms which vanish in the case of 
 a symmetrical distribution of the ether. He then goes on to 
 say 
 
 "Nos formules ainsi generalisees representent les phe- 
 nomenes de 1'absorption de la lumiere ou de certains rayons, 
 produite par les verres colores, la tourmaline, &c., le phe- 
 nomene de la polarisation circulaire produite par le cristal 
 de roche, 1'huile de terebenthine, &c. (Voir les experiences 
 de MM. Arago, Biot, Fresnel). Elles servent meme a de- 
 terminer les conditions et les lois de ces phenomenes; elles 
 montrent que generalement, dans un rayon de lumiere po- 
 larisee, une molecule d' ether decrit une ellipse. Mais dans 
 certains cas particuliers, cette ellipse se change en une droite, 
 et alors on obtient la polarisation reetiligne." " Enfin le cal- 
 cul prouve que, dans le cristal de roche, 1'huile de tereben- 
 thine, &c., la polarisation, des rayons transmis parallelement 
 a 1'axe (s'il s'agit du cristal de roche) n'est pas rigoureusement 
 circulaire, mais qu' alors 1'ellipse differe tres peu du cercle." 
 
 Thus, fco say nothing for the present of the questions of 
 dispersion and absorption, it appears that M. Cauchy conceived 
 he had completely accounted for the facts of circular and 
 elliptic polarization, and that he had deduced the formulas 
 "which serve to determine the conditions and laws of these 
 phenomena." But neither in this letter, nor in any subse- 
 
 See the Comptes Rcndus des Seances de V Academic des Sciences, torn. ii. p. 182. 
 
 o2 
 
196 Notes on some Points in the Theory of Light. 
 
 quent version* of his theory, has he given the formulas them- 
 selves. Nor has he told us the nature of the calculations by 
 which he was enabled to correct the received opinion, and to 
 prove that the vibrations in a ray transmitted along the axis 
 of quartz, or through oil of turpentine, are not rigorously 
 circular, as Fresnel and others have supposed, but slightly 
 elliptical. Now to take the case of quartz if we consider 
 that the vibrations of a ray passing along the axis are in a 
 plane perpendicular to it, and if we admit, as M. Cauchy 
 always does in the case of other uniaxal crystals, that there 
 is a perfect optical symmetry all round the axis, we shall find 
 it hard to conceive on what grounds he could have come to 
 the conclusion that the vibrations of such a ray are performed 
 in an ellipse. For if all planes passing through the axis of 
 the crystal be alike in their optical properties, there will be 
 absolutely nothing to determine the position and ratio of the 
 axes of the ellipse ; there will be no reason why its major axis, 
 for example, should lie in one of these planes, rather than in 
 any other. But, whatever may be thought of this case inde- 
 pendently of observation, it is manifestly absurd to suppose 
 that the vibrations are elliptical in the case of a ray passing 
 through oil of turpentine, or any other liquid possessing the 
 property of rotatory polarization; for, in a liquid, all planes 
 drawn through the ray itself are circumstanced alike. From 
 these simple considerations it is evident that the theory of 
 M. Cauchy is unsound; but a closer examination will show 
 that it is entirely without foundation, and that it is directly 
 opposed to the very phenomena which it professes to explain. 
 To make this appear, however, in the easiest way that the 
 abstruseness of the subject will allow, it will be necessary to 
 
 * From some statements that have been made within the last few days by 
 Professor Powell (Phil. Mag. VOL. xix. p. 374), at the request of M. Cauchy 
 himself, it appears that the latter republished his views about circular and 
 elliptic polarization, in a lithographed memoir of the date of August, 1836; 
 but I do not find that he published, either then or since, the detailed calcula- 
 tions which he seems to have made. 
 
Notes on some Points in the Theory of Light. 197 
 
 advert to some former researches of my own, which have a 
 direct bearing on the question. 
 
 The same day on which M. Cauchy's letter was read to 
 the French Academy, I had the honour of reading to the 
 Eoyal Irish Academy a Paper " On the Laws of Double 
 Eefraction in Quartz"* wherein I showed that everything 
 which we know respecting the action of that crystal upon 
 light is comprised mathematically in the following equations: 
 
 (1) 
 
 dt* " dz 
 
 which differ from the common equations of vibratory motion 
 by the two additional terms containing third differential co- 
 efficients multiplied by the same constant (7, this constant 
 having opposite signs in the two equations. The quantities 
 % and r] are, at any time , the displacements parallel to the 
 axes of x and y, which are supposed to be the principal di- 
 rections in the plane of the wave, one of them being there- 
 fore perpendicular to the axis of the crystal. The constants 
 A and B are given by the expressions 
 
 where a and b are the principal velocities of propagation, 
 ordinary and extraordinary, and i/< is the angle made by the 
 wave-normal (or the direction of z) with the axis of the crys- 
 tal. The only new constant introduced is (7, which, though 
 the peculiar phenomena of quartz depend entirely on its ex- 
 istence, is almost inconceivably small : its value is determined 
 in the Paper just referred to. The equations are there proved 
 to afford a strict geometrical representation of the facts; not 
 only connecting together all the laws discovered by the dis- 
 tinguished observers to whom M. Cauchy refers, and includ- 
 
 * See Transactions, R. I. A., VOL. xvn. p. 461 (supra, p. 63). 
 
198 Notes on some Points in the Theory of Light. 
 
 ing the subsequent additions for which we are indebted to 
 Mr. Airy, but leading to new results, one of which establishes 
 a relation between two different classes of phenomena, and 
 is verified by the experiments of M. Biot and Mr. Airy. 
 Having, therefore, such conclusive proofs of the truth of these 
 equations, we are entitled to assume them as a standard 
 whereby to judge of any theory ; so that any mechanical 
 hypothesis which leads to results inconsistent with them may 
 be at once rejected. 
 
 Now I assert that the mechanical hypothesis of M. Cauchy 
 contradicts these equations, and therefore contradicts all the 
 phenomena and experiments which he supposed it to repre- 
 sent. But before we proceed to the proof of this assertion, 
 it may perhaps be proper to remark, that previously to the 
 date of M. Cauchy's communication, and of my own Paper, I 
 had actually tried and rejected this identical hypothesis, and 
 had even gone so far as to reject along with it the whole of 
 M. Cauchy's views about the mechanism of light. For though, 
 in my Paper, I have said nothing of any mechanical investiga- 
 tions, yet, as a matter of course, before it was read to the Aca- 
 demy, I made every effort to connect my equations in some 
 way with mechanical principles; and it was because I had 
 failed in doing so to my own satisfaction, that I chose to 
 publish the equations without comment,* as bare geometrical 
 assumptions, and contented myself with stating orally to the 
 Academy, as I did some months after to the Physical Section 
 of the British Association in Bristolf that a mechanical account 
 of the phenomena still remained a desideratum which no attempts 
 of mine had been able to supply. I am not sure that on the first 
 occasion I stated the precise nature of these attempts, though I 
 
 * The circumstances here related will account for what Mr. Whewell (History 
 of the Inductive Sciences, VOL. n. p. 449) calls the "obscure and oracular form " in 
 which those equations were published. Having, at that time, no good explanation 
 of them to give, I thought it better to attempt none. But in the general view which 
 I have since taken (see p. 224 of this volume), they do not offer any peculiar diffi- 
 culty. 
 
 f See " Transactions of the Sections," p. 18. 
 
Notes on some Points in the Theory of Light. 199 
 
 incline to think I did ; but I have a distinct recollection of having 
 done so on the second occasion, in reply to questions that were 
 asked me by some Members of the Association.* Now, my first 
 attempt to explain those equations, which was made almost as soon 
 as I discovered them, actually turned upon the very idea which 
 about the same time found entrance into the mind of M. Cauchy 
 I mean the idea of an unsymmetrical arrangement of the ether. 
 For as it was generally believed, at that period, that the hypothe- 
 sis of ethereal molecules symmetrically distributed had led, in the 
 hands of M. Cauchy, to a complete theory of rectilinear polari- 
 zation in crystals,f the notion of endeavouring to account for the 
 phenomena of elliptic polarization, by freeing the hypothesis from 
 any restriction as to the distribution of the ether, would natu- 
 rally occur to anyone who was thinking on the subject, no less 
 than to M. Cauchy himself. And though, for my own part, 
 I never was satisfied with that theory, which seemed to me to 
 possess no other merit than that of following out in detail the 
 extremely curious, but (as I thought) very imperfect, analogy 
 which had been perceived to exist between the vibrations of the 
 luminif erous medium and those of a common elastic J solid (for 
 
 * At the period of this meeting, M. Cauchy 's letter on Elliptic Polarization had 
 been published for some months ; but I was not then aware of its existence. Indeed 
 the letter appears not to have attracted any general notice ; for the theory which it 
 contains was afterwards advanced in England as a new one, and M. Cauchy has 
 been lately obliged to assert his prior claim to it, through the medium of Professor 
 Powell. See notes, pp. 196, 202-3. 
 
 f See his Exercices de Mathematiques, Cinquieme Annee, Paris, 1830, and the 
 Memoires de I'Institut, torn. x. p. 293. 
 
 J The analogy was suggested by the hypothesis of transversal vibrations, which, 
 when viewed in its physical bearing, was considered by Dr. Young to be "perfectly 
 appalling in its consequences," as it was only to solids that a "lateral resistance" 
 tending to produce such vibrations had ever been attributed. (Supplement to the 
 Encyclopaedia JBritannica, VOL. vi. p. 862, Edinburgh, 1824.) He admits, how- 
 ever, that the question, whether fluids may not "transmit impressions by lateral 
 adhesion, remains completely open for discussion, notwithstanding the apparent 
 difficulties attending it." As far as I am aware, Fresnel always regarded the 
 ether as a, fluid. M. Poisson affirms that it must be so regarded, and attributes its 
 apparent peculiarities to the immense rapidity of its vibrations, which does not 
 
2OO Notes on some Points in the Theory of Light. 
 
 it is usual to regard such a solid as a rigid system of attracting 
 or repelling molecules, and M. Cauchy has really done nothing 
 more than transfer to the luminiferous ether both the constitu- 
 tion of the solid and differential formulas of its vibration), still 
 I should have been glad, in the absence of anything better, to 
 find my equations supported by a similar theory, and their form 
 at least countenanced by the like mechanical analogy. Besides, 
 I recollected that Fresnel himself, in his Memoir on Double Re- 
 fraction, had indicated a "helicoidal arrangement," or something 
 of that sort, as a probable cause of circular polarization* and as 
 this was an hypothesis of the same kind as the other, only not so 
 general, I was prepared to find that the supposition of an arbitrary 
 arrangement, whatever might be thought of its physical reality, 
 would lead to equations of the same form as those which I had 
 assumed. Upon trial, however, the very contrary proved to be 
 the case ; for though it was possible to obtain additional terms, con- 
 taining differential co-efficients of the third order, multiplied by 
 the same constant (7, yet this constant always came out with 
 the same sign in both equations, whereas a difference of sign 
 was essential for the expression of the phenomena. I had no 
 sooner arrived at this result than I perceived it to be fatal to 
 the theory of M. Cauchy, and to afford a demonstration of its 
 insufficiency, not only in the particular application which I had 
 made of it, but in all its applications. For the hypothesis 
 
 allow the law of equal pressure to hold good in the state of motion (Annales de 
 Chimie, torn. xliv. p. 432). M. Cauchy calls the ether a fluid, though he treats it 
 as a solid. My own impression is, that the ether is a medium of a peculiar kind, 
 differing from all ponderable hodies, whether solid or fluid, in this respect, that it 
 absolutely refuses, in any case, to change its density, and therefore propagates to 
 a distance transversal vibrations only ; while ordinary elastic fluids transmit only 
 normal vibrations, and ordinary solids admit vibrations of both kinds. This hypo- 
 thesis also includes the supposition that the density of the ether is unchanged by 
 the presence of ponderable matter. As to M. Cauchy's third ray, with vibrations 
 nearly normal to the wave, there is no reason to believe that it has even the 
 faintest existence; but it is necessarily introduced by his identification of the 
 vibrations of light with those of an indefinitely extended elastic solid. 
 * Memoires de V Institut, torn. vii. p. 73. 
 
Notes on some Points in the Theory of Light. 201 
 
 which I used was, in fact, identical with that theory, in the 
 most general form of which it is susceptible, when unrestricted 
 by any particular supposition as to the arrangement of the 
 ethereal molecules; and therefore the fundamental conception 
 of the theory could not be true, as it not merely failed to ex- 
 plain a large and most remarkable class of phenomena those 
 of circular and elliptical polarization but absolutely excluded 
 them, and left no room for their existence. It followed from 
 this, that the mechanical explanation, which the same theory 
 was supposed to have given, of the phenomena of rectilinear 
 polarization and double refraction in crystals, could not be well 
 founded : indeed, as I have said, I had always distrusted it, and 
 that for various reasons, of which one has been already men- 
 tioned, and another was suggested by the forced relations which 
 M. Cauchy had found it necessary to establish among the con- 
 stants of his theory, and by which he had compelled, as it were, 
 his complicated formulas to assume the appearance of an agree- , 
 ment (though, after all, a very imperfect one) with the simple 
 laws of Fresnel. 
 
 Such were the conclusions at which I arrived, and the re- 
 flections which they forced upon me, nearly six years ago. 
 They have been frequently mentioned in conversation to those 
 who took an interest in such matters, and their general tenor 
 may be gathered from what I have elsewhere written ;* but I 
 did not think it worth while to publish them in detail, because 
 it seemed probable that juster notions would prevail in the 
 course of a few years, and that the ingenious speculations to 
 which I have alluded would gradually come to be estimated at 
 their proper value. But from whatever cause it has arisen 
 whether from the real difficulties of the subject, or the extreme 
 vagueness of the ideas that most persons are content to form of 
 it, or from deference to the authority of a distinguished mathe- 
 matician certain it is that the doctrines in question have not 
 only been received without any expression of dissent, but 
 have been eagerly adopted, both in this country and abroad, by 
 
 * Transactions, R. I. A., VOL. xvm. p. 68 (supra, p. 129). 
 
202 Notes on some Points in the Theory of Light. 
 
 a host of followers ; and even the extraordinary error, which it 
 is my more immediate object to expose, has been continually 
 gaining ground up to the very moment at which I write, and 
 has at last begun to be ranked among the elementary truths of 
 the undulatory theory of light. Notwithstanding my un- 
 willingness, therefore, to be at all concerned in such discus- 
 sions, I do not think myself at liberty to remain silent any 
 longer. There are occasions on which every consideration of 
 this kind must give way to a regard for the interests of science. 
 To show that the principles of M. Cauchy contradict, in- 
 stead of explaining, the phenomenon of elliptic polarization, 
 let us take the axes of co-ordinates as before ; and let us sup- 
 pose, for the sake of simplicity, and to avoid his third ray, 
 that the normal displacements vanish. Then his fundamental 
 equations take the form 
 
 r/ 2 
 
 = S/A? + 
 
 where /, g, h are quantities depending on the law of force 
 and the mutual distances of the molecules.* If, therefore, 
 
 * I have not thought it necessary to transcribe the original equations of M. 
 Cauchy, which are rather long. He has presented them in different forms ; but 
 the system marked (16) at the end of 1 of his Memoir on Dispersion, already 
 quoted, is the most convenient, and it is the one which I have here used. The 
 directions of the co-ordinates being arbitrary, I have supposed the axis of z to be 
 perpendicular to the wave-plane. Then, on putting = 0, A = 0, in order to get 
 rid of the normal vibration, the last equation of the system becomes useless, and 
 the other two are reduced to the equations (2), given above; the letters /, g, h, 
 being written in place of certain functions depending on the mutual actions of the 
 molecules. It will be proved, further on, that this simplification does not at all 
 affect the argument. As the directions of x and y still remain arbitrary, I have 
 made them parallel to the axes of the supposed elliptic vibration. 
 
 It may be right to observe, for the sake of clearness, that, when the medium is 
 arranged symmetrically, it is always possible to take the directions of x and y such 
 that the two sums depending on the quantity h may disappear from the equations 
 (2), and then the vibrations are rectilinear. But when the arrangement is unsym- 
 metrical, this is no longer possible. 
 
Notes on some Points in the Theory of Light. 203 
 
 we assume that each molecule describes an ellipse, the axes of 
 which are parallel to those of x and y\ that is to say, if we 
 make 
 
 5 = p cos 0, r\ = q sin 0, 
 
 (3) 
 
 and consequently, 
 
 A =p (sin 20 sin - 2 sin 2 cos 0), 
 Arj = - q (sin 20 cos + 2 sin 2 sin </>), 
 
 where = , we shall find, by substituting these values in the 
 
 A 
 
 equations (2), which must hold good independently of 0, 
 
 s 2 = A + C'k, s 2 = B' - ^, (4) 
 
 K 
 
 S/sin 20 - 2k Zh sin 2 = 0, 
 S# sin 20 + \ SA sin 2 = 0, 
 
 K 
 
 The equations (2) are precisely the same as those which have been employed 
 by Mr. Tovey and by Professor Powell, the latter of whom, in his lately published 
 work, entitled, "A General and Elementary View of the Undulatory Theory, as ap- 
 plied to the Dispersion of Light, and other Subjects" has dwelt at great length on 
 the theory of elliptic polarization which they have been supposed to afford, and 
 M r hich he regards as a most important accession to the Science of Light. Professor 
 Powell has also made some communications on the subject to the British Asso- 
 ciation, and has written two Papers about it in the Philosophical Transactions 
 (1838, p. 253 : and 1840, p. 157), besides several others in the Philosophical Ma- 
 gazine. He, however, always attributed this theory of elliptic polarization to 
 Mr. Tovey, until his attention was directed, by a letter from M. Cauchy, to some 
 investigations of the latter which he had not previously seen (Phil. Mag. VOL. xix. 
 p. 374). Mr. Tovey set out with the principles of M. Cauchy, and therefore 
 naturally struck into the same track, in pursuit of the same object, apparently 
 quite unconscious that anyone had preceded him. It was, indeed, an obvious 
 reflection, that these principles, when generalized to the utmost, ought to include, 
 not only the laws of elliptic polarization, but (as really has been thought by M. 
 Cauchy and his followers) of dispersion and absorption, and, in short, of all the 
 phenomena of optics. 
 
2O4 Notes on some Points in the Theory of Light. 
 
 wherein k = - expresses the ratio of the semiaxes of the elliptic 
 
 P 
 vibration, and 
 
 A' = ^ Vf sin 2 0, V = ^ ^g sin'0, 
 
 C f = - SA sin 20. 
 4ir z 
 
 Equating the two values of s 2 , we get, for the determination of 
 k, the following quadratic : 
 
 F + A ~ B + 1 = 0. (5) 
 
 G 
 
 Now making the substitutions (3) in equations (1), page 197, 
 we have 
 
 o _ A ^ Tlf Q 2 - 7? ^ (K\ 
 
 s -A-^Ck, s --B-yp 
 
 and thence 
 
 A-B)k-l = 0, (7) 
 
 a result which is perfectly inconsistent with the former, since 
 the two roots of (5) have the same sign, if they are not imagi- 
 nary, while those of (7) have opposite signs, and cannot be 
 imaginary. If, therefore, one equation agrees with the phe- 
 nomena, the other must contradict them. The last equation 
 indicates that, in the double refraction of quartz, the two 
 elliptic vibrations are always possible, and performed in oppo- 
 site directions, which is in accordance with the facts ; whereas 
 the equation (5), deduced from M. Cauchy's theory, would 
 inform us that the vibrations of the two rays are either im- 
 possible or in the same direction.* 
 
 To apply the results to a particular instance, let us con- 
 ceive a circularly polarized ray passing along the axis of 
 quartz, or through one of the rotatory liquids, such as oil of 
 
 * This conclusion, which shows that M. Cauchy's Theory is in direct opposition 
 to the phenomena, might have been obtained without any reference to the equa- 
 tions (1). But these equations are necessary in what follows. 
 
Notes on some Points in the Theory of Light. 205 
 
 turpentine ; the position of the co-ordinates x and y, in the 
 plane of the wave, being now, of course, arbitrary. In each 
 of these cases we have k = 1, and A = B = a 2 , so that the 
 value of s 2 in equation (6) is expressed by the constant a 2 , 
 plus or minus a term which is inversely proportional to the 
 wave-length A ; the sign of this term depending on the direc- 
 tion of the circular vibration. Now it will not be possible 
 to obtain a similar value of s 2 from the formulas (4), unless 
 we suppose A' = B' = a?, since it is only in the expansion of 
 (7 that a term inversely proportional to A can be found ; but 
 on this supposition the formulas are inconsistent with each 
 other, nor can they be reconciled by any value of k. Indeed, 
 when A' = B', the equation (5) give k = */ - 1. Thus it 
 appears that circular vibrations, such as are known to be propa- 
 gated along the axis of quartz, and through certain fluids, can- 
 not possibly exist on the hypothesis of M. Cauchy. It was 
 probably some partial perception of this fact that caused M. 
 Cauchy to assert that the vibrations, in these cases, are not ex- 
 actly circular, but in some degree elliptical a supposition which, 
 if it were at all conceivable, which we have seen it is not (p. 196), 
 would be at once set aside by what has just been proved; for no 
 assumed value of #, whether small or great, will in any way help 
 to remove the difficulty. 
 
 But this is not all. Eectilinear vibrations are excluded as 
 well as circular ; for we cannot suppose k = in the equations 
 (4), so long as the quantity C', resulting from the hypothesis of 
 unsymmetrical arrangement, has any existence. Thus the in- 
 consistency of that hypothesis is complete, and the equations to 
 which it leads are utterly devoid of meaning. 
 
 The foregoing investigation does not differ materially from 
 that which I had recourse to in the beginning of the year 1836. 
 To render the proof more easily intelligible, and to get rid of 
 M. Cauchy's " third ray," which has no existence in the nature 
 of things, I have suppressed the normal vibrations ; a procedure 
 which is not, in general, allowable on the principles of M. 
 Cauchy. It will readily appear, however, that this simplifica- 
 
206 Notes on some Points in the Theory of Light. 
 
 tion still leaves the demonstration perfectly rigorous in the case 
 of circular vibrations, and does not affect its force when the 
 vibrations are elliptical. For in the rotatory fluids it is obvious 
 that the normal vibrations, supposing such to exist, must, by 
 reason of the symmetry which the fluid constitution requires, be 
 independent of the transversal vibrations, and separable from 
 them, so that the one kind of vibrations may be supposed to 
 vanish when we wish merely to determine the laws of the other. 
 The equations (2) are, therefore, quite exact in this case; and 
 they are also exact in the case of a ray passing along the axis 
 of quartz, since such a ray is not experimentally distinguishable 
 from one transmitted by a rotatory fluid, and its vibrations must 
 consequently be subject to the same kind of symmetry. In these 
 two cases, therefore, it is rigorously proved that the values of A*, 
 which ought to be equal to plus and minus unity, are imaginary, 
 and equal to f - 1. And if we now take the most general 
 case with regard to quartz, and suppose that the ray, which was 
 at first coincident with the axis of the crystal, becomes gradually 
 inclined to it, the values of k must evidently continue to be ima- 
 ginary, until such an inclination has been attained that the two 
 roots of equation (5) become possible and equal, in consequence 
 of the increased magnitude of the co-efficient of the second term. 
 Supposing the last term of that equation to remain unchanged, 
 this would take place when the co- efficient of k (without regard- 
 ing its sign) became equal to the number 2, and the values of k 
 each equal to unity, both values being positive or both negative. 
 The vibrations which before were impossible would, at this in- 
 clination, suddenly become possible ; they would be circular, 
 which is the exclusive property of vibrations transmitted along 
 the axis ; and they would have the same direction in both rays, 
 which is not a property of any vibrations that are known to 
 exist. At greater inclinations the vibrations would be ellip- 
 tical, but they would still have the same direction in the two 
 rays. These results would not be sensibly altered by regard- 
 ing the equation (5) as only approximate in the case of rays 
 inclined to the axis ; for the last term of that equation, if it 
 
Notes on some Points in the Theory of Light. 207 
 
 does not remain the same, can never differ much from unity ; 
 since it must become exactly equal to unity, ivhatever be the 
 direction of the ray, when the crystalline structure is sup- 
 posed to disappear, and the medium to become a rotatory 
 fluid. 
 
 That a theory involving so many inconsistencies should 
 have been advanced by a person of M. Cauchy's reputation 
 would, perhaps, appear very extraordinary, if we did not re- 
 collect that it was unavoidably suggested by the general prin- 
 ciplas which he had previously adopted, and which were 
 supposed, not merely by himself, but by the scientific world 
 generally, to have already afforded the only satisfactory ex- 
 planation of the laws of double refraction in the common 
 and well-known case where the vibrations are rectilinear. 
 This supposed explanation was obtained, as has been said, 
 by restricting the application of M. Cauchy's principles to 
 the hypothesis of a vibrating medium arranged symmetrically, 
 in which case it was shown that the vibrations were neces- 
 sarily rectilinear; and of course the removal of this restric- 
 tion was the only way in which it was possible, on those 
 principles, to account for the existence of circular and ellip- 
 tical vibrations. Accordingly, when M. Cauchy perceived 
 that, on the hypothesis of unsymmetrical arrangement, the 
 existence of rectilinear vibrations became impossible, and that 
 of elliptic vibrations, generally speaking, possible, he found 
 it very easy to persuade himself that he had obtained a new 
 proof of the correctness of his views, and a new and most im- 
 portant application of the fundamental equations by which 
 his general principles were analytically expressed. To have 
 supposed otherwise would have been to admit that his general 
 principles were false. If the elliptical or qitasi-circula,? vibra- 
 tions which he was now contemplating were not capable of 
 being identified with those which had been recognized in the 
 phenomena presented by quartz and the rotatory fluids if 
 their laws were essentially or very considerably different his 
 theory would be inconsistent with a wide range of well-known 
 
208 Notes on some Points in the Theory of Light. 
 
 facts, and, notwithstanding its so-called explanations of other 
 laws, should be finally abandoned. Under these circumstances, 
 therefore, he very naturally supposed that his new results 
 must be in complete harmony with the phenomena discovered 
 by M. Arago, and analyzed so successfully by MM. Biot and 
 Fresnel ; although, had he taken the precaution of acquiring 
 such a clear notion of the phenomena as would have enabled 
 him to translate them into analytical language, he must have 
 perceived that they were entirely opposite to his results, and 
 that this opposition furnished an argument which swept away 
 the very foundations of his theory. For, if the constitution 
 of the luminiferous medium were such as M. Cauchy sup- 
 poses, the well-known phenomena of circular and elliptic 
 polarization would, as we have seen, be absolutely impossible. 
 Thus the argument which overturns the particular theory 
 of elliptical polarization destroys at the same time all the 
 other optical theories of M. Cauchy, because they are all 
 built on the principles which we have now demonstrated to be 
 false. But though the principles of M. Cauchy are now, for 
 the first time, formally refuted, they were objected to, on 
 general grounds, so long ago as the year 1830, by a person 
 whose opinion, on a question of mechanics, ought to have had 
 considerable weight. This was M. Poisson, who, having de- 
 duced from the equations of motion of an elastic solid the con- 
 sequence that such a body admitted vibrations perpendicular 
 to the direction of their propagation, thought it right to re- 
 mark that this conclusion could not be supposed to account 
 for transversal vibrations in the theory of light, because (as 
 he expressed himself) "the same equations of motion could 
 not possibly apply to two systems [of molecules] so essen- 
 tially different from each other" as the ethereal fluid and 
 an elastic solid.* (See the Annales de Chimie^ torn. xliv. 
 
 * As the theory of M. Cauchy (Mem. de Vlnstitut, torn, x.) had been communi- 
 cated to the Academy of Sciences some months before the period (October, 1830) at 
 which M. Poisson wrote, there can be no doubt that M. Poisson' s remark was di- 
 rected against that theory, though he did not expressly mention it. 
 
Notes on some Points in the Theory of Light. 209 
 
 p. 432). The remark, however, did not meet with much at- 
 tention from mathematicians, who were, perhaps, not dis- 
 posed to scrutinize too closely any hypothesis which gave 
 transversal vibrations as a result. Besides, the hypothesis 
 appeared to go much further, as it offered prima facie expla- 
 nations of a great variety of phenomena ; it was one to which 
 calculation could be readily applied, and therefore it naturally 
 found favour with the calculator ; and as to M. Poisson's objec- 
 tion, it was easily removed by a change of terms, for wlien the 
 elastic solid was called an " elastic system" there was no longer 
 anything startling in the announcement that the motions of the 
 ether are those of such a system. The hypothesis was there- 
 fore embraced by a great number of writers in every part of 
 Europe, who reproduced, each in his own way, the results of 
 M. Cauchy, though sometimes with considerable modifications. 
 Every day saw some new investigation purely analytical some 
 new mathematical research uncontrolled by a single physical 
 conception put forward as a "mechanical theory" of double 
 refraction, of circular polarization, of dispersion, of absorption ; 
 until at length the Journals of Science and Transactions of 
 Societies were filled with a great mass of unmeaning formulas. 
 This state of things was partly occasioned by the great number 
 of " disposable " constants entering into the differential equa- 
 tions of M. Cauchy and their integrals ; for it was easy to in- 
 troduce, among the constants, such relations as would lead to 
 any desired conclusion; and this method was frequently 
 adopted by M. Cauchy himself. Thus, in his theory of double 
 (or rather triple) refraction, given in the works already cited 
 (p. 145), he supposes three out of his nine constants to vanish, 
 and assumes, among the other six, three very strange and im- 
 probable relations, by means of which each of the principal 
 sections of his wave-surface (considering only two out of its 
 three sheets) is reduced to the circle and ellipse of Fresnel's 
 law; and the three principal sections being thus forced to coin- 
 cide, it would not be very surprising if the two sheets were 
 found to coincide in every part with the wave-surface of Fres- 
 
2io Notes on some Points in the Theory of Light. 
 
 nel. The coincidence, however, is only approximate ; but 
 M. Cauchy is so far from being embarrassed by this circum- 
 stance, that he does not hesitate to regard his own theory as 
 rigorously true, and that of Fresnel as bearing to it, in point of 
 accuracy, the same relation which the elliptical theory of the 
 planets, in the system of the world, bears to that of gravita- 
 tion.* Nor is he at all embarrassed by the supernumerary ray 
 belonging to the third sheet of his wave-surface ; he assumes 
 at once that such a ray exists, though it was never seen, and 
 promises, for the satisfaction of philosophers, to make known 
 the means of ascertaining its existence. f But he afterwards 
 contented himself with observing that as its vibrations are in 
 the direction of propagation they probably make no impression 
 on the eye, and he then gave it the name of the "invisible 
 ray."J 
 
 In these investigations, the suppositions which M. Cauchy 
 had made respecting the constants led to the result that the 
 vibrations of a polarized ray are parallel to its plane of pola- 
 rization ; but in the year 1836 he changed his opinion on this 
 point, and then, by reinstating the constants that he had be- 
 fore supposed to vanish, and establishing proper relations 
 amongst them and the rest, he arrived at the conclusion that 
 the vibrations are perpendicular to the plane of polarization. 
 All his other results, of course, underwent some correspond- 
 ing change ; and it is this new theory which must now be 
 regarded as rigorous, while that of Fresnel is to be looked 
 on as approximate. But it is needless to say, that if the 
 accuracy of Fresnel's law of double refraction is to be disputed, 
 it must be on much better grounds than these ; and the results 
 of M. Cauchy are certainly too far removed from that law to 
 have any chance of being consonant with truth. Although, for 
 example, his new views respecting the direction of the vibra- 
 
 * Memoires de VInstitut, torn. x. p. 313. 
 
 t Ibid. p. 305. 
 
 % Nouveaux Exercices, p. 40. 
 
 Comptes Rendus, torn. ii. p. 342. 
 
Notes on some Points in the Theory of Light. 2 1 1 
 
 tions agree, in a general way, with those of Fresnel, there is 
 yet, in one particular, an important difference between them ; 
 for, according to Fresnel, the vibrations are always exactly in 
 the surface of the wave, while, according to M. Cauchy (in his 
 old theory as well as the new), they are only so in ordinary 
 media. In a biaxal crystal he finds and this is one of the 
 ways in which the "invisible ray" manifests its influence that 
 the direction of vibration, in each of the two rays that are 
 visible, is inclined at a certain angle to the wave-plane ; but 
 this angle, though small, is by no means inconsiderable, as 
 M. Cauchy seems to intimate, overlookiug the fact, which ap- 
 pears from his own equations, that it is of the same order of 
 magnitude as the quantities on which the double refraction 
 depends. It is true, the deviation measured by this angle can- 
 not, if it exists, be directly observed in the refracted light ; but 
 its indirect effects on reflected light ought to be very great, 
 since the action of the crystal on a ray reflected at its surface 
 differs from that of an ordinary medium by a quantity of the 
 same order merely as the aforesaid angle ; and as the problem 
 of crystalline reflexion has been already solved * on the suppo- 
 sition (which is an essential one in the solution) that the 
 vibrations are exactly in the plane of the wave, it is highly 
 improbable, considering the complex nature of the question, 
 that it will be solved, in any satisfactory way, on a supposition 
 so different as that which is required by the theory of M. Cauchy. 
 However, as the laws of such reflexion are now well known, by 
 means of the solution alluded to, it is possible that M. Cauchy 
 may, as in the case of double refraction, succeed in deducing the 
 same laws, or, if not the same, what may seem to be more exact 
 laws, from certain principles t of his own, helped out, if need be, 
 
212 Notes on some Points in the Theory of Light. 
 
 by proper relations among his constants ; especially if, to allow 
 greater scope for such relations, the number of constants be in- 
 creased by the hypothesis of two coexisting systems of mole- 
 cules, an hypothesis which M. Cauchy has already considered 
 with his usual generality, but without making any precise ap- 
 plication of it.* 
 
 Perhaps one cause why M. Cauchy 's views on the subject 
 of double refraction have met with such general acceptance 
 may be found in the fact, that a theory setting out from the 
 
 total reflexion for the substance of which it is composed, a ray incident perpendi- 
 cularly on one of the faces will emerge, making a very small angle with the other 
 face ; and as the reflexion at the latter face is nearly total, it is self-evident that 
 the intensity of the emergent light, as compared with that of the incident, must he 
 very small. M. Cauchy, however, finds by an elaborate analysis that a prodigious 
 multiplication of light \_"tme prodigieuse multiplication de a lumiere"~\ takes place, 
 the emergent ray being nearly six times more intense than the incident when the 
 prism is made of glass, and nearly nine times when the prism is of diamond. This 
 result was, in a general way, actually verified experimentally by himself and ano- 
 ther person ; so easy it is, in some cases, to see anything that we expect to see. 
 Had the result been true, it would have been a very brilliant discovery indeed ; for 
 then we should have been able, by a simple series of refractions, to convert the 
 feeblest light into one of any intensity we pleased ; but the very absurdity of such 
 a supposition should have taught M. Cauchy to distrust both his theory and his ex- 
 periment. Far from doing so, however, he considers the fact to be perfectly esta- 
 blished, and to afford a new argument against the system of emission, " Ici," says 
 he, " un rayon, reflechi en totalite, est de plus transmis avec accroissement de 
 lumiere; ce qui est un nouvel argument contre le systeme d'emission." The sys- 
 tem of emission has at least this advantage, that by no possible error could such a 
 conclusion be deduced from it. For if all the particles of light be reflected, cer- 
 tainly none of them can be refracted. 
 
 The truth is, that M. Cauchy mistook the measure of intensity in the hypothe- 
 sis of undulations, supposing it to be proportional simply to the square of the 
 amplitude of vibration ; whereas it is really measured by the vis viva, or by that 
 square multiplied by the quantity of ether put in motion, a quantity which in the 
 present case is evanescent, since the corresponding volumes of ether, moved by the 
 ray within in the prism and by the emergent ray, are to each other as the sine of 
 twice the angle of the prism to the sine of twice the very small angle which the 
 emergent ray makes with the second face of the prism. The intensity of the emer- 
 gent light is therefore very small, as it ought to be, though the amplitude of its 
 vibrations is considerable. 
 
 * Exercices d* Analyse et de Physique Mathenwtique, torn. i. p. 33. 
 
Notes on some Points in the Theory of Light. 213 
 
 same principles, and leading, by the same relations among con- 
 stants, to formulas identical in every respect with his earlier 
 results, was advanced independently, and nearly at the same 
 time, by M. Neumann of Konigsberg.* A coincidence so re- 
 markable would be looked upon, not unreasonably, as a strong 
 argument in favour of the theory ; though it must be allowed 
 that, in the effort to extend the knowledge of any subject, 
 there is a tendency in different minds to adopt the same errors 
 respecting it, as well as the same truths ; a fact of which we 
 have seen other examples in the course of the present article. 
 
 According to M. Neumann,f the " third ray," not being 
 perceived as light, must manifest its existence as radiant heat, 
 or as a chemical power, or as some other agent [" als strahlende 
 Wdrme, oder chemisch wirkend, oderals irgend ein anderes Agem"~\ 
 and he thinks that the nature of this ray will be more easily 
 investigated, if the laws of reflexion shall be deduced from the 
 aforesaid theory. But we have seen that the laws of reflexion 
 are, to all appearance, at variance with the theory, and they 
 take no account whatever of the third ray. Besides, the dis- 
 coveries which have been made of late years respecting the po- 
 larization of radiant heat, and the strong analogies that have 
 been traced between it and light, amount to a demonstration 
 that its vibrations are transversal, and of course essentially dif- 
 ferent from those of the supposed third ray, which are normal, 
 or nearly so. There is every reason to believe that the vibrations 
 of the chemical rays are also transversal ; and we may confi- 
 dently assert, that the three species of rays those of light and 
 heat, and the chemical rays are produced not only by vibra- 
 tions of the same medium, but by the same kind of vibrations, 
 propagated with nearly the same velocities. If, therefore, the 
 third ray of MM. Cauchy and Neumann has any existence, it 
 must be referred to " some other agent," the nature of which it 
 is impossible to conjecture. 
 
 Enough has now been said to show that the optical theory 
 
 * Poggendorff's Annals, Vol. xxv. p. 418. 
 f Ibid. p. 454. 
 
214 Notes on some Points in the Theory of Light. 
 
 which we have examined, and which has passed current in the 
 scientific world for a considerable period, is quite inadequate to 
 explain the leading phenomena of light, and that it is based 
 upon principles which are altogether inapplicable to the subject. 
 M. Cauchy states, in the memoir so often quoted,* that the first 
 application which he had made of his principles was to the 
 theory of sound, and that the formulas which he had deduced 
 from them agreed remarkably well with the experiments of 
 Savart and others on the vibrations of elastic solids. As I have 
 already intimated, it is in the solution of such questions (which, 
 however, have long been familiar to mathematicians) that the 
 fundamental equations of M. Cauchy may be most advanta- 
 geously employed ; and had he pursued his researches in this 
 direction, his labours would doubtless have been attended with 
 more success, and with greater benefit to science. 
 
 II. On Fresnel's Formula for the Intensity of Reflected Light, 
 with Remarks on Metallic Reflexion. 
 
 When Mr. Potter discovered, by experiment, 'that more 
 light is reflected by a metal at a perpendicular incidence than 
 at any oblique incidence (at least as far as 70), the fact was 
 looked upon, by himself and others, as contrary to all received 
 theories ; and certainly the universal opinion, up to that time, 
 was, that the intensity of reflexion always increases with the 
 incidence. It may therefore be worth while to remark, that the 
 formula given by Fresnel for reflexion at the surface of a trans- 
 parent body, though not of course applicable, except in a very 
 rude way, to the case of metals, would yet lead us to expect, for 
 highly refracting bodies, as the metals are supposed to be, pre- 
 cisely such a result as that obtained by Mr. Potter. For when 
 the index of refraction exceeds the number 2 + v/3, or the tan- 
 gent of 75, the expression for the intensity of reflected light 
 will be found to have a minimum value at a certain angle of in- 
 
 * Mem. de I'lnstitut, torn. x. p. 294.. 
 
Notes on some Points in the Theory of Light. 215 
 
 cidence ; while for all less values of the refractive index the in- 
 tensity will be least at the perpendicular incidence. 
 
 Let i and i' be the angles of incidence and refraction, and 
 put" 
 
 , _ sin i cos i 
 
 M = r,, a = r, J 
 
 sm ^ cos i 
 
 then if / be the intensity of the reflected light, when common 
 light is incident, FresnePs expression 
 
 = JL I sin2 (* " tan 2 (t - Q J 
 ? ( sm^ + + tan 2 (t + f) )' 
 
 in which the intensity of the incident light is taken for unity, 
 may be put under the form 
 
 1 V /I ^ 
 
 (---Ml + i?--^ 
 
 J _y j u ^y yjf y 
 
 \ 2 ' 
 
 which has a minimum value when 
 
 + l = M+ 8 
 
 ^ + u ~ + M 1 ; 
 
 the value of I being in that case 
 I vi 
 
 and the corresponding angle of incidence being given by the 
 formula 
 
 H , !/.. 1\ 
 
 sin z = x , where = J Jt 4 ^ . 
 
 , t V?-i V ^/ 
 
 Since ju + - cannot be less than 2, it is easy to see that, when 
 
216 Notes on some Points in the Theory of Light. 
 
 there is a minimum, M + -==. cannot be less than 4, and therefore 
 
 M cannot be less than 2 + \/3, or 3732. 
 
 As an example, let M + = 6. Then, at a perpendicular 
 
 incidence, one-half the incident light will be reflected. The 
 minirilum will be when i = 65 36', and at this angle only -j^- of 
 the incident light will be reflected. The value here assumed 
 for the refractive index is that which Sir J. Herschel* assigns 
 to mercury ; but if my ideas be correct, it is far too low for that 
 metal. 
 
 The only person who supposes that the refractive index of a 
 metal is not a large number is M. Cauchy, It has always 
 been held as a maxim in optics, that the higher the reflective 
 power of any substance, the higher also is its refractive index. 
 But M. Cauchy completely reverses this maxim ; for, as I have 
 elsewhere shown, f it follows from his theory that the most re- 
 flective metals are the least refractive, and even that the index 
 of refraction, which for transparent bodies is always greater 
 than unity, may for metals descend far below unity. Thus, 
 according to his formula, the index of refraction for pure silver 
 is the fraction J, so that the dense body of the silver actually 
 plays the part of a very rare medium with respect to a vacuum. 
 It appears to me that such a result as this is quite sufficient to 
 overturn the theory from which it is derived. The formulas, 
 however, which he gives for the intensity of the reflected light, 
 are identical with the empirical expressions which I had given 
 long before, and are at least approximately true. 
 
 In framing my own empirical theory,* two suppositions re- 
 lative to the value of the refractive index presented themselves. 
 Putting M for the modulus, and x f or ^ ne characteristic, I had to 
 
 choose between the values M cos Y and . The latter value 
 
 GO-SX 
 is that which I adopted ; the former, which is M. Cauchy's, was 
 
 * Treatise on Light, Art, 594. 
 
 t Comptes Eendus, torn. viii. p. 964 ; vid. note at the end of this volume. 
 
 See Proceedings, VOL. i. p. 2 (supra, p. 58). 
 
Notes on some Points in the Theory of Light. 217 
 
 rejected, because I saw that it would lead to the result above 
 mentioned. 
 
 Another result of M. Cauchy's, which he has given twice in 
 the Oomptes Rendus* requires to be noticed. When a polarized 
 ray is reflected by a metal, the phase of its vibration is altered ; 
 and if the incidence be oblique, the change of phase is different 
 according as the light is polarized in the plane of incidence, or 
 in the perpendicular plane. But when the ray is reflected at a 
 perpendicular incidence, it is manifest that the change is a con- 
 stant quantity, whatever be the plane of polarization. In fact, 
 the distinction between the plane of incidence and the perpen- 
 dicular plane no longer exists, and the phenomena must be the 
 same in all planes passing through the ray. Yet M. Cauchy, 
 in the two places above quoted, asserts it to be a consequence of 
 his theory, that in this case the alterations of phase are different 
 for two planes of polarization at right angles to each other, and 
 that the difference of the alterations amounts to half an undula- 
 tion. The same singular hypothesis had been previously made 
 by M. Neumann,f whom M. Cauchy appears to have followed ; 
 but M. Neumann has since admitted it to be erroneous. $ 
 
 * Tom. ii. p. 428, and torn. viii. p. 965. 
 t Poggendorff s Annals, Vol. xxvi. p. 90. 
 J Ibid. VOL. XL. p. 513. 
 
( 218 ) 
 
 XYIIL ON THE PROBLEM OF TOTAL REFLEXION. 
 
 [Proceedings of the Royal Irish Academy, VOL. n. Read November 30, 1841.] 
 
 PROFESSOR MAC CULLAGH communicated to the Academy a 
 very simple geometrical rule, which gives the solution of the 
 problem of total reflexion, for ordinary media, or for uniaxal 
 crystals. 
 
 First, let the total reflexion take place at the common sur- 
 face of two ordinary media, as between glass and air, and let it 
 be proposed to determine the incident and reflected vibrations, 
 when the refracted vibration is known. It is to be observed, 
 that the refracted vibration (which is in general elliptical) can- 
 not be arbitrarily assumed ; for, as may be inferred from what 
 has been already stated,* it must be always similar to the sec- 
 tion of a certain cylinder, the sides of which are perpendicular 
 to the plane of incidence, and the base of which is an ellipse 
 lying in that plane, and having its major axis perpendicular to 
 the reflecting surface, the ratio of the major to the minor axis 
 being that of unity to the constant r. The value of r, as deter- 
 mined by the general rule in p. 191, is 
 
 < " J 1 o- 
 
 \ n* si 
 
 sin 1 
 
 where i is the angle of incidence, and n the index of refraction 
 out of the rarer into the denser medium. The ellipse is greatest 
 
 * Proceedings of the Academy, VOL. n. p. 102 (supra, p. 192). 
 
On the Problem of Total Reflexion. 219 
 
 for a particle at the common surface of the media ; and for a 
 particle situated in the rarer medium, at the distance z from that 
 surface, its linear 'dimensions are proportional to the quantity 
 
 2-rrz 
 
 e K ; so that for a very small value of z the refracted vibra- 
 tion becomes insensible. 
 
 Now, taking any plane section of the aforesaid cylinder to 
 represent the refracted vibration for a particle situated at the 
 common surface of the two media, let OP and OQ be the semi- 
 axes of the section, and let them be drawn, with their proper 
 lengths and directions, from the point of incidence ; through 
 which point also let two planes be drawn to represent the inci- 
 dent and reflected waves. Then conceive a plane passing 
 through the semiaxis OP, and intersecting the two wave- 
 planes, to revolve until it comes into the position where the 
 semiaxis makes equal angles with the two intersections ; and in 
 this position let the intersections be made the sides of a parallel- 
 ogram, of which the semiaxis OP is the diagonal. Let OA 
 and OA', which are of course equal in length, denote these two 
 sides. Make a similar construction for the other semiaxis OQ, 
 and let OB, OB', which are also equal, denote the two sides of 
 the corresponding parallelogram. Then will the incident vi- 
 bration be represented by the ellipse of which OA and OB are 
 conjugate semidiameters, and the reflected vibration by the 
 ellipse of which OA and OB' are conjugate semidiameters. 
 And the correspondence of phase in describing the three ellipses 
 will be such that the points A, A, P will be simultaneous posi- 
 tions, as also the points B, B', Q. 
 
 The same construction precisely will answer for the case of 
 total reflexion at the surface of a uniaxal crystal, which is 
 covered with a fluid of greater refractive power than itself. It 
 is to be applied successively to the ordinary and extraordinary 
 refracted vibrations, and we thus get the uniradial incident and 
 reflected vibrations, or rather the ellipses which are similar to 
 them. And as any incident vibration may be resolved into two 
 which shall be similar to the uniradial ones, we can find the re- 
 
220 On the Problem of Total Reflexion. 
 
 fleeted vibration which corresponds to it, by compounding the 
 uniradial reflected vibrations. 
 
 It may be well to mention that, in a uniaxal crystal, the 
 plane of the extraordinary refracted vibration is always perpen- 
 dicular to the axis, and therefore the ellipse in which the vibra- 
 tion is performed may be easily determined by the remark in 
 p. 192. The plane of the ordinary vibration has no fixed po- 
 sition in the crystal; but if we conceive the auxiliary quantities 
 i, TJI, 1 (p. 188), to be compounded into an ellipse (as if they 
 were displacements), the plane of this auxiliary ellipse will be 
 perpendicular to the axis of the crystal. 
 
 Whether the preceding very simple construction, for finding 
 the incident and reflected vibrations by means of the refracted 
 vibration, extends also to the case of liaxal crystals, is a point 
 which has not yet been determined, on account of the compli- 
 cated operations to which the investigation leads, at least when 
 attempted in any way that obviously suggests itself. 
 
( 221 ) 
 
 XIX. 01$ THE DISPERSION OF THE OPTIC AXES, AND OF 
 THE AXES OF ELASTICITY, IN BIAXAL CEYSTALS. 
 
 [From the Philosophical Magazine, VOL. xxi., October, 1842.] 
 
 IN the last number of the Philosophical Magazine (p. 228), 
 there appeared an extract from the Proceedings of the Royal 
 Irish Academy, containing a notice of a memoir which I had 
 the honour of reading to that body on the 24th of May, 1841 ; 
 and in the concluding paragraph of the notice a brief allusion 
 is made to a "mathematical hypothesis" by which I had con- 
 nected the laws of dispersion and those of the elliptic polari- 
 zation of rock-crystal with the other laws that were there 
 announced. My present object is to indicate the development 
 of that hypothesis, with reference more particularly to the sub- 
 ject of dispersion in crystals, and to communicate a very simple 
 result which I have lately had occasion to obtain from it. The 
 result is remarkable as embracing and explaining a class of in- 
 tricate phenomena which hitherto have not been connected with 
 any theory, or rather have stood in opposition to all theories ; 
 I mean the phenomena of the dispersion of the optic axes, and 
 of the axes of elasticity (as they are called) in biaxal crystals. 
 
 The name of axes of elasticity was given by Fresnel to 
 three rectangular directions, which, according to his theory, 
 exist in every crystallized medium, and which are distinguished 
 by the property, that if a particle of the medium be slightly 
 displaced in the direction of any one of them, the elastic force 
 thereby called into play will act precisely in the line of the dis- 
 
222 On the Dispersion of the Optic Axes, and 
 
 placement. These directions coincide with the axes of the 
 ellipsoid by which he constructs his wave-surface ; and the po- 
 sition of the axes being thus fixed, it is only their lengths that 
 can be supposed to vary for the differently coloured rays. 
 Such is the view taken by Fresnel with regard to crystalline 
 dispersion, and it is obviously the only view that his theory 
 admits. Succeeding theorists, in their numerous attempts to 
 deduce Fresnel's beautiful laws from dynamical principles, have 
 always been obliged to assume that the medium is symmetri- 
 cally arranged with respect to three rectangular planes ; and 
 as, in this hypothesis, the axes of elasticity, or of optical sym- 
 metry, necessarily coincide with those of symmetrical arrange- 
 ment, their directions are fixed, as before, independently of 
 colour. 
 
 From these principles it follows that the optic axes for dif- 
 ferent colours all lie in the same plane, namely, the plane of 
 the greatest and least axes of the ellipsoid, and that they are 
 equally inclined to each of the latter axes, so that the angle 
 made by any pair, to whatever colour they belong, is always 
 bisected by the same right line. This was accordingly, for a 
 long time, believed to be the case ; and the earlier experiments 
 of Sir J. Herschel,* which are appealed to by Fresnel, as well 
 as the observations of Sir David Brewster, seemed to establish 
 it as a general law. But it was afterwards discovered by Sir J". 
 Herschel that, in borax, the optic axes for different colours lie 
 in different planes inclined at very sensible angles to each 
 other ; and the same discovery was made about the same time 
 (1832) by M. Norrenberg. The latter observer further ascer- 
 tained, that even when the optic axes all lie in the same plane 
 there are cases, as in sulphate of lime, wherein their angles are 
 not bisected by the same right line. These facts, and others of 
 a like nature that have been since observed, show the falsehood 
 of the supposition that the lines called the axes of elasticity 
 have always the same directions whatever be the colour of the 
 
 * Phil. Trans. 1820. 
 
of the Axes of Elasticity, in Biaxal Crystals. 223 
 
 light ; they are inconsistent with all received notions, and con- 
 tradict every theory that has been hitherto proposed. No 
 person, as far as I am aware, has even attempted to explain 
 them. 
 
 But in the theory which I have constructed to represent the 
 laws of the action of crystallized bodies upon light, and which 
 has already brought so much within its grasp, the phenomena 
 in question do not offer any difficulty whatever: on the con- 
 trary, they are of a kind that would naturally be looked for, 
 antecedently to experiment. For, in this theory, I make no 
 hypothesis as to the constitution of the ether, or the arrange- 
 ment of its molecules ; nor any hypothesis, like that of Fresnel, 
 respecting the mechanical signification of the axes of elasticity. 
 The existence of three rectangular axes possessing peculiar pro- 
 perties is not a principle, but a result, of theory : their direc- 
 tions are determined by conditions perfectly analogous to those 
 which determine the principal axes of an ellipsoid from its ge- 
 neral equation ; and these directions are functions of certain 
 quantities which are constant when differentials of the second 
 and subsequent orders are neglected, but which vary when these 
 are taken into account. The differentials of higher orders in- 
 troduce terms depending on the wave-length ; and thus the 
 directions, as well as the lengths, of the principal lines depend on 
 the colour of the light, or, to speak more accurately, on the 
 length of the wave. 
 
 All this will be easily understood if we recur to the first 
 principles of the theory. According to these, everything de- 
 pends on the form assigned to the function Y in the general 
 dynamical equation 
 
 JJJ dxdydz 8$ + &, + dt = f dxdydzW, 
 
 from which the motion of the ether is deduced. In my first 
 memoir on the subject (read to the Academy on the 9th of De- 
 cember, 1839), I showed that when differentials of the first 
 order only are preserved, the function Y which may perhaps 
 
224 On the Dispersion of the Optic Axes, and 
 
 with propriety be called the potential, since the motion of the 
 system is potentially, or virtually, included in it is a function 
 of the second degree, composed of the three quantities X, Y, Z, 
 which are connected with the/ displacements , rj, , by the fol- 
 lowing relations : 
 
 = _ _ - 
 
 dz dy* dx dz 9 dy dx 
 
 To show this, I make use simply of the consideration that the 
 motion must be such as to satisfy the condition 
 
 T+T+f--> 
 dx dy dz 
 
 which seems to be characteristic of the vibrations of light. But 
 the same condition allows us to suppose that the potential con- 
 tains not only the quantities X, F, Z, but their differential co- 
 efficients of any order with respect to the co-ordinates. This 
 supposition, however, is too general, and requires to be limited 
 by other considerations. Now the most natural restriction 
 which can be imposed consists in the assumption that the quanti- 
 ties of all orders are formed on the same type, those of any 
 order being derived from the preceding in the same way that 
 the quantities X, Y, Z are derived from , 17, : there are par- 
 ticular reasons also which go to strengthen this hypothesis, and 
 have led me to adopt it. Putting therefore 
 
 d_Y_dZ V _<^_Z 7 _^_ 
 Al ~ dz dy 9 * " dx dz 9 ' ~ dy dx '* 
 
 dYi d& v _ dZi dX l 7 _ dX, dT\ 
 
 ^~~dz~~dy 9 **~ dx dz' ^-dy"dx> 
 
 and so on, I suppose the potential to be a function of the second 
 degree, composed of all the quantities X, Y, Z, Xi, Pi, Z^ X 3t 
 P 2 , Z-i, &c. ; and this is the " mathematical hypothesis" alluded 
 to in the beginning of this article. The hypothesis occcurred to 
 me more than three years ago (June, 1839), but I did not ven- 
 
of the Axes of Elasticity, in Biaxal Crystals. 225 
 
 ture to communicate it to the Academy until the date of my 
 second memoir (May, 1841) ; and even then I had not studied 
 it with the attention which I now conceive it merits. It was 
 only very lately, in fact, in some conversations which I had 
 with M. Babinet during a short visit to Paris, that my attention 
 was strongly drawn to the subject of dispersion in crystals, par- 
 ticularly the dispersion of the axes of elasticity. My thoughts 
 then naturally reverted to the hypothesis which I have men- 
 tioned, and since my return I have found that it affords a com- 
 plete explanation of all the phenomena.* 
 
 I have also found that it gives the general law, extended to 
 biaxal crystals, of that elliptic and circular polarization which 
 has hitherto been detected only in quartz and in certain fluids ; 
 while for the case of rectilinear polarization it gives a law (very 
 possibly a true one) more general than that of Fresnel, but 
 quite as elegant, and differing very slightly from it. The hy- 
 pothesis, therefore, is still too general for our present purpose. 
 To make it include only those crystals to which the law of 
 Fresnel is rigorously applicable, the alternate derivatives Xi, 
 Fi, ZD X 3 , F 3 , Z z , &c., must be supposed to vanish in the func- 
 tion which represents the potential. Then, the axes of co- 
 ordinates having any fixed directions within the crystal, the 
 axes of elasticity will be the principal axes of an ellipsoid re- 
 presented by an equation of the form 
 
 Ax* + By* + <7s 2 + 2Dyz + ZExz + 2Fxy = 1, 
 
 in which each of the six coefficients the first, for example ex- 
 presses a series of the form 
 
 A -&-\ AI AS p 
 
 ^ + v + * + v + *" 
 
 where X denotes the wave-length, and all the other quantities 
 
 * I am indebted, for my information on the subject, to a short article, drawn 
 up by MM. Quetelet and Babinet, in the Bulletin of the Royal Academy of 
 Brussels, VOL. n. p: 150 ; as also to Poggendorff' s Annals, YOL. xxvi. p. 309 ; 
 VOL. xxxv. p. 81. 
 
 Q 
 
226 On the Dispersion of the Optic Axes, &c. 
 
 are constant. The ellipsoid itself is the reciprocal of that ellip- 
 soid by which the wave-surface is constructed, and its semiaxes 
 are the three principal indices of refraction. As X is supposed 
 to vary, not only the lengths but the directions of the principal 
 axes vary, and thus we have a different wave-surface for every 
 different wave-length within the crystal. 
 
 The optic axes are perpendicular to the circular sections of 
 the above ellipsoid, and describe, in general, two fragments of a 
 cone, the equation of which may be found by supposing X to be 
 variable in the equation of the ellipsoid. But only very par- 
 ticular cases have been hitherto observed, and I shall not stop 
 to discuss them. 
 
 J. MACCULLAGH. 
 THIN. COLL., PUBLIN, September, 1842. 
 
XX. ON THE LAW OF DOUBLE REFRACTION. 
 
 [From the Philosophical Magazine, VOL. xxi., 1842]. 
 
 HAVING mentioned, in an article * which I sent a few days ago 
 for insertion in the Philosophical Magazine, that I had been led, 
 in following out an hypothesis, to a law of double refraction 
 more general than that of Fresnel, I think it may be well to 
 state very briefly the nature of that law, and to point out the 
 difference between it and the law of Fresnel, especially as I 
 have since observed that the difference is one of a very extra- 
 ordinary kind, and one which, if it has a real existence (a 
 question which experiment only can decide), may serve to 
 account for phenomena that have seemed hitherto inexpli- 
 cable. 
 
 I have said, in the article referred to, that when the poten- 
 tial Y, which is a function of the second degree, is supposed 
 to contain only the squares and products of the derivatives 
 X, Y 9 Z, X. z , Y 2 , Z^ 3T 4 , &c., we get the law of Fresnel, as well 
 as the law of crystalline dispersion ; but if we make the more 
 general, and apparently the more natural supposition, that it 
 contains also the squares and products of the alternate deriv- 
 atives Xi, Fi, Zi, X 9 , F 3 , Z 3 , &c., then we get, of course, a dif- 
 ferent law. Now I find that there will still be two optic axes 
 for each colour, and that the two directions of vibration in a 
 given wave-plane will have the same relation to them as be- 
 
 * " On the Dispersion of the Optic Axes, and of the Axes of Elasticity, in Biaxal 
 Crystals" (supra, p. 221). 
 
 Q2 
 
228 On the Law of Double Refraction. 
 
 fore ; while the difference of the squares of the two velocities of 
 propagation will continue proportional to the product of the 
 sines of the angles which the wave normal makes with the optic 
 axes ; but the sum of the squares of these velocities will be in- 
 creased or diminished by a quantity proportional to the square 
 of a perpendicular let fall from the centre on the tangent plane 
 of a certain very small ellipsoid, this tangent plane being sup- 
 posed parallel to the wave. Such is the general result for biaxal 
 crystals ; but its bearing will be best perceived by taking the 
 case of a uniaxal crystal, wherein the law of Fresnel reduces 
 itself to that of Huyghens. 
 
 In this case the wave-surface will, instead of the sphere and 
 spheroid of Huyghens, consist of two ellipsoids touching each 
 other at the extremities of a common diameter, which coincides 
 with the axis of the crystal; one ellipsoid differing slightly 
 from a sphere, the other slightly from a spheroid. Neither of 
 the rays will be refracted according to the ordinary law, nor 
 will the wave-surface be symmetrical round the axis. As the 
 law of refraction is unsymmetrical, that of reflexion will be so 
 likewise, and thus we may perhaps obtain an explanation of 
 the extraordinary phenomena observed by Sir David Brewster 
 in reflexion at the common surface of oil of cassia and Iceland 
 spar. 
 
 It will no doubt appear strange to call in question the ac- 
 curacy of the Huyghenian law, which is generally, considered 
 to be established beyond dispute by the experiments of Wol- 
 laston and Malus. But the fact is, that no exact experiments 
 have ever been made on the refraction of the ordinary ray. 
 Neither of those philosophers seems to have entertained any 
 suspicion that the ordinary law might be inapplicable to it; 
 they both took for granted that it followed the law of Snellius. 
 But their results seem to be quite consistent with the suppo- 
 sition that the ordinary index, for rays passing in different 
 directions through Iceland spar, may vary in the third place 
 of decimals, perhaps even in the second. The experiments of 
 Eudberg throw no light upon the question, for it happens, 
 
On the L aw of Double Refraction. 229 
 
 oddly enough, that though he had two prisms in every other 
 case, he used only one of Iceland spar : he could not there- 
 fore compare the velocities of rays passing in different direc- 
 tions. On comparing his numbers, however, with those of 
 Wollaston and Malus, there is, as Sir David Brewster has 
 observed,* a " surprising discrepancy," so great indeed as to 
 be quite "alarming." After remarking the difficulty of find- 
 ing any explanation of it, Sir David concludes that it must 
 arise from the different refractive powers possessed by different 
 specimens. But though this cause must operate in some 
 degree, we cannot tell to what extent it is effective, and the 
 discrepancy may notwithstanding be occasioned, in a great 
 measure, by a deviation from the Huyghenian law. The whole 
 question must therefore be reopened, and the ordinary in- 
 dices for the fixed lines of the spectrum must be determined by 
 means of different prisms cut out of the same piece of Iceland 
 spar. 
 
 Whatever the result may be whether it shall confirm the 
 law of Huyghens, or show that another must be substituted for 
 it it will at least be useful for science, by removing the uncer- 
 tainty in which the subject is at present involved. 
 
 * Phil Mag., S. 3, VOL. i. p. 8. 
 TEIN. COLL., DUBLIN, Sept. 24, 1842. 
 
XXI. OK THE LAWS OF METALLIC BEFLEXION, AM) 
 ON THE MODE OF MAKING EXPEBIMENTS UPON 
 ELLIPTIC POLARIZATION. 
 
 [Proceedings of the Royal Irish Academy, VOL. n. p. 376. Read, May 8, 1843.] 
 
 SEVERAL years ago, as the Academy are aware, I made an at- 
 tempt to investigate the laws according to which light is re- 
 flected at the surface of metals, and I then proposed certain 
 formulae which represented, with sufficient accuracy, all the 
 facts and experiments which I was able to collect upon the sub- 
 ject.* But in order to test these formulas satisfactorily, it was 
 necessary to obtain measurements far more exact than any that 
 had previously been made ; and for this end I devised an in- 
 strument, which was constructed for me by Mr. Grubb, and of 
 which a brief description has been given in the Proceedings.-^ I 
 regret to say, however, that nothing of much consequence has 
 yet been done with the instrument. Some preliminary trials of 
 its performance were indeed made in the summer of 1837, and 
 the results of one of these shall presently be given ; but an ac- 
 cidental strain which it suffered, while I was preparing to un- 
 dertake a series of experiments, caused me to discontinue the 
 observations at the time ; and being then obliged to superintend 
 the printing of my essay on the " Laws of Crystalline Reflexion 
 and Refraction,":}: my attention was drawn afresh to this latter 
 
 * See the Proceedings of the Academy, VOL. i. p. 2, October, 1836 (supra, p. 58) ; 
 Transactions, VOL. xvm. p- 71, note (supra, p. 133). 
 + VOL. i. p. 159 (supra, p. 138). 
 % Transactions R. I. A. (supra, p. 87). 
 
Laws of Metallic Reflexion. 231 
 
 subject, respecting which some new questions suggested them- 
 selves, which I thought it right to discuss in notes appended to 
 the essay. I was not afterwards at leisure to take up the ex- 
 perimental inquiry, until the beginning of the year 1839, when 
 I began to think of putting the instrument in order for that 
 purpose. The strain which it had suffered rendered some slight 
 alterations necessary ; and I now resolved to make additions to 
 it also, with the view of operating upon the fixed lines of the 
 spectrum, as a few trials had convinced me that measures suffi- 
 ciently precise could not be obtained without employing light 
 of definite refrangibility. I wished, moreover, to take the oppor- 
 tunity which the nature of the proposed experiments presented, 
 of verifying the theory of Fresnel's rhomb, or rather of verify- 
 ing, by means of the rhomb, the formulae which Fresnel has 
 given for computing the effects of total reflexion, when it takes 
 place at the common surface of two ordinary media. I wrote 
 therefore to Munich for several articles which I wanted ; among 
 others, for a set of rhombs cut at different angles, out of dif- 
 ferent kinds of glass. But while I was waiting for these some 
 months elapsed, and in the meantime I got sight of a new 
 theory, which, from its connexion with my former researches, 
 possessed more immediate interest, and the pursuit of which, in 
 conjunction with other studies and various engagements, caused 
 me again to suspend the inquiry respecting the laws of metallic 
 reflexion. I allude to the " Dynamical Theory of Crystalline 
 Reflexion and Befraction," communicated to the Academy in 
 December, 1839.* This was followed soon after by a general 
 " Theory of Total Beflexion,"f founded on the same principles. 
 The latter theory, forming a new department of physical optics, 
 and involving the solution of questions not previously attempted, 
 was analytically complete when it was communicated to the 
 Academy in May, 1841 ; but its geometrical development has 
 since required my attention from time to time, and has not yet 
 been brought to that degree of simplicity of which it appears to 
 
 * Proceedings, VOL. i. p. 374, (supra, p. 145), 
 f Ibid. VOL. n. p. 96 (sitpra, p. 187). 
 
232 Laws of Metallic Reflexion, and Mode of 
 
 be susceptible.* Indeed I have found that, in this instance, 
 the geometrical laws of the phenomena are by no means obvious 
 interpretations of the equations resulting from the analytical 
 solution of the problem ; and in endeavouring to verify such 
 supposed laws I have often been led to algebraical calculations 
 of so complicated a nature that it has been impracticable to 
 bring them to any conclusion, and I have been obliged, from 
 mere weariness, to abandon them altogether. On returning, 
 however, to the investigation, after perhaps a long interval of 
 time, I have usually perceived some mode of eluding the calcu- 
 lations, or of directly deducing the geometrical law ; and, when 
 the theory comes to be published in its final form, no trace of 
 these difficulties will probably appear. 
 
 From the causes above mentioned, combined with frequent 
 absence from Dublin, the researches which I had entered upon, 
 respecting the action of metals upon light, have been hitherto 
 interrupted ; and as it may still be some time before they are 
 resumed, I venture, in the meanwhile, to submit to the Aca- 
 demy the results already spoken of, which were obtained on the 
 first trial of the instrument, and which afford the best data 
 that can yet be had for comparison with theory. 
 
 The results, it must be confessed, are those of very rough 
 experiments, made one evening (about the month of July, 1837) 
 in company with Mr. Grrubb, before I had received the instru- 
 ment from his hands, and merely with the view of showing him, 
 when it was finished, the kind of phenomena that I proposed to 
 observe with it, and the mode of observing them. But the in- 
 strument was so far superior (in workmanship at least) to any 
 apparatus previously employed for this sort of experiments, 
 that it was impossible, without great negligence in using it, not 
 to obtain measures of considerable accuracy. I did not, how- 
 ever, at the time, set much value on these measures, because I 
 expected shortly to possess a series of observations made with 
 every possible precaution ; but having chanced to preserve the 
 paper on which they were noted down, I was tempted, a few 
 
 * Proceedings, VOL- n. p. 174 (supra, p. 218). 
 
making Experiments upon Elliptic Polarization. 233 
 
 days ago, to try how far they agreed with my formulae ; and the 
 agreement turns out to be so close, that I think myself justified 
 in publishing them. Besides, it will be curious hereafter to 
 compare them with more careful measurements. 
 
 Before we proceed, however, to the details of the experi- 
 ments, it may be well to give the formulae in a state fitted for 
 immediate application. The light incident on the metal being 
 polarized in a certain plane, let a denote the azimuth of this 
 plane, or the angle which it makes with the plane of incidence ; 
 and as the reflected light will be elliptically polarized, or, in 
 other words, will perform its vibrations in ellipses all similar 
 and equal to each other, as well as similarly placed, put for 
 the angle which either axis of any one of these ellipses makes 
 with the plane of incidence, and let ]3 be another angle, such 
 that its tangent may represent the ratio of one axis of the 
 ellipse to the other. Then when the optical constants M and x 
 (of which I suppose the first to be a number greater than unity, 
 and the second an angle less than 90) are known for the par- 
 ticular metal, the angles 9 and ]3 may be computed for any 
 value of a, at any given angle of incidence, by the following 
 formulas : 
 
 (i/ - v) sin 2a 
 
 tan 20 
 
 sin 
 
 2/+ (v' + v) cos2a' 
 
 2g sin 2a 
 v' + v < + 2/cos2a' 
 
 in which /and g are constant quantities given by the expres- 
 sions 
 
 - -^J cos x , ff = (^ f+ jj\ sin X> ( B ) 
 
 and v, i/ are quantities depending on the angle of incidence *, 
 in the following way. Let i' be an angle such that 
 
 sin i M 
 
 ~, = 9 ( C ) 
 
 sin \ cos 
 
234 Laws of Metallic Reflexion, and Mode of 
 and put 
 
 cos i 
 
 c^ 
 then will 
 
 The angles and |3 are given by immediate observation with 
 the instrument ; and from their values at any incidence, and 
 for any azimuth a of the plane of primitive polarization, we can 
 find the constants M and x> which we may afterwards use to 
 calculate the values of 6 and /3 for all other incidences and 
 azimuths, in order to compare them with the observed values. 
 It is indifferent, in the formulae, whether be referred to the 
 major or the minor axis of the elliptic vibration, as also whether 
 tan /3 be the ratio of the minor to the major axis, or the reci- 
 procal of that ratio ; but in what follows we shall suppose to 
 be the inclination of the plane of incidence to that axis, which, 
 when a is 45 or less, is always the major axis ; and ]3 shall be 
 supposed less than 45, in order that its tangent may represent 
 the ratio of the minor axis to the major. 
 
 When the azimuth a is equal to 45, the formulae (A) become 
 
 n/i \>' - v . ~~ 2g . 
 
 tan 20 = - , sm 2/3 = , ; (F) 
 
 2/ v + v 
 
 from which we may deduce the remarkable relation 
 
 tan 2/3 = g 
 cos 20 f 
 
 showing that, in the case supposed, the ratio of tan 2/3 to 
 cos 20 is independent of the angle of incidence. In the experi- 
 ments which I made with Mr. Grrubb this azimuth was always 
 45 ; and the following Table contains the results of observation 
 compared with those obtained by calculation from formulae (F). 
 The experiments were made upon a small disk of speculum 
 metal ; and in the calculations I have taken 
 
 M = 2-94, x - 64 25'. 
 
making Experiments upon Elliptic Polarization 235 
 
 
 Value of 0. Value of ft. 
 
 Angle of 
 
 
 
 Incidence. 
 
 
 
 
 
 
 Observed. 
 
 Calculated. 
 
 Observed. 
 
 Calculated. 
 
 65 
 
 27 55' 
 
 27 53' 
 
 28 0' 
 
 28 0' 
 
 70 
 
 15 41 
 
 15 44 
 
 33 7 
 
 33 1 
 
 75 
 
 - 8 46 
 
 - 9 16 
 
 34 10 
 
 34 6 
 
 80 
 
 _ 30 15 
 
 -29 25 
 
 27 
 
 26 53 
 
 84 
 
 - 37 22 
 
 -37 25 
 
 16 47 
 
 17 17 
 
 The light used in these observations was that of a candle 
 placed at a short distance, and was admitted through small 
 apertures at the ends of the tubes.* The Nicol's prism in the 
 first tube having been secured in a position in which its princi- 
 pal plane was inclined 45 to the plane of incidence, and the 
 two arms having been set at the proper angle with the surface 
 of the metal, the Fresnel's rhomb and the Nicol's prism in the 
 second tube were moved simultaneously, until the image of the 
 candle became as faint as possible. Had light perfectly homo- 
 geneous been employed, the image could have been made to 
 vanish altogether ; but instead of vanishing, it became highly 
 coloured ; and our rule in observing was to make the blue at 
 one side of it, and the red at the other, equally vivid, so as to get 
 results which should belong, as nearly as possible, to the mean 
 ray of the spectrum. When this was done, the angles and |3 
 (subject, however, to certain corrections which will be hereafter 
 explained) were respectively read off from the divided circles 
 belonging to the rhomb and the prism. The observations were 
 made at large incidences, because it is within the last thirty de- 
 grees of incidence that the phenomena go through their most 
 rapid changes. 
 
 If we now cast our eyes on the above Table, making due 
 allowance for the uncertainty arising from the dispersion of the 
 metal, we shall be struck with the agreement between the cal- 
 culated and observed numbers. The differences are greatest in 
 
 * See the description of the instrument in the Proceedings, VOL. i. p. 159 
 p. 138). 
 
2 3 6 Laws of Metallic Reflexion, and Mode of 
 
 the last two observations, which, however, were really the first ; 
 for the observations were made in the inverse order of the inci- 
 dences, and their accuracy may have improved as they went on. 
 However that may be, the differences are quite within the 
 limits of the errors of observation ; and they are actually less 
 than those which Fresnel found to exist between calculation 
 and experiment in the much simpler case of reflexion at the 
 surface of a transparent ordinary medium, when he proceeded 
 to verify the formula which he had discovered for computing 
 the effect of such reflexion.* 
 
 It may seem extraordinary that these experiments should 
 have been in my possession for nearly six years before I became 
 aware of their close agreement with my formulae ; but the fact 
 is, that I did not regard them with much interest, because, 
 from the circumstances in which they were made, I did not ex- 
 pect more than a general accordance with theory. And even 
 now I am in no haste to infer the absolute exactitude of the 
 formulae, though they are found to represent the phenomena 
 so well. It was far more allowable to infer that the formula oi 
 Fresnel was exact in the case just mentioned, though it appeared 
 to represent the phenomena less perfectly. For, to say nothing 
 of the small number of our experiments, the present is a much 
 more complicated case, and the phenomena depend on two con- 
 stants instead of one, so that the formulse might be slightly 
 altered, and yet perhaps continue to agree very well with rough 
 experiments. Where there is only one constant this is not so 
 probable. Again, there is one of the quantities in the preceding 
 formulas which may be greatly altered without producing more 
 than a slight effect on the values of 9 and ]3. This quantity is 
 the ratio of sin i to sin i', which, according to the value in for- 
 mula (c), is a number so large as to make the angle i' always 
 small, so that its cosine never differs much from unity; and 
 therefore if the above ratio were taken equal to any other large 
 number, the value of ju in formula (D) would remain nearly the 
 
 * See the Table which he has given in the Annales de Chimie, torn, xviii. 
 p. 3H. 
 
making Experiments upon Elliptic Polarization. 237 
 
 same, and consequently the values of 6 and |3 would be but 
 slightly changed. 
 
 It is with regard to the value of /u. as a function of the inci- 
 dence that I entertain the greatest doubts, and if any defect 
 shall be found in the formulae I think it will be here. The re- 
 lations (c) and (D), from which ju may be deduced in terms of ', 
 were not indeed adopted without strong reasons ; but I am not 
 entirely satisfied with them ; because, when we reverse the prob- 
 lem, and seek to determine the constants M and x from the ob- 
 served values of 6 and ft at a given incidence, the results are 
 rather complicated and involved, though the approximate deter- 
 mination is easy enough. As the formulae are in a great mea- 
 sure built upon conjecture, we must not be disposed to receive 
 them without the strongest experimental proofs ; and it will 
 certainly require experiments of no ordinary accuracy to decide 
 some of the questions which may be raised respecting them. 
 
 When plane-polarized light is incident on a metal, if its vi- 
 brations be resolved in directions parallel and perpendicular to 
 the plane of incidence, the effect of the reflexion is to change 
 unequally the phases of the resolved vibrations ; and it may be 
 useful to have the formulae which express the difference of 
 phase after reflexion, and the ratio of the amplitudes of vibra- 
 tion. Put <p for the difference of phase ; and supposing, for 
 simplicity, the incident light to be polarized in an azimuth of 
 45, let o- be an angle less than 45, such that tan a may represent 
 the ratio of the reflected amplitudes respectively perpendicular 
 and parallel to the plane of incidence ; then we shall have 
 
 tn d> = -r^, cos 2<r = -r ; (H) 
 
 v v v + v 
 
 from which we may infer that 
 
 sin tan 2<r = -, (i) 
 
 or that the product on the left side of the last equation is inde- 
 pendent of the angle of incidence. It is to be observed that the 
 
238 Laws of Metallic Reflexion, and Mode of 
 
 relations (G) and (i) are independent of the value of //, and may 
 hold good though that value should require to be changed. 
 
 All the preceding formulae are merely mathematical conse- 
 quences of those which I published long ago in the Transactions* 
 of the Academy. The formulas which I had previously given 
 in the Proceedings, f are slightly different, and, I think, less 
 likely to be exact, because they are less simple, and do not lead 
 to any of the remarkable relations which may be deduced from 
 the others. 
 
 Having had occasion, in the course of the few experiments 
 which I made with the instrument before mentioned, to study 
 the nature of Fresnel's rhomb, which constitutes an important 
 part of it, I shall here describe the method which must be fol- 
 lowed in order to obtain true results, when the rhomb is em- 
 ployed in observations on light elliptically polarized. A ray in 
 which the vibrations are supposed to be elliptical is given, and 
 what we want is to determine the ratio of the axes of the 
 elliptic vibration, and their directions with respect to a fixed 
 plane passing through the ray ; in other words, to determine 
 the angles which we have denoted by |3 and in the case of a 
 ray reflected from a metal. For this purpose the ray is ad- 
 mitted perpendicularly to the surface at one end of the rhomb, 
 and after having suffered two total reflexions within, passes out 
 perpendicularly to the surface at the other end. Then causing 
 the rhomb to revolve about the ray, we shall find two positions 
 of it in which the emergent light will be the plane-polarized, 
 these positions being readily indicated by a Nicol's prism inter- 
 posed between the rhomb and the eye ; for such a prism, by 
 being turned round the ray, can make the light totally disap- 
 pear when it is plane-polarized, but not otherwise. These two 
 positions of the rhomb will be exactly 90 from each other ; in 
 one of them the principal plane of the rhomb (the plane of re- 
 flexion within it) will be parallel to the major axis of the elliptic 
 vibration, and the angle which it makes with the plane of inci- 
 
 *VoL. xvni. p. 71 (supra-, p. 133). f VOL. i. p. 2 (supra, p. 58). 
 
making Experiments upon Elliptic Polarization. 239 
 
 dence on the metal will be equal to 9 : while in the same posi- 
 tion the angle which the principal plane makes with the plane 
 of polarization of the emergent ray (as given by the Nicol's 
 prism) will be equal to j3. In the other position, the principal 
 plane will be parallel to the minor axis of the elliptic vibration, 
 and the corresponding angles will be equal to 90 - 6 and 
 90 - j3 respectively. This, however, proceeds on the supposi- 
 tion that the rhomb is exact. When it is not so, which is of 
 course the proper supposition, and a very necessary one in the 
 experiments with which we are concerned, there will still be, 
 generally speaking, two positions of it in which the emergent 
 ray will be plane-polarized, or in which a disappearance of the 
 light may be produced by the Nicol's prism ; but these positions 
 will no longer be 90 from each other, nor will the principal 
 plane, in either of them, coincide with an axis of the elliptic 
 vibration. If we now measure the angles between the different 
 planes as before, and denote them by 0', /3' in the first position, 
 and by 90 - 0", 90 - )3" in the second, we shall find that 0' 
 and 0" are unequal, but we shall have j3 r equal to /3". The va- 
 lues of 6 and )3 will then be given by the formulae 
 
 0' + 0" cos 2/3' 
 
 The error of the rhomb may easily be found. Supposing 
 the vibrations to be resolved in directions parallel and perpen- 
 dicular to its principal plane, the rhomb is intended to produce 
 a difference of 90 between the phases of the resolved vibrations, 
 or to alter by that amount the difference of phase which may 
 already exist ; but the effect really produced is usually different 
 from 90, and this difference, which I call c, is the error of the 
 rhomb. The value of e is given by the formula 
 
 sin (0' - 0") 
 tan= tan2/3 ' 
 
 and as the error of the rhomb is a constant quantity, we have 
 
240 Laws of Metallic Reflexion, and Mode of 
 
 thus an equation of condition which must always subsist between 
 the angles 6' - 0" and |3. For any given rhomb the sine of the 
 first of these angles is proportional to the tangent of twice the 
 second, and therefore 0' - 0" constantly increases as )3 increases 
 towards 45, that is, as the axes of the elliptic vibration approach 
 to equality. When j3 is equal to 45 - $ e , we have 0' - 0" = 90 ; 
 and for values of j3 still nearer to 45, the value of sin (8 f - 0") 
 becomes greater than unity, that is to say, it becomes impos- 
 sible, by means of the rhomb, to reduce the light to the state of 
 plane-polarization. This is a case that may easily happen with 
 an ordinary rhomb in making experiments on the light re- 
 flected from metals ; because at a certain incidence, and for a 
 certain azimuth of the plane of primitive polarization, the re- 
 flected light will be circularly polarized. 
 
 The rhomb which I used in the experiments tabulated above 
 was made by Mr. Dollond, and was perhaps as accurate as 
 rhombs usually are ; it was cut at an angle of 54^, as pre- 
 scribed by Fresnel. Its error was about 3, and the value of 
 0' - 0", at the incidence of 75, was about 7. But in another 
 rhomb, also procured from Mr. Dollond, and cut at the same 
 angle, the value of 8' - 8", under the same circumstances, was 
 about 20, and the value of e was therefore about 8. The 
 angle given by Fresnel was calculated for glass of which the 
 refractive index is 1'51 ; and the errors of the rhombs are to be 
 attributed to differences in the refractive powers of the glass. I 
 was not at all prepared to expect errors so large as these when 
 I began to work with the rhomb, and they perplexed me a good 
 deal at first, until I found the means of taking them into ac- 
 count, and of making the rhomb itself serve to measure and to 
 eliminate them. The value of the rhomb as an instrument of 
 research is much increased by the circumstance that it can thus 
 determine its own effect, and that it is not at all necessary to 
 adapt its angle exactly to the refractive index of the glass. It 
 may also be remarked, that this circumstance affords a method 
 of directly and accurately testing the truth of the formulae 
 which. Fresnel has given for the case of total reflexion at the se- 
 
making Experiments upon Elliptic Polarization. 241 
 
 parating surface of two ordinary media ; for we have only to 
 measure the angle of the rhomb and the refractive index of the 
 glass, and to compute, by FresnePs formula, the alteration 
 which the rhomb ought to produce in the difference between the 
 phases of the resolved vibrations; which alteration of phase 
 we may then compare with that deduced, by means of the for- 
 mulae (K) and (L), from direct experiment. 
 
 If, in each position of the rhomb, we measure the angle 
 which the plane of polarization of the emergent ray makes with 
 the plane of incidence on the metal, and call the two angles re- 
 spectively y', y", we shall have 
 
 y^ff-p, 7 " = r + /3', (M) 
 
 and therefore 
 
 y' + y" = 0' + 0" = 29, 2/3' = y" - y' + 0' - 0" ; (N) 
 
 from which it appears that if the rhomb were perfectly exact, 
 that is, if 0' and 6" were equal to each other, the angle would 
 be half the sum of y', y", and the angle ]3 half their difference. 
 It would then be sufficient to measure the angles y' and y", in 
 order to get 6 and |3 accurately. And if the rhomb were erro- 
 neous, the true value of would still be half the sum of y', 
 y" ; but the true value of ]3 would not be discoverable without 
 measuring the angles 0', 0", by the help of which it can be de- 
 duced from the second of formulae (N), combined with the 
 second of formulae (K). Nor can we discover whether the 
 rhomb is erroneous or not, without measuring the angles 0', 
 0" ; and therefore as these angles must be measured in any case, 
 the former method of determining and |3 is to be preferred. 
 
 In making experiments on elliptically polarized light, a 
 plate of mica, or any other doubly-refracting crystal, placed per- 
 pendicular to the ray, may be used instead of Fresnel's rhomb. 
 If the thickness of the crystalline plate be such that the interval 
 between the two rays which emerge from it is equal to the 
 fourth part of the length of a wave, for light of a given refran- 
 gibility, the plate will, for such light, perform all the functions 
 
242 Laws of Metallic Reflexion, and Mode of 
 
 of the rhomb ; the principal plane of the rhomb being repre- 
 sented by the plane of polarization of one of the emergent rays. 
 But unless the light be perfectly homogeneous, this method is 
 liable to great inaccuracy in practice, since the effect of the plate 
 in producing or altering the difference of phase between the two 
 rays which interfere on their emergence from it is inversely 
 proportional to the length of a wave, and will therefore be ex- 
 tremely different for light of different colours, and will change 
 very perceptibly even within the limits of the same colour. It 
 is true, the effect of the rhomb always varies with the colour of 
 the light : but this variation is trifling compared with that which 
 exists in the other case. It was for this reason that I employed 
 the rhomb in my experiments, instead of a crystalline plate. 
 The apparatus, however, is much simplified by using such a 
 plate ; and if any one chooses to do so, and to work with homo- 
 geneous light, he must take care to f oUow, in every respect, the 
 directions which I have given for conducting experiments with 
 the rhomb. The two cases are precisely similar; and if it be 
 necessary not to neglect the errors of the rhomb, it is certainly 
 not less necessary to take into account those which may arise 
 from a want of accuracy in the thickness of the plate, consider- 
 ing how difficult it is to make the thickness correspond exactly 
 to the particular ray which we wish to observe. 
 
 I have been induced to enter into these particulars, respect- 
 ing the mode of making experiments on elliptic polarization, 
 because the subject is one which has not hitherto been studied; 
 nor does it seem to have occurred to anyone that any precaution 
 was requisite beyond that of getting the rhomb cut as nearly as 
 possible at the proper angle, or the crystalline plate made as nearly 
 as possible of the proper thickness. This, indeed, was quite suf- 
 ficient for ordinary purposes. For example, light polarized in a 
 plane inclined 45 to the principal plane of the rhomb or of the 
 plate, would, as far as the eye could judge, be circularly pola- 
 rized after passing through either of them. Notwithstanding a 
 certain error in the angle of the one, or in the thickness of the 
 other, such light would, when analyzed by a rhomboid of Iceland- 
 
making Experiments upon Elliptic Polarization. 243 
 
 spar, give two images always sensibly equal in intensity. But 
 an error which could not be at all detected in this way might 
 produce a very great effect in such experiments as those upon 
 the metals, and, for the purpose of comparison with theory, 
 might render them entirely useless, if in the first method of 
 observing we relied upon one set of observations, taking (sup- 
 pose) the values of & and |3' for the true values of and /3 ; or 
 if, in the second method, we contented ourselves with merely 
 measuring the angles y' and y". 
 
 The necessity of attending to the foregoing rules and re- 
 marks will appear from an examination of the experiments of 
 M. de Senarmont, published in the Annales de Chimie* In 
 these very elaborate experiments, which were made upon light 
 reflected at various incidences from steel and speculum metal, 
 the author followed a plan similar to that which I have adopted, 
 and which, in a general way, I had previously sketched in the 
 Proceedings f of the Academy. There was this difference, how- 
 ever, that he used a plate of mica instead of Fresnel's rhomb. 
 Now as he worked with common white light, the use of the 
 mica plate must have rendered two kinds of errors unavoidable. 
 In the first place, it would be impossible always to take the ob- 
 servations for the same ray of the spectrum ; and next, as a 
 consequence of this, the thickness of the plate would be gene- 
 rally inexact for the particular ray to which the observations 
 happened to correspond. If the thickness of the plate were 
 exact for a certain ray, it would be very sensibly inexact even 
 for the neighbouring parts of the spectrum ; and as the part of 
 the spectrum to which the observations belonged was continually 
 changing, the results obtained for different incidences and azi- 
 muths would not be comparable with each other, even though, 
 in each separate case, the error of the plate were allowed for 
 and eliminated. The values of 9, however, as determined by 
 M. de Senarmont, would be correct, so far as this error is con- 
 cerned ; those of j3 alone would be erroneous. For the values 
 of were determined in two ways : by measuring the angles 
 
 * Trnn. Ixxiii. iro. 351-358. t VOL. I. p. 159, 
 
244 Laws of Metallic Reflexion, and Mode of 
 
 0', 0", and taking their sum for 29 ; also by measuring the 
 angles 7', 7", and taking their sum for the same quantity. 
 Now each of these methods gives a true value of 0, because by 
 the preceding formulae we have 20 = 6' + 6" = 7' + 7" ; and 
 this accounts for the agreement, shown by the tables of M. de 
 Senarmont, between the values* of 20 obtained by these different 
 methods. But the values of /3 were deduced from the angles 
 7'," 7", by simply making their difference equal to 2/3 ; and we 
 see by the second of formulae (N) that, when the plate is not of 
 the proper thickness, this value of 2/3 is erroneous by the whole 
 amount of the angle 0' - 0", the difference between /3' and j3 
 being supposed so small that it may be neglected. As M. de 
 Senarmont proceeded on the common assumption that when the 
 thickness of the plate has been adjusted to that part of the 
 spectrum to which the observations are intended to refer, it 
 may afterwards, through the whole series of experiments, be re- 
 garded as exact, he necessarily conceived 0' and 6" to be the 
 same angle ; and it was on the principle of taking an average 
 between two measures of the same quantity that he made the 
 supposition 20 = / + 0", which happened to be correct. When 
 therefore he found 0' and 0" to be different, he of course looked 
 upon the difference as merely an error of observation, which it 
 would be superfluous to tabulate. Not having the values of 
 this difference, therefore, we have not the means of immediately 
 correcting the values of 2/3. But as observations were made 
 for several azimuths at each angle of incidence, we may use the 
 values of to determine those of /3 ; for when at any inci- 
 dence (except that of maximum polarization, where = for all 
 azimuths) the values of are known for two given values of a, 
 we can deduce the corresponding values of /3, without any other 
 theory than that of the composition of vibrations. The values 
 
 * Or rather the values of 180 + 26 ; because the angle w, the double of which 
 appears in the tables of M. de Senarmont, is equal to 90 + 0. The angles which 
 he calls 71 and 72 are equal to 90 + 7" and 90 + 7' respectively. It therefore 
 comes to the same thing, whether the one set of angles or the other is supposed to 
 be measured. The letter # has the same signification in both notations. 
 
making Experiments upon Elliptic Polarization. 245 
 
 of j3 so deduced must indeed be expected to be very inaccurate, 
 partly because of errors in the observed values of 0, partly be- 
 cause the observations in different azimuths do not answer to 
 the same ray of the spectrum ; but they will be accurate enough 
 to show the great amount of the error committed by neglecting 
 the difference 0'-0". For example, putting and j3 for the 
 values of 9 and ]3 when a = 45, M. de Senarmont gives, at the 
 incidence of 60 upon steel, 20 = 64 15' (taking the mean of 
 his two determinations), and for the azimuths 55, 30, 25, he 
 gives 20 equal to 88 5', 37 2', and 29 36' respectively. Com- 
 bining these values of 29 in succession with that of 20 , we get 
 for 2j3 the series of values 32 38', 33 28', 34 30'; the dif- 
 ferences between which are to be attributed to the causes above 
 stated. The mean value of 2j3 thus found is 33 32' ; while 
 its value, as given by M. de Senarmont, is only 28 41'. The 
 difference 4 51' is the value of 0' - 0", which, divided by the 
 tangent of 2|3 , gives 7 19' for the mean value of t, the error of 
 the mica plate corresponding to that part of the spectrum which 
 was observed at the incidence of 60. 
 
 At incidences nearer the angle of maximum polarization 
 the errors are probably much greater. Beyond that angle 
 they again diminish, and in some cases they almost vanish. 
 Thus, at the incidence of 85 upon steel, with the value of 20 
 and the value of 20 corresponding to a = 20, we get, by com 
 putation, a value of 2j3 , which differs only by a few minutes 
 from that given by M. de Senarmont. Nearly the same thing 
 happens at the same incidence when we take a = 25. In these 
 cases, therefore, the results belong to that particular ray for 
 which the thickness of the plate was exact. 
 
 The observations of M. de Senarmont on speculum metal 
 were not carried beyond the incidence of 60. He states that 
 he was unable to observe at higher incidences, on account of the 
 uncertainty arising from the dispersion of the metal ; but though 
 this cause operated in some degree, his embarrassment must 
 have been really occasioned by the increasing magnitude of the 
 difference 0' - 0", as he ^approached the angle of maximum po- 
 
246 Laws of Metallic Reflexion, and Mode of 
 
 larization ; that difference being perhaps twice as great as in 
 the case of steel. My own experiments on speculum metal were 
 all made, as has been seen, at incidences greater than 60. 
 
 The experiments of M. de Senarmont do not at all agree 
 with the formulae ; and therefore I have been obliged to analyze 
 his method of observation, and to show that it could not lead to 
 correct results. It is to be regretted that his method was de- 
 fective, as the zeal and assiduity which he has displayed in the 
 inquiry would otherwise have put us in possession of a large 
 collection of valuable data. 
 
 I shall conclude by saying a few words respecting the in- 
 tensity of the light reflected by metals. The formulae for com- 
 puting this intensity have been given in the Transactions of the 
 Academy, in the place already referred to ; but they may be 
 here stated in a form better suited for calculation. If we sup- 
 pose i// and i// to be two angles, such that 
 
 ootan i// = , cotan i// = M^ (o) 
 
 and then take two other angles w, a/, such that 
 
 cos o> = sin 2i// cos x> cos a/ = sin 2$' cos x> (*) 
 
 we shall have 
 
 T = tan iw, r = tan ^o/, (Q) 
 
 where T is the amplitude of the reflected rectilinear vibration, 
 when the incident light is polarized in the plane of incidence, 
 and T is the amplitude of the reflected vibration when the 
 incident light is polarized perpendicularly to that plane ; the 
 amplitude of the incident vibration being in each case supposed 
 to be unity. Hence when common light is incident, if its in- 
 tensity be taken for unity, the intensity I of the reflected light 
 will be given by the formula 
 
 I=\ (tan 2 u + tan 2 >') . (R) 
 
 If with the values of M and x determined by my experi- 
 
making Experiments upon Elliptic Polarization. 247 
 
 inents we compute, by the last formula, the intensity of re- 
 flexion for speculum metal at a perpendicular incidence, in 
 which case /j. = 1, we shall find I = *583. This is considerably 
 lower than the estimate of Sir William Herschel, who, in the 
 Philosophical Transactions for 1800 (p. 65), gives "673 as the 
 measure of the reflective power of his specula. The same num- 
 ber, very nearly, results from taking the mean of Mr. Potter's 
 observations.* It might seem therefore that the formula is in 
 fault ; but I am inclined to think that the metal which I em- 
 ployed had really a low reflective power. Its angle of maxi- 
 mum polarization was certainly much less than that of the specu- 
 lum metal used by Sir David Brewster,f who states the angle 
 to be 76, whereas in my experiments it was only about 73^- ; 
 and |any increase in this angle, by increasing the value of M, 
 raises the reflective power. On the other hand, the maximum 
 value of ]3 (when a = 45) was greater than that given by Sir 
 David Brewster, namely, 32 ; and any increase in |3 tends also 
 to increase the reflective power. Now it is not unreasonable to 
 suppose that the highest values of both angles may be most 
 nearly those which belong to the best specula ; and accordingly, 
 if we take 76 for the incidence of maximum polarization, and 
 retain the maximum value of ]3, namely, 34 37', which results 
 from my experiments, we shall get M = 3 '68, x = 66 16', and the 
 value of I at the perpendicular incidence will come out equal to 
 662, which scarcely differs from the number given by Herschel. 
 It is clear from what precedes that the optical constants are 
 different for different specimens of speculum metal, and this is 
 no more than we should expect, from the circumstance that the 
 metal is a compound, and therefore liable to vary in its optical 
 properties from variations in the proportion of its constituents ; 
 but I am disposed to believe that the same thing is generally 
 true, though of course in a less degree, of the simple metals ; 
 so that in order to render the comparison satisfactory, the 
 measures of intensity should always be made on the same spe- 
 
 * Edinburgh Journal of Science, New Series, VOL. in. p. 280. 
 t Philosophical Transactions, 1830, p. 324. 
 
248 Laws of Metallic Reflexion, &c. 
 
 cimen which has furnished the values of M and x- There is 
 one metal, however, with respect to which there can be no 
 doubt that the experiments of different observers are strictly 
 comparable, when it is pure, and at ordinary temperatures I 
 mean mercury. For this metal Sir David Brewster states the 
 angle of maximum polarization to be 78 27', and the maximum 
 value of /3, when a = 45, to be 35 ; from which I find M = 4*616, 
 X = 68 13', and, at the perpendicular incidence, J = *734. 
 Now Bouguer observed the quantity of light reflected by mer- 
 cury, but not at a perpendicular incidence. His measures were 
 taken at the incidences of 69 and 78, for the first of which 
 he gives, by two different observations, '637 and -666 ; for the 
 second, by two observations, *754 and *703, as the intensity of 
 reflexion.* If we make the computation from the formula, 
 with the above values of M and x> we fi n( i ^ ne quantities of 
 light reflected at these two incidences to be, as nearly as pos- 
 sible, equal to each other, and to seven-tenths of the incident 
 light, the intensity of reflexion being a minimum at an inter- 
 mediate incidence ; and if we suppose these quantities to be 
 really equal at the incidences observed by Bouguer, we must 
 take the mean of all his numbers, which is *69, as the most pro- 
 bable result of observation. This result differs but little from 
 one of the two numbers given by him at each incidence, and 
 scarcely at all from the result of calculation. 
 
 The angle at which the intensity of reflexion is a minimum, 
 when common light is incident, may be found from the formula 
 
 which gives the values of ju, and thence that of i. This inci- 
 dence for mercury is, by calculation, 75 15', and the minimum 
 value of /is '693, which is less than its value at a perpendicular 
 incidence by about one-eighteenth of the latter. According 
 to the formulae, the reflexion is always total at an incidence 
 of 90. 
 
 * See his Traite d'Optique sur la Gradation de la Lumiere : Paris, 1760 ; pp. 124, 
 126. 
 
249 
 
 XXII, -ON THE ATTEMPT LATELY MADE BY M. LAURENT 
 TO EXPLAIN, ON MECHANICAL PRINCIPLES, THE 
 PHENOMENON OF CIRCULAR POLARIZATION IN 
 LIQUIDS. 
 
 [Abstract of a Communication addressed to the BRITISH ASSOCIATION, 
 in the year 1843. Report, p. 7.] 
 
 THE Author showed that this attempt had not succeeded. M. 
 Laurent supposes the particles of the luminiferous ether not 
 to be simply material points, but to have dimensions which 
 are not insensible when compared with their distances; and on 
 this hypothesis he deduces a system of differential equations, 
 the integrals of which he conceives to represent the phenome- 
 non in question. The integrals given by M. Laurent are, 
 however, altogether erroneous, though this circumstance was 
 not noticed by M. Cauchy in the remarks and comments 
 which he made on M. Laurent's Memoir. The true integrals 
 of these equations (supposing the equations to be correctly 
 deduced) were shown by Professor Mac Cullagh to indicate 
 motions of the ether which do not correspond to the observec 
 phenomenon. The account of M. Laurent's theory, with M. 
 Cauchy's remarks upon it, will be found in the eighteenth 
 volume of the Comptes Eendus of the Academy of Science 
 of Paris. 
 
( 250 ) 
 
 XXIII. OK TOTAL BEFLEXION. 
 
 Proceedings of the Royal Irish Academy, VOL. in. p. 49. Read, January 13, 1845. 
 
 PROFESSOR MAC CULLAGH made a communication on the subject 
 of Total Eeflexion. 
 
 In the case of total reflexion the vibrations which take place 
 in the rarer medium are in general elliptical, and when this 
 medium is a crystal, the equations by which the ellipse of vi- 
 bration is determined are very complicated. The projection 
 of this ellipse upon the plane of incidence may, however, be 
 easily found by the remark in p. 12 of the present volume ; 
 the projecting cylinder is therefore known, and as the ellipse of 
 vibration is a section of this cylinder, the question of deter- 
 mining the ellipse is reduced to that of determining its plane. 
 For this purpose Mr. Mac Cullagh gave the following rule. 
 Having constructed the ellipsoid of indices (that whose axes 
 are parallel to the axes of elasticity, and inversely proportional 
 to the three principal velocities of propagation in the crystal), 
 let its two planes of circular section intersect the aforesaid 
 cylinder. The curves of intersection will be ellipses, which 
 shall be supposed to have a common centre in the axis of 
 the cylinder. Let OP, OP' be the greater semiaxes of these 
 ellipses, and OQ, OQf the less semiaxes; the lengths of the 
 two former being denoted by p, p', and the lengths of the two 
 latter by q, q'. Join the extremities P, P' of the greater 
 semiaxes, and the extremities Q, Q! of the less semiaxes; and 
 
On Total Reflexion. 251 
 
 divide each of the right lines PP, QQ', in the ratio of 
 ^p* _ <f to v// 2 - q'\ Then a plane drawn through the 
 centre and the two points of division will be the plane of 
 vibration. In the application of this rule some precautions 
 are to be observed, but they need not here be insisted on. 
 
 The foregoing rule was deduced (in the year 1843) from 
 the general equations by a peculiar use of imaginary quanti- 
 ties, after the author had several times tried in vain to obtain a 
 geometrical interpretation of those equations by considerations 
 of a more obvious and ordinary kind. This use of imaginaries 
 is founded on a remarkable theorem relative to the ellipse, by 
 which it appears, that the plane of an ellipse and its species 
 (that is, the directions and the ratio of its axes) ^ may be ex- m 
 pressed by two imaginary constants, just as the direction of a 
 right line in space is expressed by two real constants. By 
 means of this theorem which it is unnecessary to repeat, as 
 it has been published in the University Calendar* we may find 
 such properties of elliptical vibrations as are analogous to those 
 of rectilinear vibrations; and it was in this way that the above 
 rule was discovered. It is analogous (though it scarcely appears 
 so at first sight) to the rule by which, in the theory of Fresnel, 
 the direction of rectilinear vibrations is determined, when the 
 plane of the wave is given. 
 
 * "Examination Papers" of the year 1842, p. Ixxxiv. [The theorem is as 
 
 follows : 
 
 Given an ellipse in space, the origin of co-ordinates being taken at the centre. 
 Let the points *, y, , ^ /> *> be the extremities of conjugate diameter. Then 
 the imaginary quantities 
 
 y + 
 
 are 
 
 constant for every system of conjugate diameters.] 
 
PART II. 
 
 GEOMETRY. 
 
I. GEOMETEICAL THEOEEMS ON THE EECTIFICATION 
 OP THE CONIC SECTIONS. 
 
 [Transactions of the Royal Irish Academy, VOL. XTI. p. 79. Read, June 21, 1830.] 
 
 Lemma 1. Let T and t be two points indefinitely near each 
 other on any given curve AT, and let tangents at Tand t meet 
 in the points P and p any 
 other given line MN, 
 straight or curved, and 
 draw Pq perpendicular to 
 tp\ then the difference 
 between the arc AT and 
 the tangent TP will ex- 
 ceed or fall short of the Fig> L 
 difference between the arc At and the tangent tp by a quantity 
 which is ultimately to pq in a ratio of equality. 
 
 For the increment of TP, or the difference of TP and tp, is 
 ultimately equal to the sum of pq, VT, and Vt ( V being the 
 intersection of pt and PT produced) ; and Tt, or the increment 
 of the arc AT, is ultimately equal to the sum of FT and Vt', 
 therefore the difference of the increments is ultimately equal to 
 pq. Whence the proposition is manifest. 
 
 PROBLEM. To find the Length of the Arc of a Parabola. 
 
 Let F be the focus and A the vertex of a parabola AT: 
 draw AK perpendicular to AF, and let a tangent at the point 
 T meet it in P; take p indefinitely near to P, and let Fp 
 
256 
 
 Geometrical- Theorems on the 
 
 intersect PT in q : then since FP and Fp are perpendicular to 
 the tangents PT and^tf, it follows (by the preceding lemma) that 
 Pq is ultimately equal to the increment 
 "of the difference between the arc AT 
 and the tangent TP. With the centre 
 F and semiaxis FA describe the equi- 
 lateral hyperbola GAH, and bisect 
 the angles AFP, AFp by the straight 
 lines FL, Fl. Then (since the square 
 of the radius vector of an equilateral 
 hyperbola is inversely as the cosine of 
 twice the angle which it makes with 
 the axis) the square of LF will be 
 equal to AF x PF; and because the 
 angle LFl is half the angle PFq, there- 
 
 Fig. 2. 
 
 fore the area LFl will be equal to one-fourth of the rectangle 
 under AF and Pq. 
 
 Hence the hyperbolic sector AFL is equal to one-fourth of 
 the rectangle under AF and the difference between the para- 
 bolic arc AT and the tangent TP. 
 
 Lemma 2. If on either axis of an ellipse a semicircle be 
 described, of which CD and CH 
 are two radii at right angles to 
 each other, and if DN and HM 
 be drawn perpendicular to the 
 axis aA, and meeting the ellipse 
 in E and L ; then CE and CL 
 will be conjugate semidiameters. 
 
 For tangents at H and L will meet in a point T in Aa pro- 
 duced ; the triangles HM. T and DNC will be similar, and DN 
 and HM will be similarly divided in E and L] therefore CE 
 and TL will be parallel, and consequently CE and CL will be 
 conjugate semidiameters. 
 
 Lemma 3. Take in the ellipse a point I indefinitely near to 
 L, and draw through it mh parallel to MH\ join (7A, and with 
 the centre C and a radius equal to CE describe the arc Kk 
 
 Fig. 3. 
 
Rectification of the Conic Sections. 
 
 257 
 
 meeting OH and Ch in K and k : then Kk will be ultimately 
 equal to LI 
 
 For LI: Hh : : LT: HT:: CE : CD : : CK : CH : : Kk : Eh. 
 Therefore LI = Kk. 
 
 Lemma 4. If the semiaxes A C and BC of an ellipse be equal 
 to the sum and difference of the sides PQ, QR, of a triangle PQJR, 
 and if the angle BCD be 
 
 equal to half the contained / ~ \ s 
 
 angle PQR (ADa being the 
 semicircle on Aa) ; then, 
 DEN being drawn perpen- 
 dicular to Aa, CE will be 
 
 Fig. 4. 
 
 equal to the base PR. Take Q^and QS equal to QR-, then 
 PS and PU will be equal to AC and (7J5, and the angle CDN 
 to T&P ; therefore drawing PT perpendicular to T8, DN and 
 NC will be equal to TS and TJP. But Z72&S being a right angle, 
 7.ft is parallel to PT, and therefore TS : TR : : PS : PU : : AC 
 :BC \iDN\EN\ but TS = DN, therefore TR = EN', and 
 since PT = CN, it foUows that PR = CE. 
 
 THEOREM. 
 
 Let ^Tand AT' be an ellipse and hyperbola, the semiaxis 
 (CA or C'A) of either being equal 
 to (C'F' or CF) the distance between 
 the focus and centre of the other; and 
 let tangents at the points T and T f 
 meet in P and P f the circles described 
 on the axes, so that FP = FP\ let Fig. 5. 
 
 also a" B" A" be another ellipse whose semiaxes (A" C" and 
 B" C") are equal to ^and .E4, 
 and take in its circumference a 
 point L so that the semidiameter 
 conjugate to that passing through 
 L may be equal to FP or F'P' ; 
 then will the excess of the ellip- 
 tic arc A T above its tangent TP be greater than the excess of 
 
258 
 
 Geometrical Theorems on the 
 
 the hyperbolic arc A'T' above its tangent T'P', by twice the 
 elliptic arc A"L. 
 
 Take the point L so that when the ordinate MLH is drawn 
 to meet in H the semicircle described on the axis, the angle 
 HC"M may be equal to 
 half the angle PCF (or 
 P'C'F', for the triangles 
 PCF and P'C'F' have 
 all their sides and angles 
 equal); draw C"D per- 
 pendicular to C"H, and 
 DE to A"a", meeting the Fig. 7. 
 
 ellipse in E ; then, by km. 2, C"E will be conjugate to C"L, 
 and by km. 4 it will be equal to FP, since the angle B"C"D is 
 equal to HC"M, and is therefore half of PCF, whilst the semi- 
 axes C"A", C"B", are the sum and difference of PC and CF. 
 Hence the point L thus found is that required by the enuncia- 
 tion. Take p, p f , h, indefinitely near to P, P', H, and similarly 
 related to each other ; let Fp and Fp f intersect TP and T'P' in 
 q and q', and with the centre C" and a radius equal to C"E de- 
 scribe the evanescent arc Kk. Then .FP, F'P' are always per- 
 pendicular to TP, T'P', and therefore Pq is ultimately the 
 increment of the difference between the arc AT and the tangent 
 TP (km. 1.) and P'<?' the increment of the difference between 
 the arc A'T' and the tangent T'P'; also by km. 3..2&C is ulti- 
 mately equal to LI, the increment of the arc A"L. Now the 
 angles CFP and CFp are equal to C'P'F' and C'tfF \ therefore 
 PFp is equal to the sum of P'F'p' and P'C'p, or to P'Py with 
 twice HC"h : but FP, F'P', and 0"JTare aU equal, and there- 
 fore Pq is equal to P'q' and twice Kk, or to PY and twice LI. 
 Whence the proposition is manifest. 
 
 Schol The angle B"C"H is half the angle fCP oifC'P', 
 and therefore by km. 4, the semidiameter C"L is equal to the 
 straight line fP or/'P 7 . This gives another and easier method 
 of finding the point L. 
 
 Hence every arc of a hyperbola may be found by means of 
 
Rectification of the Conic Sections. 259 
 
 two elliptic arcs. This beautiful theorem was discovered by 
 Landen, and is now for the first time demonstrated geometri- 
 cally ; but the manner in which it is stated by him and by suc- 
 ceeding writers differs from the above, and is much more 
 complicated. The different forms under which they have pre- 
 sented it may be easily deduced from the preceding, by means 
 of the theorem of Fagnani, of which a geometrical demonstra- 
 tion was first given in vol. ix. of the Transactions of the Royal 
 Irish Academy, by the Lord Bishop of Cloyne. 
 
 23, 1829. 
 
 UNIVERSITY 
 
( 260 ) 
 
 II. ON THE SURFACES OF THE SECOND ORDER. 
 
 [Proceedings of the Royal Irish Academy, VOL. n. p. 446. Bead Nov. 30, 1843.] 
 
 THERE is hardly any geometrical theory which more requires to 
 be studied, or which promises to reward better whatever thought 
 may be bestowed upon it, than that of the surfaces of the second 
 order. My attention was drawn to it, many years ago, by the 
 consideration of mechanical and physical questions. In the 
 dynamical problem of the Eotation of a Solid body, and in the 
 investigation of the properties of the Wave-Surface of Fresnel, 
 I found, so long since as the year 1829, that the ellipsoid could 
 be employed with very great advantage ; while the discussion of 
 these questions, but especially of the former,* suggested proper- 
 ties of the ellipsoid and its kindred surfaces which I might not 
 otherwise have perceived. In this manner I was led to consider 
 systems of confocal surfaces, and thence to notice the focal 
 curves, which I discovered to be analogous, in the theory of the 
 surfaces of the second order, to the foci in that of the plane 
 conic sections. That theory now began to interest me on its 
 own account, and, guided by analogy, I struck out the leading 
 properties possessed by the surfaces in relation to their focal 
 curves ; but the interference of other matters prevented me from 
 continuing the inquiry. I had done enough, however, in this 
 
 * The Theory of Rotation, here spoken of, was completed in the year 1831 ; but, 
 from causes which need not be mentioned at present, it was not published. The 
 investigations relative to Fresnel' s Wave -Surf ace will be found in the Transactions 
 of the Royal Irish Academy, VOL. xvi. p. 65 ; VOL. xvn. p. 241. See also VOL. 
 xxi. p. 32, of the same Transactions. 
 
On the Surfaces of the Second Order. 261 
 
 and other parts of the theory, to open new views respecting it ; 
 and the results at which I had arrived seemed so fitted for in- 
 struction, that, when I was appointed Professor of Mathematics 
 in the University, I made them the subject of the first lectures 
 which I gave in that capacity, in the beginning of the year 1836. 
 Next year the heads of these lectures were communicated to this 
 Academy, in a Paper of which a very short abstract appeared 
 in the Proceedings* The subject soon became a favourite one 
 among the more advanced students in the University, who are, 
 for the most part, excellent geometers, and in the present Article 
 very little will be found which is not well known amongst them ; 
 very little, indeed, which was not communicated to the Academy 
 on the occasion just mentioned, or which may not be gathered, 
 in the shape of detached questions, out of the Examination 
 Papers published in the University Calendar. But as nothing 
 has yet been published on the subject in a connected form, 
 except the brief notice in the Proceedings of the Academy, and 
 as mathematicians in other countries attach some importance to 
 researches of this kind, and appear to be in quest of certain 
 principles which are familiar to us here, it seems proper to col- 
 lect together the chief results that have already been obtained, 
 in order that persons wishing to pursue these speculations may 
 be better able to judge where their inquiries should begin, and 
 in what direction further progress is most likely to be made. 
 
 PART I. GENERATION OF SURFACES OF THE SECOND ORDER. 
 
 1. The different species of surfaces of the second order are 
 obtained, as is usually shown in elementary treatises, by the 
 discussion of -the general equation of the second degree among 
 three co-ordinates ; but it is necessary that we should also be 
 able to derive these surfaces from a common geometrical origin, 
 if we would bring them completely within the grasp of geo- 
 metry. Now as the different conic sections may (with the 
 
 * Proceedings of the Royal Irish. Academy, VOL. i. p. 89. 
 
262 On the Surfaces of the Second Order. 
 
 exception of the circle), be described in piano by the motion of a 
 point whose distance from a given point bears a constant ratio 
 to its distance from a given right line,* it is natural to suppose 
 that there must be some analogous method by which the sur- 
 faces of the second order may be generated in space. Accord- 
 ingly I have sought for such a method, and I have found that 
 (with certain analogous exceptions) every surface of the second 
 order may be regarded as the locus of a point whose distance 
 from a given point bears a constant ratio to its distance from a 
 given right line, provided the latter distance be measured pa- 
 rallel to a given plane ; this plane being, in general, oblique to 
 the right line. The given point I call, from analogy, a focus, 
 and the given right line a directrix; the given plane may be 
 called a directive plane, and the constant ratio may be termed 
 the modulus. 
 
 To find the equation of the surface so defined, let the axis 
 of z be parallel to the directrix ; let the plane of xy pass through 
 the focus, and cut the directrix perpendicularly in A, the co- 
 ordinates being rectangular, and their origin arbitrarily assumed 
 in that plane ; and let the axis of y be parallel to the intersec- 
 tion of the plane xy with the directive plane, the angle between 
 the two planes being denoted by 0. Then if we put x^ y l for 
 the co-ordinates of the focus, and # 2 , y* for those of the point A, 
 while the co-ordinates of a point 8 upon the surface are denoted 
 by %, y, z, the distance of this last point from the focus will be 
 the square root of the quantity 
 
 and if a plane drawn through S, parallel to the directive plane, 
 be conceived to cut the directrix in D, the distance SD will be 
 the square root of the quantity 
 
 (x - x*Y sec 2 ^ + (y - ytf ; 
 
 * This method of describing the conic sections is due to the Greek geometers. 
 It is given by Pappus at the end of the Seventh Book of his Mathematical Collec- 
 tions. 
 
On the Surfaces of the Second Order. 263 
 
 so that, m being the modulus, the locus of the point S will be a 
 surface of the second order, represented by the equation 
 
 (x - xtf + (y - 2/0 2 + * 2 = w 3 [v - x*J sooty + (y - y 2 ) 2 } , (1) 
 which, by making 
 
 A = I - m z sec 2 ^, B = 1 - m 2 , 
 
 G = m*x* sec 2 ^ - x, y H = m^ - y l9 (2) 
 
 K - m z (x* sec 2 + / 2 2 ) - x? - y?, 
 
 may be put under the form 
 
 Ax* + By* + s 2 + 2Gv +. 2JTy = JT, (3) 
 
 showing that the plane of xy is one of the principal planes of 
 the surface, and that the planes of xz and yz are parallel to 
 principal planes. 
 
 Before we proceed to discuss this equation it may be well to 
 observe, that as it remains the same when is changed into - 0, 
 or into 180 - 0, the directive plane may have two positions 
 equally inclined to the plane of xy, and therefore equally in- 
 clined to the directrix. Indeed it is obvious that, if through 
 the point 8 we draw two planes making equal angles with the 
 directrix, and cutting it in the points D and H respectively, the 
 distances &D and SD' will be equal. Every surface described in 
 this way has consequently two directive planes ; and as each of 
 these planes is parallel to the axis of /, their intersection is 
 always parallel to one of the axes of the surface. This axis 
 may therefore be called the directive axis. The directive planes 
 have a remarkable relation to the surface, as may be shown 
 in the following manner : 
 
 Suppose a section of the surface to be made by a plane which 
 is parallel to one of the directive planes, and which cuts the 
 directrix in D ; then the distance of any point S of the section 
 from the focus F will have a constant ratio to its distance SD 
 
264 On the Surfaces of the Second Order. 
 
 from the point D ; and, as the locus of a point whose distances 
 from the two points .Fand D are in a constant ratio to each 
 other is a plane or a sphere, according as the ratio is one of 
 equality or not, it follows that the section aforesaid will be a 
 right line in the one case, and a circle in the other. Hence 
 it appears that all directive sections, that is, all sections made in 
 the surface by planes parallel to either of the directive planes, 
 are right lines when the modulus is unity, and circles when the 
 modulus is different from unity. 
 
 Since the equation (3) is not altered by changing the sign of 
 0, or by changing $ into its supplement, we may suppose this 
 angle (when it is not zero) to be always positive and less than 
 90 ; for the supposition = 90 is to be excluded, as it would 
 make the secant of $ infinite, and the directive planes parallel 
 to the directrix. In the discussion of the equation there are two 
 leading cases to be considered, answering to two classes of sur- 
 faces. The first case, when neither A nor B vanishes, gives the 
 ellipsoid, the two hyperboloids, and the cone ; the second, when 
 either or each of these quantities is zero, includes the two para- 
 boloids and the different kinds of cylinders. 
 
 2. First Class of Surfaces. When neither A nor B vanishes, 
 we may make both G and H vanish, by properly assuming the 
 origin of co-ordinates. Supposing this done, we have 
 
 sec 2 ^, yi = nfyz, . (4) 
 
 the equation of the surface being then 
 
 Ax 2 + By* + z* = K, (5) 
 
 in which the axes of co-ordinates are of course the axes of the 
 surface. When K is not zero, the surface is an ellipsoid or hy- 
 perboloid, having its centre at the origin of co-ordinates ; when 
 K = 0, the surface is a cone having its vertex at the origin. 
 
 Eliminating # 2 , y* from the value of K y by means of the re- 
 lations (4), we get 
 
On the Surfaces of the Second Order. 265 
 
 and eliminating #1, y in like manner, we get 
 
 K=A (1 -A)^ + B(l-B) ^ ; (7) 
 
 from which expressions it appears that, everything else remain- 
 ing, the focus and directrix may be changed without changing 
 the surface described. For in order that the surface may re- 
 main unchanged, it is only necessary that K should remain con- 
 stant, since A and B are supposed constant. This condition 
 being fulfilled, the focus may be any point F whose co-ordinates 
 #1, yi satisfy the equation (6), and A (the foot of the directrix) 
 may be any point whose co-ordinates a? 2 , y* satisfy the equation 
 (7) ; it being understood, however, that when one of these points 
 is chosen, the other is determined. The locus of F (supposing 
 K not to vanish) is therefore an ellipse or a hyperbola,* which 
 may be called the focal curve , or the focal line; and the locus of 
 A is another ellipse or hyperbola, which may be called the diri- 
 gent curve or line : the centre of each curve is the centre of the 
 surface, and its axes coincide with the axes of the surface which 
 lie in the plane of xy. Moreover, as the quantities 1 - A and 
 1 - B are essentially positive, the two curves are always of the 
 same kind, that is, both ellipses, or both hyperbolas ; and when 
 they are hyperbolas, their real axes have the same direction. 
 The directrix, remaining always parallel to the axis of *, de- 
 scribes a cylinder which may be called the dirigent cylinder. 
 
 Since, by the relations (4), the corresponding co-ordinates 
 of F and A have always the same sign, these points either lie 
 within the same right angle made by the axes of x and y, or lie 
 on the same axis, at the same side of the centre. And as these 
 relations give 
 
 it is easy to see that the right line AF is a normal to the focal 
 
 * In the Proceedings of the Academy, VOL. i. p. 90, it was stated inadvertently 
 that " if we confine ourselves to the central surfaces, the locus of the foci will be 
 an ellipse." 
 
266 On the Surfaces of the Second Order. 
 
 curve ; for the quantities x z - x and y z - y l are proportional to 
 the cosines of the angles which that right line makes with the 
 axes of x and y respectively, while the values just given for 
 these quantities are, in virtue of the equation (6), proportional 
 to. the cosines of the angles which the normal to the focal curve 
 at the point F makes with the same axes. 
 
 It may also be shown, that if the directrix prolonged through 
 A intersects a directive plane in a certain point, and if a right 
 line drawn through F, parallel to the directrix, intersect the 
 same plane in another point, the right line joining those points 
 will be a normal to the curve described in that plane by the first 
 point. 
 
 3. To find in what way the focal and diligent curves are 
 connected with the surface, let the equations (5), (6), (7) (when 
 K does not vanish), be put under the forms 
 
 so that the quantities P, Q, R may represent the squares of the 
 semiaxes of the surface, and Pi, Qi, P 2 , Qz the squares of the 
 semiaxes of the curves, these quantities being positive or ne- 
 gative, according as the corresponding semiaxes are real or 
 imaginary. Then we have 
 
 (1-B), (11) 
 
 whence it follows that 
 
 P! P 2 = P 2 , Qi Q 2 = Q 2 , (12) 
 
 and also that 
 
On the Surfaces of the Second Order. 267 
 
 From equations (12) we see that PI and P 2 have always the same 
 sign, as also Qi and Q 2 ; and that, neglecting signs, the semi- 
 axes of the surface are mean proportionals between the corre- 
 sponding semiaxes of the focal and dirigent curves. These curves 
 are therefore reciprocal polars with respect to the section made 
 in the surface by the plane of xy ; and it would be easy to show 
 that the points F and A are reciprocal points, or that a tangent 
 applied at one of them to the curve which is its locus has the 
 other for its pole. 
 
 The focal curve, when we know in which of the principal 
 planes it lies, is determined by the conditions (13) ; and as it 
 depends . on the relative magnitudes of the quantities P, Q, It, 
 it will be convenient to distinguish the axes of the surface, with 
 relation to these magnitudes. Supposing, therefore, the quan- 
 tities P, Q, R to be taken with their proper signs, as they are 
 in the equation (9), that axis to which the greatest of them 
 (which is always positive) refers shall be called the primary 
 axis ; and that to which the quantity algebraically least has re- 
 ference shall be termed the secondary axis ; while the quantity 
 which has an intermediate algebraic value shall mark the middle 
 or mean axis. Then, since Pi and Qi will be negative, if R be 
 the greatest of the quantities aforesaid, the focal curve cannot 
 lie in the plane of the mean and secondary axes. Its plane 
 must therefore pass through the primary axis: it will be the 
 plane of the primary and mean axes, if R be the least of the 
 three quantities ; but the plane of the primary and secondary 
 axes, if R be the intermediate quantity. In the former case the 
 curve will be an ellipse, in the latter a hyperbola ; and we shall 
 extend the name of focal curves to both the curves so deter- 
 mined, though it may happen that only one of them can be 
 used in the generation of the surface by the modular method, as 
 the method of which we are treating may be called, from its 
 employment of the modulus. A focal curve which can be so 
 used shall be distinguished as a modular focal; but each focal, 
 whether modular or not, shall be supposed to have a dirigent 
 curve and a dirigent cylinder connected with it by the relations 
 
268 On the Surfaces of the Second Order. 
 
 Since P l - Qj = P - Q, the foci of a focal curve are the same 
 as those of the principal section in the plane of which it lies, 
 and they are therefore on the primary axis of the surface. It 
 will sometimes contribute to brevity of expression, if we also 
 give the name of primary to the major axis of an ellipse and 
 to the real axis of a hyperbola. "We may then say that the 
 primary axes of the surface and of its two focal curves are 
 coincident in direction ; and that (as is evident) the foci of 
 either curve are the extremities of the primary axis of the 
 other. 
 
 If K be supposed to approach gradually to zero, while A and 
 B remain constant, the focal and dirigent ellipses will gradually 
 contract, and the focal and dirigent hyperbolas will approach 
 to their asymptotes, which remain fixed. When K actually 
 vanishes, the surface becomes a cone ; the two ellipses are each 
 reduced to a point coinciding with the vertex of the cone, and 
 each hyperbola is reduced to the pair of right lines which were 
 previously the asymptotes. The dirigent cylinder, in the one 
 case, is narrowed into a right line ; in the other case it is con- 
 verted into a pair of planes, which we may call the dirigent 
 planes of the curve. 
 
 4. "We have now to show how the different kinds of sur- 
 faces belonging to the first class are produced, according to the 
 different values of the modulus and other constants concerned 
 in their generation. 
 
 I. When m is less than cos 0, the quantities A,B,K,P, Q, R 
 are all positive, and Q is intermediate in value between P and 
 JR. The surface is therefore an ellipsoid, and its mean axis is 
 the directive. As the quantities 1 - A and 1 - B are always 
 positive, the focal and dirigent curves are ellipses. 
 
 Here we cannot suppose K to vanish, as the surface would 
 then be reduced to a point. 
 
 When $ = 0, that is, when the directive planes coincide with 
 each other, and therefore with a plane perpendicular to the 
 directrix, so that SD is the shortest distance of the point S from 
 the directrix, the surface is a spheroid produced by the revo- 
 
On the Surfaces of the Second Order. 269 
 
 lution of an ellipse round its minor axis, and the focal and diri- 
 gent curves are circles. 
 
 II. "When m is greater than unity; A and B are negative ; 
 and if K be finite, it is also negative ; whence P and Q are po- 
 sitive, and K is negative. Also, supposing $ not to vanish, Q 
 is greater than P. The surface is therefore a hyperboloid of 
 one sheet, with its real axes in the plane of xy ; and the direc- 
 tive axis is the primary. The focal and dirigent curves are 
 ellipses. But when = 0, the surface is that produced by the 
 revolution of a hyperbola round its imaginary axis, and the 
 focal and dirigent* are circles. 
 
 If K = 0, which implies, since A and B have the same sign, 
 that a?!, i/i, o? 2 , 2/2 are each zero, the surface is a cone having the 
 axis of z for its internal axis ; and the focal and dirigent are 
 each reduced to a point. The focus and directrix are conse- 
 quently unique ; the focus can only be the vertex of the cone ; 
 the directrix can only be the internal axis ; and the directrix 
 therefore passes through the focus. The directive axis, which 
 coincides with the axis of y, is one of the external axes ; that 
 one, namely, which is parallel to the greater axes of the elliptic 
 sections* made in the cone by planes perpendicular to its internal 
 axis. This is on the supposition that is finite ; for, when 
 = 0, the cone becomes one of revolution round the axis of *. 
 
 III. When m is greater than cos 0, but less than unity, we 
 have A positive and B negative, and the species of the surface 
 depends on K. It is inconsistent with these conditions to sup- 
 pose = 0, and therefore the surface cannot, in this case, be one 
 of revolution. The value of K may be supposed to be given 
 by the formula 
 
 K= L^ (^ - -,.,). + i^? (y z - ytf, (14) 
 
 which contains only the relative co-ordinates of the focus and 
 the foot of the directrix, and is a consequence of the equations 
 (6) and (7). 
 
 * When the term dirigent stands alone, it is understood to mean a dirigent line. 
 
270 On the Surfaces of the Second Order. 
 
 1. If K is a positive quantity, the surface is a hyperboloid 
 of one sheet, with its secondary axis in the direction of x ; the 
 primary axis, as before, in the directive, but the focal and diri- 
 gent are now hyperbolas. 
 
 2. If K is a negative quantity, the surface is a hyperboloid 
 of two sheets, having its primary axis coincident with that of oc. 
 The secondary axis is the directive ; the focal and dirigent are 
 hyperbolas. 
 
 3-. If K = 0, the surface is a cone, having the axis of x for 
 its internal axis ; the directive axis being, as before, that exter- 
 nal axis to which the greater axes of the elliptic sections, made 
 by planes perpendicular to the internal axis, are parallel. The 
 axis of & is the other external axis, which may be called the 
 mean axis of the cone, because it coincides with the mean axis of 
 any hyperboloid to which the cone is asymptotic. As A and B 
 have different signs, it is evident, from the equations (6) and 
 (7), that the focal and dirigent are each a pair of right lines 
 passing through the vertex, each pair making equal angles with 
 the internal axis. Two planes, each of which is drawn through 
 the mean axis and a dirigent line, are the dirigent planes of 
 the cone. 
 
 The corresponding focal and dirigent lines are those which 
 lie within the same right angle made by the internal and direc- 
 tive axes ; and since by the equations (6) and (8) the value of K 
 may be written 
 
 K = x l (x z - Xi] + y l (y z - yj, 
 
 we see that, as K now vanishes, the right line joining corre- 
 sponding points F and A upon these lines is perpendicular to 
 the focal line. Of the two sides of the cone which are in the 
 plane xy, one lies between each focal and its dirigent ; and 'it 
 may be inferred from the equations, that the tangents of the 
 angles which the internal axis makes with a focal line, with one 
 of these sides of the cone, and with a dirigent line, are in con- 
 tinued proportion, the proportion being that of the cosine of 
 to unity. And hence it follows, that these two sides of the cone, 
 
On the Surfaces of the Second Order. 271 
 
 with a focal line and its dirigent, eu harmonically any right 
 line which crosses them. 
 
 5. From this discussion it appears, that the ellipsoid and 
 the hyperboloid of two sheets can be generated modularly, each 
 in one way only, the modular focal being the ellipse for the 
 former, and the hyperbola for the latter ; but that the hyperbo- 
 loid of one sheet can be generated in two ways, each of its focals 
 being modular, and each focal having its proper modulus. The 
 cone also admits two modes of generation,* in one of which, 
 however, the focus is limited to the vertex of the cone, and the 
 directrix to its internal axis. But when the hyperboloid of one 
 sheet, or the cone, is a surface of revolution, it has only one 
 mode of modular generation. In cases of double generation, 
 the directive planes of course remain the same, as they have a 
 fixed relation to the surface. A modular focal, it may be ob- 
 served (and the remark applies equally to surfaces of the second 
 class), is distinguished by the circumstance that it does not in- 
 tersect the surface. The only exception to this rule are the 
 focal lines of the cone; which pass through its vertex. A focal 
 which is not modular may be called umbilicar, because it inter- 
 sects the surface in the umbilics ; an umbilic being a point on 
 the surface where the tangent plane is parallel to a directive 
 plane. Thus the focal hyperbola of the ellipsoid, and the focal 
 ellipse of the hyperboloid of two sheets, are umbilicar focals, 
 and pass through the umbilics of these surfaces ; but the hyper- 
 
 * The double generation of the cone, when its vertex is the focus, may be proved 
 synthetically by the method indicated in the Examination Papers of the year 1838, 
 p. xlvi. (published in the University Calendar for 1839). Supposing the cone to 
 stand on a circular base (one of its directive sections), and to be circumscribed by a 
 sphere, the right lines joining its vertex with the two points where a diameter per- 
 pendicular to the plane of the base intersects the sphere, will be its internal and 
 mean axes. Then if P be either of these points, Fthe vertex, C the point where 
 the axis PFcuts the plane of the base, and B any point in the circumference of 
 the base, the triangles PVB and PB C will be similar, since the angles at Fand 
 are equal, and the angle at P is common to both triangles ; therefore V will be to 
 JBO as PV to PB, that is, in a constant ratio. It is not difficult to complete the 
 demonstration, when the focus is supposed to be any point on one of the focal 
 lines. 
 
272 On the Surfaces of the Second Order. 
 
 boloid of one sheet has no umbilics, and accordingly both its 
 focals are modular, and neither of them intersects the surface. 
 The umbilicar focals and dirigents have properties which shall 
 be mentioned hereafter. 
 
 An umbilicar focal and the principal section whose plane 
 coincides with that of the focal are curves of different kinds, 
 the one being an ellipse when the other is a hyperbola ; but a 
 modular focal is always of the same kind with the coincident 
 section of the surface, being an ellipse, a hyperbola, or a pair of 
 right lines, according as the section is an ellipse, a hyperbola, 
 or a pair of right lines ; and when the section is reduced to a 
 point, so likewise is the modular focal. 
 
 The plane of a modular focal always passes through the 
 directive axis. When the directive axis is the primary, as in 
 the hyperboloid of one sheet, both focals are modular. But in 
 the ellipsoid and the hyperboloid of two sheets, where the pri- 
 mary axis is not directive, only one of the focals can be modular. 
 The plane of an umbilicar focal is always perpendicular to the 
 directive axis ; and therefore, when that axis is the primary, 
 there is no umbilicar focal.* 
 
 When the surface is doubly modular, the two moduli m, m' 
 aje connected by the relation 
 
 *,l. (16) 
 
 * 
 
 m 
 
 * If the first of the equations (10), when P\ and Qi are both negative, he sup- 
 posed to express an imaginary focal, there will, in a central surface, be three focals, 
 two modular and one umbilicar; the two modular focals being in the principal 
 planes which pass through the directive axis, and the umbilicar focal in the remain- 
 ing principal plane. Then, when we know which of the axes is the directive axis, 
 we know which of the three focals is imaginary, because the plane of the imagi- 
 nary focal is perpendicular to the primary axis. A modular focal may be imagi- 
 nary, and yet have a real modulus ; this occurs in the hyperboloid of two sheets. 
 In the ellipsoid, the imaginary focal has an imaginary modulus. In all cases the 
 two moduli are connected by the relation (16). 
 
 It will appear hereafter that the vertex of the cone is an umbilicar focus. The 
 cone has therefore three focals, none of which is imaginary ; but two of them are 
 single points coinciding with the vertex. 
 
On the Surfaces of the Second Order. 273 
 
 where is the angle made by a directive plane with the plane 
 of the focal to which the modulus m belongs. One modulus is 
 greater than unity ; the other is less than unity, but greater 
 than the cosine of the angle which the plane of the correspond- 
 ing focal makes with a directive plane. In the hyperboloid of 
 one sheet, the less modulus is that which belongs to the focal 
 hyperbola. In the cone, the less modulus belongs to the focal 
 lines. Of the two moduli of a cone, that which belongs to the 
 focal lines may be termed the linear modulus ; and the other, to 
 which only a single focus corresponds, may be called the singu- 
 lar modulus. 
 
 6. Second Class of Surfaces. In this class of surfaces, one 
 of the quantities A, B vanishes, or both of them vanish. 
 
 I. When m = cos 0, and is not zero, A vanishes, but B 
 does not ; and the surface is either a paraboloid or a cylinder. 
 
 1. If the surface is a paraboloid, we may suppose the origin 
 of co-ordinates to be at its vertex, in which case both H and K 
 vanish, and we have the relations 
 
 G = %.> -a?!, y l = y 2 cos 2 0, 
 
 (17) 
 # 2 2 + y 2 2 cos 2 0-0 1 2 -y l 2 = 0; 
 
 the equation of the surface being 
 
 0, (18) 
 
 which shows that the paraboloid is elliptic, having its axis in the 
 direction of #, and the plane of xy for that of its greater prin- 
 cipal section. From the relations (17) we obtain the following : 
 
 <? = 0, 
 
 (19) 
 - G* = ; 
 
 from which we see that the focal and dirigent curves are para- 
 bolas, having their axes the same as that of the surface ; and 
 their vertices equidistant from the vertex of the surface, but at 
 opposite sides of it. The cavity of each curve is turned in the 
 
274 On the Surfaces of the Second Order. 
 
 same direction as that of the section xy. The focus of the focal 
 parabola is the focus of the section xy, and its vertex is the 
 focus of the section xz of the surface ; its parameter being the 
 difference of the parameters of these two sections. The para- 
 meter of the section xy is a mean proportional between the para- 
 meters of the focal and dirigent parabolas. 
 
 2. If the surface is a cylinder, we may make G and H 
 vanish, by taking the origin on its axis. We then have 
 
 x z = x^ i/i = y* cos 2 0, 
 
 (20) 
 K ' = 
 
 the equation of the cylinder, which is elliptic, being 
 
 ^ sin 2 + s 2 = K. (21) 
 
 Here the focal and dirigent are each a pair of right lines 
 parallel to the axis -of the cylinder, and passing through the foci 
 and directrices of a section perpendicular to the axis. The cor- 
 responding focal and dirigent lines lie at the same sides of the 
 axis. 
 
 II. When m = 1, and is not zero, B vanishes, but A does 
 not. 
 
 1. If the surface is a paraboloid, and the origin of co-ordi- 
 nates at its vertex, the quantities G and K vanish ; and the 
 equation of the surface becomes 
 
 (22) 
 and we have the relations 
 
 H" = 5 - \< %i = 
 
 (23) 
 + y? - x? - y? = 0. 
 
 The paraboloid is therefore hyperbolic, its axis being that of y, 
 which is also the directive axis ; and as the tangent of may 
 have any finite value, the plane of xy, which is that of the focal 
 
On the Surfaces of the Second Order. 275 
 
 curve, may be either of the principal planes passing through the 
 axis of the surface. The relations (23) give 
 
 (24) 
 x? tan 2 ^ sec 2 < - 
 
 for the equations of the focal and dirigent, which are therefore 
 parabolas, having their axes the same as those of the surface, 
 and their concavities turned in the same direction as that of the 
 section xy ; their vertices being equidistant from the vertex of the 
 surface, and at opposite sides of it. The focus of the focal para- 
 bola is the focus of the section xy, and its vertex is the focus of 
 the section y%, its parameter being the sum of the parameters of 
 these two sections. The parameter of the section xy is a mean 
 proportional between the parameters of the focal and dirigent 
 parabolas. 
 
 2. If the surface is a cylinder, and the origin on its axis, 
 Gr and H vanish, and we have 
 
 (25) 
 - K = x? sin 2 = x 
 
 the equation of the cylinder, which is hyperbolic, being 
 
 a? 2 tan 2 ^ - s 2 = - K. (26) 
 
 The focal and dirigent are each a pair of right lines parallel 
 to the axis of the cylinder; the corresponding lines passing 
 through a focus and the adjacent directrix of any section per- 
 pendicular to the axis. The directive planes are parallel to the 
 asymptotic planes of the cylinder. 
 
 In this case, if JT= 0, the surface is reduced to two directive 
 planes, and the focal and dirigent to the intersection of these 
 planes. 
 
 Ill, When m = l, and <f> = 0, both A and B vanish, and the 
 surface is the parabolic cylinder. If, as is allowable, we sup* 
 
276 On the Surfaces of the Second Order. 
 
 pose G and K to vanish, the equation of the cylinder becomes 
 
 s 2 + %Hy = 0, (27) 
 
 and we have 
 
 H=y z - y l9 x l = x-t, 
 
 (28) 
 
 x? + y?. - x? - y? = ; 
 whence 
 
 (29) 
 
 The focal and dirigent are each a right line parallel to the axis 
 of #, the former passing through the focus, the latter meeting 
 the directrix of the parabolic section made by the plane of i/z. 
 The plane of ocy is the directive plane. 
 
 7. We learn from this discussion that, among the surfaces of 
 the second class, the hyperbolic paraboloid is the only one which 
 admits a twofold modular generation ; the modulus, however, 
 being the same for both its focals. In the elliptic paraboloid 
 the modular focal is restricted to the plane of that principal 
 section which has the greater parameter ; we shall therefore 
 suppose a parabola to be described in the plane of the other 
 principal section, according to the law of the modular focals ; 
 the law being, that the focus of the parabola shall be the focus 
 of the principal section in the plane of which the parabola lies, 
 and its vertex the focus of the principal section in the per- 
 pendicular plane. The parabola so described will have its 
 concavity opposed to that of the surface ; it will- cut the sur- 
 face in the umbilics, and will be its umbilicar focal, the only 
 such focal to be found among the surfaces of the second 
 class. We shall of course suppose further, that this focal has 
 a dirigent parabola connected with it by the same law as in 
 the other cases, the vertices of the focal and dirigent being equi- 
 distant from that of the surface and at opposite sides of it, while 
 the parameter of the dirigent is a third proportional to the para- 
 meters of the focal and of the principal section in the plane of 
 which the curve lies. The two focals of a paraboloid are so re- 
 lated, that the focus of the one is the vertex of the other. The 
 cylinders have no other focals than those which occur above. 
 
On the Surfaces of the Second Order. 277 
 
 8. In this, as in the first class of surfaces, the right line 
 FA, joining a focus F with the foot of its corresponding direc- 
 trix, is perpendicular to the focal line ; and the focal and diri- 
 gent are reciprocal polars with respect to the section xy of the 
 surface. These properties are easily inferred from the preced- 
 ing results ; but, as they are general, it may be well to prove 
 them generally for both classes of surfaces. Supposing, there- 
 fore, the origin of co-ordinates to be anywhere in the plane of 
 xy, and writing the equation of the surface in the form 
 
 (* - *i) 2 + (y - y$ + z* = L(x- x,}* + M(y- y,}*, (30) 
 which, when identified with (3), gives the relations 
 A = 1-L, B = l-M, 
 
 G = Lx,- x^ H=My,- y l9 (31) 
 
 K = LxJ + My? - x? - y? y 
 
 we find, by differentiating the values of the constants 6r, H, 
 and K, 
 
 Ldx, = d%i 9 Mdy, = dy^ 
 
 Lx, dx, + My, dy, - #1 dxi - y\ dy = 0. (32) 
 
 Hence we obtain 
 
 (# a - #1) dsK 1 + (y* - yi) dy l = ; (33) 
 
 an equation which expresses that the right line joining the 
 points F and A is perpendicular to the line which is the locus 
 of the point F. 
 
 Again, the equation of the section xy of the surface^being 
 
 Ax 2 + By* + 2Gx + %Hy = K, (34) 
 
 the equation of the right line which is, with respect to this 
 section, the polar of a point A whose co-ordinates are o? 3 , y 3 , is 
 
 (Ax, + G)x+ (By, + H)y = K- Gx 2 - Hy,; (35) 
 
 but the relations (31) give 
 
 Ax, + G = x, - #!, By, + H= y, - y l9 
 
 (36) 
 K- Gx, - Hy, = ^ (x, - a?,) + y (y, - y^ ; 
 
278 On the Surfaces of the Second Order. 
 
 and hence the equation (35) becomes 
 
 (a? a - *i) (x - 0i) + (y 2 - yi) (y - yO = 0, (37) 
 
 which, as is evident from (33), is the equation of a tangent 
 applied to the focal at the point F corresponding to A. This 
 shows that the focal and dirigent are reciprocal polars with 
 respect to the section #y, and that in this relation, as well as in 
 the other, the points F and A are corresponding points. 
 
 Supposing F' and A' to be two other corresponding points 
 on the focal and dirigent, if tangents applied to the focal at F 
 and F' intersect each other in T, the point T will be the pole of 
 the right line A A' with respect to the section a?y, as well as the 
 pole of the right line FF' with respect to the focal ; and hence 
 if any right line be drawn through T, and if P be the pole of 
 this right line with respect to the section, and N its pole with 
 respect to the focal, the points P and N will be on the right 
 lines A A' and FF' respectively. Now it is useful to observe 
 that the distances A A' and FF' are always similarly divided 
 (both of them internally or both of them externally) by the 
 points P and N, so that we have AP to AT as FN to F'N. 
 This property may be proved directly by means of the fore- 
 going equations ; or it may be regarded as a consequence of the 
 following theorem : If through a fixed point in the plane of 
 two given conies having the same centre, or of two given para- 
 bolas having their axes parallel, any pair of right lines be 
 drawn, and their poles be taken with respect to each curve, 
 the distance between the poles relative to one curve will be in 
 a constant ratio to the distance between the poles relative to 
 the other curve.* In fact, the poles of the right lines TF, TF', 
 with respect to the focal, are F, F' ; and their poles with respect 
 to the section xy are A, A'; therefore, since the focal and the 
 section xy may be taken for the given curves, and the point T 
 
 * There is an analogous theorem for two surfaces of the second order which 
 have the same centre, or two paraboloids which have their axes parallel. If 
 through a fixed right line any two planes be drawn, and their poles be taken with 
 respect to each surface, the distance between the poles relative to the one surface 
 will be in a constant ratio to the distance between the poles relative to the other. 
 
On the Surfaces of the Second Order. 279 
 
 for the fixed point, the ratio of FF' to A A' is the same as the 
 ratio of FN to AP, or of F'N to AT ; and consequently the 
 distances FF' and AA' are similarly divided in the points N, 
 
 and P. 
 
 9. In the equation (30), considered as equivalent to the 
 equation (1), the constants L and M are both positive; but the 
 properties which have been deduced from the former equation 
 are independent of this circumstance, and equally subsist when 
 one of these constants is supposed to be negative (for they can- 
 not both be negative). This leads us to inquire what surfaces 
 the equation (30) is capable of representing when the constants 
 L and M have different signs; as also, for a given surface, what 
 lines are traced in the plane of xy by points F and A, of which 
 #!, yu and # 2 , 11* are the respective co-ordinates. After the ex- 
 amples already given, this question is easily discussed, and the 
 result is, that the only surfaces which can be so represented are 
 the ellipsoid, the hyperboloid of two sheets, the cone, and the 
 elliptic paraboloid that is to say, the umbilicar surfaces to- 
 gether with the cone ; and that, for an umbilicar surface, the 
 locus of F is the umbilicar focal, and therefore the locus of A is 
 the corresponding dirigent; while for the cone the points F and 
 A are unique, coinciding with each other and with the vertex of 
 the cone. A geometrical interpretation of this case is readily 
 found ; for as L and M have different signs, the right-hand 
 member of the equation (30), if M be the negative quantity, is 
 the product of two factors of the form 
 
 f(x - a*) + g(y - y*), /(* -**)-g(y- y a ), 
 
 in which /and g are constant; and these factors are evidently 
 proportional to the distances of a point whose co-ordinates are 
 a?,'y, s, from two planes whose equations are 
 
 f(x - x z ] +g(y - y) = 0, f(x - a,) -g(y - y 8 ) = 0, 
 
 which planes always pass through a directrix, and are inclined 
 at equal and constant angles to the axis of x or of y. There- 
 fore, if F be the focus which belongs to this directrix, the square 
 
280 On the Surfaces of the Second Order. 
 
 of the distance of F from any point of this surface is in a con- 
 stant ratio to the rectangle under the distances of the latter 
 point from the two planes. And these planes are directive 
 planes ; because, if a section parallel to one of them be made 
 in the surface, the distance of any point of the section from the 
 other plane will be proportional to the square of the distance of 
 the same point from the focus ; and, as the locus of a point, 
 whose distance from a given plane is proportional to the square 
 of its distance from a given point, is obviously a sphere, it 
 follows that the section aforesaid is the section of a sphere, and 
 consequently a circle ; which shows that the plane to which the 
 section is parallel is a directive plane. Thus,* the square of the 
 
 r * In attempting to find a geometrical generation for the surfaces of the second 
 order, one of the first things which I thought of, before I fell upon the modular 
 method, was to try the locus of a point such that the square of its distance from a 
 given point should he in a constant ratio to the rectangle under its distances from 
 two given planes; but when I saw that this locus would not represent all the 
 species of surfaces, I laid aside the discussion of it. Some time since, however, 
 Mr. Salmon, Fellow of Trinity College, was led independently, in studying the 
 modular method, to consider the same locus ; and he remarked to me, what I had 
 not previously observed, that it offers a property supplementary, in a certain sense, 
 to the modular property; that when the surface is an ellipsoid, for example, the 
 given point or focus is on the focal hyperbola, which the modular property leaves 
 empty. This remark of Mr. Salmon served to complete the theory of the focals, 
 by indicating a simple geometrical relation between a non-modular focal and any 
 point on the surface to which it belongs. 
 
 In a memoir " On a new method of Generation and Discussion of .the Surfaces 
 of the second Order," presented by M. Amyot to the Academy of Sciences of Paris, 
 on the 26th December, 1842, the author investigates this same locus, conceiving it 
 to involve that property in surfaces which is analogous to the property of the focus 
 and directrix in the conic sections ; and the importance attached to the discovery of 
 such analogous properties induced M. Cauchy to write a very detailed report on 
 M. Amyot' s memoir, accompanied with notes and additions of his own (Comptes 
 rendus des Seances de V Academic des Sciences, torn. xvi. pp. 783-828, 885-890 ; 
 April, 1843) ; and also occasioned several discussions, principally between M. 
 Poncelet and M. Chasles, relative to that Memoir (Comptes rendus, torn. xvi. 
 pp. 829, 938, 947, 1105, 1110). But the property involved in this locus cannot 
 be said to afford a method of generation of the surfaces of the second order, since 
 it applies only to some of the surfaces, and gives an ambiguous result even where 
 it does apply. It is therefore not at all analogous to the aforesaid general property 
 of the conic sections, and moreover it was not new when M. Amyot brought it 
 
On the Surfaces of the Second Order. 281 
 
 distance of any point of the surface from an umbilicar focus 
 bears a constant ratio to the rectangle under the perpendicular 
 distances of the same point from two directive planes drawn 
 through the directrix corresponding to that focus; and it is 
 easy to see that this ratio, the square root of which we shall 
 denote by //, is equal to L - M, or, neglecting signs, to the. 
 sum of the numerical values of L and M. Of course, if the 
 distances from the directive planes, instead of being perpendi- 
 cular, be measured parallel to any fixed right line, the ratio 
 will still be constant, though different. For example, if the 
 fixed right line for each plane be that which joins the corre- 
 sponding umbilic with either focus of the section xy, the ratio 
 
 forward. Mr. Salmon had in fact proposed it for investigation to the students of 
 the University of Dublin, at the ordinary Examinations in October, 1842 ; and it 
 was published, towards the end of that year, in the University Calendar for 1843, 
 some months before the date of M. Cauchy's report, by which the contents of 
 M. Amy of s memoir were first made known. The parallelism of the two given 
 planes to the circular sections of the surface is also stated in the Calendar ; but this 
 remarkable relation is not noticed by M. Amyot, nor by M. Cauchy (see the " Exa- 
 mination Papers" of the year 1842, p. xlv., quest. 17, 18 ; in the Calendar for 1843). 
 It is scarcely necessary to add, that the analogue which M. Amyot and other 
 mathematicians have been seeking for, and which was long felt to be wanting in 
 the theory of surfaces of the second order, is no other than the modular property 
 of these surfaces, which appears to be not yet known abroad. M. Poncelet insists 
 much on the importance of extending the signification of the terms focus and direc- 
 trix, so as to make them applicable to surfaces ; and he supposes this to have been 
 effected, for the first time, by M. Amyot. These terms, however, applied in their 
 true general sense to surfaces, had been in use, several years before, among the 
 mathematical students of Dublin, as may be seen by referring to the Calendar 
 ("Examination Papers" of the year 1838, p. c. 1839, p. xxxi.). 
 
 The locus above mentioned, being co-extensive with the umbilicar property, 
 does not represent any surface which can be generated by the right line, except 
 the cone. To remedy this want of generality, M. Cauchy proposes to consider a 
 surface of the second order as described by a point, the square of whose distance 
 from a given point bears a constant ratio either to the rectangle under its distances 
 from two given planes, or to the sum of the squares of these distances. This enun- 
 ciation, no doubt, takes in both kinds of focals, and all the species of surfaces ; but 
 the additional conception is not of the kind required by the analogy in question, 
 nor has it any of the characters of an elementary principle. For the given planes, 
 according to M. Cauchy's idea, do not stand in any simple or natural relation to 
 the surface ; and besides there is no reason why, instead of the sum of the squares 
 
282 On the Surfaces of the Second Order. 
 
 of the square to the rectangle will be the square of the number 
 m sec $, where m is the modulus, and the angle which the 
 primary axis makes with a directive plane. 
 
 When the umbilicar property is applied to the cone, the 
 vertex of which is, as we have seen, to be regarded as an umbi- 
 licar focus, having the directive axis for its directrix, it indi- 
 cates that the product of the sines of the angles which any side 
 of the cone makes with its two directive planes is a constant 
 quantity. 
 
 It is remarkable that the vertex of the cone affords the only 
 instance of a focal point which is at once modular and umbi- 
 licar, as well as the only instance of a focal point which is 
 doubly modular. This union of properties it may be con- 
 ceived to owe to the circumstance that the cone is the asymp- 
 totic limit of the two kinds of hyperboloids. For if a series 
 of hyperboloids have the same asymptotic cone, and their 
 primary axes be indefinitely diminished, they will approach 
 indefinitely to the cone; and, in the limit, the focal ellipse 
 and hyperbola of the hyperboloid of one sheet will pass into 
 the vertex and the focal lines of the cone, thus making the 
 vertex doubly modular ; while the focal ellipse of the hyperbo- 
 loid of two sheets will also be contracted into the vertex, and 
 will make that point umbilicar. 
 
 of the distances from the given planes, we should not take the sum after multi- 
 plying the one square by any given positive number, and the other square by 
 another given positive number ; nor is there any reason why we should not take 
 other homogeneous functions of these distances. This conception would therefore 
 be found of little use in geometrical applications ; while the modular principle, on 
 the contrary, by employing a simple ratio between two right lines, both of which 
 have a natural connexion with the surface, lends itself with the greatest ease to the 
 reasonings of geometry. Indeed the whole difficulty, in extending the property of 
 the directrix to surfaces of the second order, consisted in the discovery of such a 
 ratio^inherent in all of them a ratio having nothing arbitrary in its nature, and 
 for which no other of equal simplicity can be substituted. 
 
 It may be proper to mention that the term modulus, which I have used for the 
 first time in the present Paper, with reference to surfaces of the second order, has 
 been borrowed from M. Cauchy, by whom it is employed, however, in a significa- 
 tion entirely different. Several other new terms are also now introduced, from the 
 necessity of the case. 
 
On the Surfaces of the Second Order. 283 
 
 When the two directive planes coincide, and become one 
 directive plane, the umbilicar property is reduced to this, that 
 the distances of any point in the surface from the point F and 
 from the directive plane are in a constant ratio to each other ; 
 and therefore the surface becomes one of revolution round an 
 axis passing through F at right angles to that plane ; the point 
 F being a focus of the meridional section, or the vertex if the 
 surface be a cone. When the directive planes are supposed to 
 be parallel, but separated by a finite interval, we get the same 
 class of surfaces of revolution, with the addition of the surface 
 produced by the revolution of an ellipse round its minor axis ; 
 the point F being still on the axis of revolution, but not having 
 any fixed relation to the surface. 
 
 10. If in the equation (30) we supposed the right-hand 
 member to have an additional term containing the product of 
 the quantities x - x z and y - y^ with a constant coefficient, all 
 the foregoing conclusions regarding the geometrical meaning 
 of that equation would remain unchanged, because the addi- 
 tionarterm could always be taken away by assigning proper 
 directions to the axes of x and y. If, after the removal of this 
 term, the coefficients of the squares of the aforesaid quantities 
 were both positive, the locus of F would be a modular focal of 
 the surface expressed by the equation ; but if one coefficient 
 were positive and the other negative, the locus of F would be 
 an umbilicar focal. The equation in its more general form is 
 evidently that which we should obtain for the locus of a point 
 S, such that the square of its distance SF from a given point F 
 should be a given homogeneous function of the second degree 
 of its distances from two given planes ; the plane of xy being 
 drawn through F perpendicular to the intersection of these 
 planes, and x^ y* being the co-ordinates of any point on this 
 intersection, while #1, y\ are the co-ordinates of F. The point 
 F might be any point on one of the focals of the surface de- 
 scribed by S; the intersection of the two planes (supposing 
 them always parallel to fixed planes) being the corresponding 
 directrix. 
 
284 On the Surfaces of the Second Order. 
 
 These considerations may be further generalized, if we 
 remark that the equation of any given surface of the second 
 order may be put under the form 
 
 -y), (38) 
 
 where Z, M, N, L', M', N f are constants, and #1, y^ Si are con- 
 ceived to be the co-ordinates of a certain point F, and # 2 , y*> s 2 
 the co-ordinates of another point A. The constants L', M' 9 N f 
 may, if we please, be made to vanish by changing the directions 
 of the axes of co-ordinates ; and when this is done, the new co- 
 ordinate planes will be parallel to the principal planes of the 
 surface. Then, by proceeding as before, it may be shown that, 
 without changing the surface, we are at liberty, under certain 
 conditions, to make the points F and A move in space. The 
 conditions are expressed geometrically by saying that the two 
 surfaces, upon which these points must be always found, are 
 reciprocal polars with respect to the given surface, the points F 
 and A being, in this polar relation, corresponding points ; and 
 that the surface which is the locus of F is a surface of the 
 second order, confocal with the given one, it being understood 
 that confocal surfaces are those which have the same focal 
 lines. The surface on which A lies is therefore also of the 
 second order, and the right line AF is a normal at F to the 
 surface which is the locus of this point. Moreover, if through 
 the point A three or more planes be drawn parallel to fixed 
 planes, and perpendiculars be dropped upon them from any 
 point S whose co-ordinates are #, y, 2, the right-hand member 
 of the equation (38) may be conceived to represent a given 
 homogeneous function of the second degree of these perpendi- 
 culars ; and the given surface may therefore be regarded as the 
 locus of a point S, such that the square of the distance SF is 
 always equal to that function. 
 
 11. In the enumeration of the surfaces capable of being 
 generated by the modular method, we miss the five following 
 
On the Surfaces of the Second Order. 285 
 
 varieties, which are contained in the general equation of the 
 second degree, but are excluded from that method of genera- 
 tion by reason of the simplicity of their forms namely, the 
 sphere, the right cylinder on a circular base, and the three 
 surfaces which may be produced by the revolution of a conic 
 section (not a circle) round its primary axis.* These three 
 surfaces are the prolate spheroid, the hyperboloid of two sheets, 
 and the paraboloid of revolution ; and the circumstance, that 
 the foci of the generating curves are also foci of the surfaces, 
 renders it easy to investigate their focal properties.! In point 
 of simplicity, the excepted surfaces are to the other surfaces of 
 the second order what the circle is to the other conic sections, 
 the circle being, in like manner, excepted from the curves 
 which can be generated by the analogous method in piano; 
 and the geometry of the five excepted surfaces may therefore 
 be regarded as comparatively elementary. These five surfaces 
 were, in fact, studied by the Greek geometers, J and, along with 
 the oblate spheroid and the cone, they make up all the surfaces 
 of the second order with which the ancients were acquainted. 
 Except the cone, the surfaces considered by them are all of 
 revolution; and there is only one surface of revolution, the 
 hyperboloid of one sheet, which was not noticed until modern 
 times. This surface is mentioned (under the name of the 
 hyperbolic cylindroid) by Wren, who remarks that it can 
 be generated by the revolution of a right line round another 
 right line not in the same plane. As to the general conception 
 of surfaces of the second order, the suggestion of it was reserved 
 for the algebraic geometry of Descartes. In that geometry the 
 
 * The case of two parallel planes is also excluded, but it is not here taken into 
 account. The case of two parallel right lines is in like manner excluded from the 
 corresponding generation of lines of the second order. 
 
 t A Paper by M. Chasles, on these surfaces of revolution, will be found in the 
 " Memoirs/' of the Academy of Brussels, torn. v. (An. 1829). 
 
 J The hyperboloid of two sheets, and the paraboloid of revolution, were known 
 by the name of conoids. Archimedes has left a treatise on Conoids and Spheroids, 
 as well as a treatise on the Sphere and Cylinder. 
 
 In the Philosophical Transactions for the year 1669, p. 961. 
 
286 On the Surfaces of the Second Order. 
 
 curves previously known as sections of the cone are all expressed 
 by the general equation of the second degree between two co- 
 ordinates ; and hence it occurred to Euler* about a century ago, 
 to examine and classify the different kinds of surfaces comprised 
 in the general equation of the second degree among three co- 
 ordinates. The new and more general forms thus brought to 
 light have since engaged a large share of the attention of geo- 
 meters ; but the want of some other than an algebraic principle 
 of connexion has prevented any great progress from being made 
 in the investigation of such of their properties as do not im- 
 mediately depend on transformations of co-ordinates. This 
 want the modular method of generation perfectly supplies, by 
 evolving the different forms from a simple geometrical concep- 
 tion, at the same time that it brings them within the range of 
 ideas familiar to the ancient geometry, and places their relation 
 to the conic sections in a striking point of view. 
 
 It may be well to remark that the excepted surfaces are 
 limits of surfaces which can be generated modularly, as the 
 circle is the limit of the ellipse in the analogous generation of 
 the conic sections. Thus the sphere is the limit of an oblate 
 spheroid, one of whose axes remains constant, while its focal 
 circle is indefinitely diminished ; and the right circular cylinder 
 is the limit of an elliptic cylinder, whose focal lines are con- 
 ceived to approach indefinitely to coincidence with each other 
 and with the axis of the cylinder, while one of the axes of the 
 principal elliptic section remains constant. In these cases the 
 dirigent lines, along with the directrices, move off to infinity. 
 The other three excepted surfaces correspond to the supposition 
 = 90, which was excluded in the discussion of the general 
 equation (1). For if we make m sec = n, the quantity which 
 constitutes the right-hand member of that equation may be 
 written 
 
 n*(x - x z y + n z (y - ytf cos 2 ; 
 
 and if we suppose n to remain finite and constant, while 
 
 * See his Introduetio in Analysin Infinitorum, p. 373. Lausanne, 1748. 
 
On the Surfaces of the Second Order. 287 
 
 approaches to 90, and m indefinitely diminishes, this quantity 
 will approach indefinitely to n* (x - # 2 ) 2 , which will be its limit- 
 ing value when <j> = 90. But x - x z is the distance of the point 
 S from a fixed plane intersecting the axis of x perpendicularly 
 at the distance x from the origin of co-ordinates ; and there- 
 fore, in the limit, the equation expresses that the distances of 
 any point S of the surface, from the focus F and from this fixed 
 plane, are to each other as n to unity, that is, in a constant 
 ratio, which is a common property of the three surfaces in 
 question. This property also belongs to the right cone, but the 
 right cone does not rank among the excepted surfaces. 
 
 12. We have seen that, when the modulus is unity, any 
 plane parallel to either of the directive planes intersects the 
 surface in a right line; whence it follows, that through any 
 point on the surface of a hyperbolic paraboloid two right lines 
 may be drawn which shall lie entirely in the surface. The 
 plane of these right lines is of course the tangent plane at that 
 point, and therefore every tangent plane intersects the surface 
 in two right lines. This is otherwise evident from considering 
 that the sections parallel to a given tangent plane are similar 
 hyperbolas, whose centres are ranged on a diameter passing 
 through the point of contact, and whose asymptotes, having 
 always the same directions, are parallel to two fixed right lines 
 which we may suppose to be drawn through that point. For 
 as the distance between the plane of section and the tangent 
 plane diminishes, the axes of the hyperbola diminish ; and they 
 vanish when that distance vanishes, the hyperbola being then 
 reduced to its asymptotes. The tangent plane therefore inter- 
 sects the surface in the two fixed right lines aforesaid. The 
 same reasoning, it is manifest, will apply to any other surface 
 of the second order which has hyperbolic sections parallel to its 
 tangent planes; and therefore the hyperboloid of one sheet, 
 which is the only other such surface,* is also intersected in two 
 
 * The double generation of these two surfaces by the motion of a right line has 
 been long known. It appears to have been discovered and fully discussed by some 
 of the first pupils of the Polytechnic School of Paris. This mode of generation had, 
 
288 On the Surfaces of the Second Order. 
 
 right lines by any of its tangent planes. These right lines are 
 usually called the generatrices of the surface. 
 
 From what has been said, it appears that the generatrices of 
 the hyperbolic paraboloid, and the asymptotes of its sections (all 
 its sections, except those made by planes parallel to the axis, 
 being hyperbolas), are parallel to the directive planes. The 
 generatrices of the hyperboloid of one sheet, and the asymp- 
 totes of its hyperbolic sections, are parallel to the sides of the 
 asymptotic cone; because any section of the hyperboloid is 
 similar to a parallel section of the asymptotic cone ; and when 
 the latter section is a hyperbola its asymptotes are parallel to 
 two sides of the cone. 
 
 PART II. PROPERTIES OF SURFACES OF THE SECOND 
 ORDER. 
 
 1. In the preceding part of this Paper it has been neces- 
 sary to enter into details for the purpose of communicating fun- 
 damental notions clearly. In the following part, which will 
 contain certain properties of surfaces of the second order, we 
 shall be as brief as possible ; giving demonstrations of the more 
 elementary theorems, but confining ourselves to a short state- 
 ment of the rest. 
 
 Many consequences follow from the principles already laid 
 down. 
 
 Through any directrix of a surface of the second order let a 
 fixed plane be drawn cutting the surface, and let S be any point 
 of the section. If the directrix and its focus F be modular, and 
 if a plane always parallel to the same directive plane be con- 
 ceived to pass through S and to cut the directrix in D, the direc- 
 tive distance SD will be always parallel to a given right line, 
 and will therefore be in a constant ratio to the perpendicular dis- 
 tance of S from the directrix. This perpendicular distance will 
 
 however, been remarked by Wren, with regard to the hyperboloid of revolution. 
 It does not seem to have been observed, that the existence of rectilinear genera- 
 trices is included in the idea of hyperbolic sections parallel to a tangent plane. 
 
On the Surfaces of the Second Order. 289 
 
 consequently bear a given ratio to SF, the distance of the point 
 S from the focus. And the same thing will be true when the 
 directrix and focus are umbilicar, because the perpendicular dis- 
 tance of the point S from the directrix will be in a constant ratio 
 to its distance from each directive plane drawn through the 
 directrix. 
 
 The fixed plane of section will in general contain another 
 directrix parallel to the former, and belonging to the same 
 focal ; and it is evident that the perpendicular distance of S from 
 this other directrix will be in a given ratio to its distance SF' 
 from the corresponding focus F', the ratio being the same as in 
 the former case. Hence, according as the point S lies between 
 the two directrices, or at the same side of both, the sum or dif- 
 ference of the distances SF and SF' will be constant. 
 
 If the plane of section pass through either of the foci, as F, 
 this focus and its directrix will manifestly be the focus and di- 
 rectrix of the section. In this case the plane of section will be 
 perpendicular to the focal at F. And if the surface be a cone, 
 the point F being anywhere on one of its focal lines, the distance 
 of the point S from the directrix will be in a constant ratio to 
 its perpendicular distance from the dirigent plane which con- 
 tains the directrix, and therefore this perpendicular distance will 
 be in a given ratio to the distance SF. Now, calling Y the 
 vertex of the cone, and taking SV for radius, the perpendicular 
 distance aforesaid is the sine of the angle which the side SY of 
 the cone makes with the dirigent plane ; and SF, which is per- 
 pendicular to YF, is the sine of the angle SYF. Consequently 
 the sines of the angles which any side of a cone makes with a 
 dirigent plane and the corresponding focal line are in a given 
 ratio to each other.] 
 
 2. Conceive a surface of the second order to be intersected 
 in two points S, S' by a right line which cuts two parallel direc- 
 trices in the points E, E', and let F, F' be the foci correspond- 
 ing respectively to these directrices. The perpendicular dis- 
 tances of the points S, S' from the first directrix and from the 
 second are to each other as the lengths SE, S'E, SK, S'E' 
 
2 go On the Surfaces of the Second Order. 
 
 respectively, and therefore the ratios of FS to SE, of FS' to S'E, 
 of F'S to SE', and of F'S' to S'E', are all equal. 
 
 Hence, the right line FE bisects one of the angles made by 
 the right lines FS and FS' ; and the right line F'E' bisects one 
 of the angles made by F'S and F'S'. 
 
 When the points S, S' are at the same side of E, the angle 
 supplemental to SFS' is that which is bisected by the right line 
 FE. Now if the point S be fixed, and S' approach to it inde- 
 finitely, the angle SFE will approach indefinitely to a right 
 angle. Therefore if a right line touching the surface meet a 
 directrix in a certain point, the distance between this point and 
 the point of contact will subtend a right angle at the focus 
 which corresponds to the directrix. And if a cone circumscrib- 
 ing the surface have its vertex in a directrix, the curve of contact 
 will be in a plane drawn through the corresponding focus at right 
 angles to the right line which joins that focus with the vertex. 
 
 When the surface intersected by the right line SS' is a cone, 
 suppose this line to lie in the plane of the focus F and its direc- 
 trix, that is, in the plane which is perpendicular at F to the focal 
 line YF (the vertex of the cone being denoted, as before, by Y) : 
 the angles made by the right lines FE, FS, FS', are then the 
 same as the angles made by planes drawn through YF and eact 
 of the right lines YE, YS, YS' ; and the last three right lines are 
 the intersections of a plane YSS' with the dirigent plane on which 
 the point E lies, and with the surface of the cone. ' Therefore iJ 
 a plane passing through the vertex of a cone intersect its sur- 
 face in two right lines, and one of its dirigent planes in anothei 
 right line, and if a plane be drawn through each of these right 
 lines respectively and the focal line which belongs to the dirigent 
 plane, the last of the three planes so drawn will bisect one oJ 
 the angles made by the other two. And hence, if a plane touch- 
 ing a cone along one of its sides intersect a dirigent plane in 
 . certain right line, and if through this right line and the side oJ 
 contact, respectively, two planes be drawn intersecting each othei 
 in the focal line which corresponds to the dirigent plane, the twc 
 planes so drawn will be at right angles to each other. 
 
On the Surfaces of the Second Order. 291 
 
 Let a right line touching a surface of the second order in S 
 meet two parallel directrices in the points E, E', and let F, F' 
 be the corresponding foci. Then the triangles FSE and F'SE' 
 are similar, because the angles at F and F' are right angles, and 
 the ratio of FS to SE is the same as the ratio of FS to SE'. 
 Therefore the tangent EE' makes equal angles with the right 
 lines drawn from the point of contact S to the foci F, F'. When 
 the surface is a cone, let the tangent be perpendicular to the 
 side YS which passes through the point of contact ; the angles 
 FSE and F'SE' are then the angles which the tangent plane 
 VEE' makes with the planes YSF and YSF', because the right 
 line FE is perpendicular to the plane YSF, and the right line 
 F'E' is perpendicular to the plane YSF'. Therefore the tangent 
 plane of a cone makes equal angles with the planes drawn 
 through the side of contact and each of the focal lines. 
 
 Supposing a section to be made in a surface of the second 
 order by a plane which cuts any directrix in the point E, if the 
 focus F belonging to this directrix be the vertex of a cone having 
 the section for its base, the right line FE will be an axis of the 
 cone. For if through FE any plane be drawn cutting the base 
 of the cone in the points S, S', one of the angles made by the 
 sides FS, FS' which pass through these points will always be 
 bisected by the right line FE ; and this is the characteristic 
 property of an axis. 
 
 3. Two surfaces of the second order being supposed to 
 have the same focus, directrix, and directive planes, so that they 
 differ only in the value of the modulus m, or of the umbilicar 
 ratio fj. (see Part I. 9) : let a right line passing through any 
 point E of the directrix cut one surface in the points S, S', and 
 the other in the points S , Si, and conceive right lines to be 
 drawn from all these points to the common focus F. Since, if 
 ratios be expressed by numbers, the ratio of FS to SE (or of 
 FS' to S'E) is to the ratio of FS to*S E (or of F& to S X E) as 
 the value of m for the one surface is to its value for the other, 
 when the fociis is modular, or as the value of ju for the one sur- 
 face is to its value for the other when the focus is umbilicar, the 
 
292 On the Surfaces of the Second Order. 
 
 sines of the angles EFS and EFS (or of the angles EF&! anc 
 EFS') are in a constant proportion to each other, because thes( 
 sines are proportional to those ratios. And since the right lin* 
 FE bisects the angles SFS' and S FSi, both internally or botl 
 externally, in which case the angles SFS and S'FSi are equal 
 or else one internally and the other externally, in which case th< 
 angles SFS and S'FSi are supplemental, it is easy to infer 
 from the constant ratio of the aforesaid sines, that in the firs 
 case the. product, in the second case the ratio of the tangents o: 
 the halves of the angles SFS and S'FS (or of the halves of th( 
 angles SFSi and S'FSi) is a consequent quantity. 
 
 If the point S' approximate indefinitely to S, the right lin< 
 passing through these points will approach indefinitely to a tan- 
 gent. Therefore when two surfaces are related as above, if i 
 right line passing through any point E of their common direc- 
 trix intersect one surface in the points S , Si, and touch the othei 
 in the point S, the chord SoSi will subtend a constant angle al 
 the common focus F, and this angle will be bisected, either in- 
 ternally or externally, by the right line FS drawn from the f ocuj 
 to the point of contact. And the angle EFS being then a righl 
 angle, the cosine of the angle SFS or SFSi will be equal to the 
 ratio of the less value of m or p to the greater.* 
 
 4. Among the surfaces of the second order, the only one 
 which has a point upon itself for a modular focus is the cone, 
 the vertex of which is such a focus, related either to the internal 
 or to the mean axis as directrix. In the latter relation the 
 vertex belongs to the series of foci which are ranged on the focal 
 lines. To see the consequence of this, let Y be the vertex oi 
 the cone, and YW its mean axis perpendicular to the plane oi 
 the focal lines. On one of the focal lines and its dirigent assume 
 any corresponding points F and A, and let AD be the directrix 
 passing through A. Then if a directive plane, drawn through 
 any point S of the surface,' cut this directrix in D and the mean 
 
 * See Exam. Papers, An. 1839, p. xxxi., questions 9, 10. These and some oi 
 the preceding theorems were originally stated with reference to modular foci only. 
 They are now extended to umbilicar foci. 
 
On the Surfaces of the Second Order. 293 
 
 axis in W, the ratio of SF to SD will be expressed by the linear 
 modulus, as will also the ratio of YF to WD, since Y is a point 
 of the surface, and WD is equal to the directive distance of Y 
 from AD. But since Y is a focus to which the mean axis is 
 directrix, the ratio of SY to SW is expressed by the same mo- 
 dulus. Thus the triangles SYF and SWD are similar, the sides 
 of the one being proportional to those of the other. Therefore 
 the angle SYF is equal to the angle SWD ; that is to say, the 
 angle which the side YS of the cone makes with the focal line 
 YF is equal to the angle contained by two right lines WD and 
 WS, of which one is the intersection of the directive plane with 
 the dirigent plane YWD corresponding to YF, and the other is 
 the intersection of the directive plane with the plane YWS 
 passing through the mean axis and the side YS of the cone. 
 
 Hence it appears that the sum of the angles (pispperly 
 reckoned) which any side of the cone makes with its two focal 
 lines is constant. For if F' be a point on the other focal line, 
 and D' the point where the directrix corresponding to F' is in- 
 tersected by the same directive plane SWD, it may be shown, as 
 above, that the angle SYF' is equal to the angle SWD', that is, 
 to the angle made by the right line WS with the right line 
 WD', in which the directive plane intersects the dirigent plane 
 corresponding to YF'. Conceiving therefore the points F, F', S, 
 and with them the points D, D', to lie all on the same side of 
 the principal plane which is perpendicular to the internal axis, 
 the right line WS will lie between the right lines WD and WD', 
 and the grim of the angles SYF and SYF' will be equal to the 
 angle DWD', which is a constant angle, being contained by 
 the right lines in which a directive plane intersects the two diri- 
 gent planes of the cone. This constant angle will be found to be 
 equal, as it ought to be, to one of the angles made by the two 
 sides of the cone which are in the plane of the focal lines, namely, 
 to the angle within which the internal axis lies. 
 
 If we conceive the cone to have its vertex at the centre of a 
 sphere, and the points F, F', S to be on the surface of this sphere, 
 the arcs of great circles connecting the point S with each of the 
 
294 On the Surfaces of the Second Order. 
 
 fixed points F, F' will have a constant sum. The curve formed 
 by the intersection of the sphere and the cone may therefore, 
 from analogy, be called a spherical ellipse, or, more generally, a 
 spherical conic, because, by removing one of its foci F, F' to the 
 opposite extremity of the diameter of the sphere, the difference 
 of the arcs SF and SF' will be constant, which shows that the 
 spherical curve is analogous to the hyperbola as well as to the 
 ellipse. Either of these plane curves may, in fact, be obtained 
 as a limit of the spherical curve when the sphere is indefinitely 
 enlarged, according as the diameter along which the enlarge- 
 ment takes place, and of which one extremity may be conceived 
 to be fixed while the other recedes indefinitely, coincides with 
 the internal or with the directive axis of the cone. The fixed 
 extremity becomes the centre of the limiting curve, which is an 
 ellipse in the first case, and a hyperbola in the second. 
 
 The great circle touching a spherical conic at any point 
 makes equal angles with the two arcs of great circles which join 
 that point with the foci, because the sum of these arcs is con- 
 stant. This is identical with a property already demonstrated 
 relative to the tangent planes of the cone. Indeed it is obvious 
 that the properties of the cone may also be stated as properties 
 of the spherical conic, and this is frequently the more convenient 
 way of stating them. 
 
 5. If the sides of one cone be perpendicular to the tan- 
 gent planes of another, the tangent planes of the former will be 
 perpendicular to the sides of the latter. For the plane of two 
 sides of the first cone is perpendicular to the intersection of the 
 two corresponding tangent planes of the second cone ; and as 
 these two sides approach indefinitely to each other, their plane 
 approaches to a tangent plane, while the intersection of the two 
 corresponding tangent planes of the second cone approaches in- 
 definitely to a side of the cone. Thus any given side of the one 
 cone corresponds to a certain side of the other ; and any side of 
 either cone is perpendicular to the plane which touches the other 
 along the corresponding side. This reasoning applies to cones 
 of any kind. 
 
On the Stirfaces of the Second Order. 295 
 
 Two cones so related may be called reciprocal cones. When 
 one is of the second order, it will be found that the other is also 
 of the second order, and that, in their equations relative to their 
 axes, which are obviously parallel or coincident, the coefficients 
 of the squares of the corresponding variables are reciprocally 
 proportional, so that the equations 
 
 (1) 
 
 express two such cones which have a common vertex. These 
 cones have the same internal axis, but the directive axis of the one 
 coincides with the mean axis of the other, and it may be shown 
 from the equations that the directive planes of the one -are per- 
 pendicular to the focal lines of the other. The two curves in 
 which these cones are intersected by a sphere, having its centre 
 at their common vertex, are reciprocal spherical conies. In 
 general, two curves traced on the surface of a sphere may be said 
 to be reciprocal to each other, when the cones passing through 
 them, and having a common vertex at the centre of the sphere, 
 are reciprocal cones. Any given point of the one curve corre- 
 sponds to a certain point of the other, and the great circle which 
 touches either curve at any point is distant by a quadrant from 
 the corresponding point of the other curve. 
 
 By means of these relations any property of a cone of the 
 second order, or of a spherical conic, may be made to produce a 
 reciprocal property. Thus, we have seen that the tangent plane 
 of a cone makes equal angles with two planes passing through 
 the side of contact and through each of the focal lines ; there- 
 fore, drawing right lines perpendicular to the planes, and planes 
 perpendicular to the right lines here mentioned, we have, in the 
 reciprocal cone, a side making equal angles with the right lines 
 in which the directive planes of this cone are intersected by a 
 plane touching it along that side. It is therefore a property of 
 the cone, that the intersections of a tangent plane with the two 
 directive planes make equal angles with the side of contact ; a 
 
296 On the Surfaces of the Second Order. 
 
 property which it is easy to prove without the aid of the reci- 
 procal cone. 
 
 The two directive sections drawn through any point S of 
 a given surface of the second order may, when they are circles, 
 be made the directive sections of a cone, and this may obviously 
 be done in two ways. Each of the two cones so determined will 
 be touched by the plane which touches the given surface at the 
 point S, because the right lines which are tangents to the two 
 circular sections at that point are tangents to each cone as well 
 as to the given surface ; therefore the side of contact of each 
 cone bisects one of the angles made by these two tangents ; and 
 hence the two sides of contact are the principal directions in the 
 tangent plane at the point S, that is, they are the directions of 
 the greatest and least curvature of the given surface at that 
 point ; for these directions are parallel to the axes of a section 
 made in the surface by a plane parallel to the tangent plane, 
 and the axes of any section bisect the angles contained by the 
 right lines in which the plane of section cuts the two directive 
 planes. 
 
 6. It has been shown that the sum of the angles which any 
 side of a cone makes with its focal lines is constant. Hence 
 we obtain the reciprocal property,* that the sum of the angles 
 (properly reckoned) which any tangent plane of a cone makes 
 with its two directive planes is constant. This property may 
 be otherwise proved as follows : 
 
 Through a point assumed anywhere in the side of contact 
 
 * This property, and that to which it is reciprocal, as well as some other pro- 
 perties of the cone, were, together with the idea of reciprocal cones and of spherical 
 conies, suggested by my earliest researches connected -with the mechanical theory 
 of rotation and the laws of double refraction. I was not then aware that the focal 
 lines of the cone had been previously discovered, nor that the spherical conic had 
 been introduced into geometry. Indeed all the properties of the cone which are 
 given in this Paper were first presented to me in my own investigations. Its double 
 modular property, related to the vertex as focus, was one of the propositions in the 
 theory of the rotation of a solid body, and was used in finding the position of the axis 
 of rotation within the body at a given time. 5 ut the modular property common to 
 all the surfaces of the second order was not discovered until some years later. 
 
On the Surfaces of the Second Order. 297 
 
 let two directive planes be drawn. As the circles in which the 
 cone is cut by these planes have a common chord, they are circles 
 of the same sphere ; and a tangent plane applied to this sphere, 
 at the aforesaid point, coincides with the tangent plane of the 
 cone, because each tangent plane contains the tangents drawn to 
 the two circles at that point. The common chord of the circles is 
 bisected at right angles by the principal plane which is perpen- 
 dicular to the directive axis, and therefore that principal plane 
 contains the centres of the two circles and the centre of the 
 sphere. Now the acute angle made by a tangent plane of a 
 sphere with the plane of any small circle passing through the 
 point of contact is evidently half the angle subtended at the 
 centre of the sphere by a diameter of that circle ; therefore the 
 acute angles, which the common tangent plane of the cone and 
 of the sphere above mentioned ma^es with the planes of the 
 directive sections, are the halves of the angles subtended at the 
 centre of the sphere by the diameters of the sections. But the 
 diameters which lie in the principal plane already spoken of, 
 and are terminated by two sides of the cone, are chords of the 
 great circle in which that plane intersects the sphere ; and the 
 halves of the angles which they subtend at its centre are equal 
 to the angles in the greater segments of which they are the 
 chords, and consequently equal to the two adjacent acute angles 
 of the quadrilateral which has these chords for its diagonals. 
 Hence, as two opposite angles of the quadrilateral are together 
 equal to two right angles, it follows that the four angles of the 
 quadrilateral represent the four angles, the obtuse as well as.the 
 acute angles, which the tangent plane of the cone makes with the 
 planes of the directive sections ; the two angles of the quadrila- 
 teral which lie opposite to the same diagonal being equal to the 
 acute and obtuse angles made by the tangent plane with the 
 plane of the section of which that diagonal is the diameter. 
 
 Thus any two adjacent angles of the quadrilateral may be 
 taken for the angles which the tangent plane of the cone makes 
 with the directive planes. If we take the two adjacent angles 
 which lie in the same triangle with the angle K contained by the 
 
298 On the Surfaces of the Second Order. 
 
 two sides of the cone that help to form the quadrilateral, the 
 sum of these two angles will be equal to two right angles 
 diminished by K ; and if we take the two remaining angles of 
 the quadrilateral, their sum will be equal to two right angles 
 increased by K ; both which sums are constant. But if we take 
 either of the other pairs of adjacent angles, the difference of the 
 pair will be constant, and equal to K. 
 
 The same conclusion may be deduced as a property of the 
 spherical conic. Let a great circle touching this curve be inter- 
 sected in two points, one on each side of the point of contact, by 
 the two directive circles, that is, by two great circles whose 
 planes are directive planes of the cone which passes through the 
 conic and has its vertex at the centre of the sphere. Since the 
 right lines in which the tangent plane of a cone intersects the 
 directive planes are equally inclined to the side of contact, the 
 arc intercepted between the points where the tangent circle of 
 the conic intersects the directive circles is bisected in the point of 
 contact ; therefore, either of the spherical triangles whose base is 
 the tangent arc so intercepted, and whose other two sides are the 
 directive circles, has a constant area ; because, if we suppose the 
 tangent arc to change its position through an indefinitely small 
 angle, and to be always terminated by the directive circles, the 
 two little triangles bounded by its two positions and by the two 
 indefinitely small directive arcs which lie between these posi- 
 tions, will have their nascent ratio one of equality, so that the 
 area of either of the spherical triangles mentioned above will 
 not* be changed by the change in the position of its base. But 
 in each of these triangles the angle opposite the base is constant ; 
 therefore the sum of the angles at the base is constant. 
 
 From this reasoning it appears that if a spherical triangle 
 have a given area, and two of its sides be fixed, the third side 
 will always touch a spherical conic having the fixed sides for its 
 directive arcs, and will be always bisected in the point of contact. 
 
 7. The intersection of any given central surface of the 
 second order with a concentric sphere is a spherical conic, since 
 the cone which passes through the curve of intersection, and has 
 
On the Surfaces of the Second Order. 299 
 
 its vertex at the common centre, is of the second order. The 
 cylinder also, which passes through the same curve and has its 
 side parallel to any of the axes of the given surface, is of the 
 second order ; and the cone, the cylinder, and the given sur- 
 face are condirective, that is, the directive planes of one of them 
 are also the directive planes of each of the other two. This may 
 be seen from the equations of the different surfaces ; for, in ge- 
 neral, two surfaces whose principal planes are parallel will he 
 condirective, if, when their equations are expressed by co-ordi- 
 nates perpendicular to these planes, the differences of the coeffi- 
 cients of the squares of the variables in the equation of the one 
 be proportional to the corresponding differences in the equation 
 of the other. 
 
 If any given surface of the second order be intersected by a 
 sphere whose centre is any point in one of the principal planes, 
 the cylinder passing through the curve of intersection, and hav- 
 ing its side perpendicular to that principal plane, will be of the 
 second order, and will be condirective with the given surface. 
 This cylinder, when its side is parallel to the directive axis, is 
 hyperbolic ; otherwise it is elliptic. If a paraboloid be cut by 
 any plane, the cylinder which passes through the curve of sec- 
 tion, and has its side parallel to the axis of the paraboloid, will 
 be condirective with that surface ; and it will be elliptic or hy- 
 perbolic, according as the paraboloid is elliptic or hyperbolic.* 
 
 If two concentric surfaces of the second order be reciprocal 
 polars with respect to a concentric sphere, the directive axis of 
 the one surface will coincide with the mean axis of the other, 
 and the directive planes of the one will be perpendicular to the 
 asymptotes of the focal hyperbola of the other. When one of 
 the surfaces is a hyperboloid, the other is a hyperboloid of the 
 same kind ; the asymptotes of the focal hyperbola of each sur- 
 
 * I have introduced the terms directive and condirective, as more general than 
 the terms cyclic and biconcyclic employed by M. Chasles. The latter terms suggest 
 the idea of circular sections, and therefore could not properly he used with refe- 
 rence to the hyperbolic paraboloid, or to the hyperbolic or parabolic cylinder, in each 
 of which surfaces a directive section is a right line. 
 
3OO On the Surfaces of the Second Order. 
 
 face are the focal lines of its asymptotic cone ; and the two 
 asymptotic cones are reciprocal. 
 
 When any number of central surfaces of the second order 
 are confocal, or, more generally, when their focal hyperbolas 
 have the same asymptotes, it is obvious that their reciprocal sur- 
 faces, taken with respect to any sphere concentric with them, are 
 all condirective. 
 
 8. If a diameter of constant length, revolving within a 
 given central surface, describe a cone having its vertex at the 
 centre, the extremities of the diameter will lie in a spherical 
 conic. And if I the cone be touched by any plane, the side of 
 contact will evidently be normal to the section which that plane 
 makes in the given surface, and will therefore be an axis of the 
 section. As the axes of a section always bisect the angles made 
 by the two right lines in which its plane intersects the directive 
 planes of the surface, and as the cone aforesaid has the same 
 directive planes with the given surface, it follows that the right 
 lines in which a tangent plane of a cone cuts its directive planes 
 are equally inclined to the side of contact a theorem which has 
 been already obtained in another way. 
 
 If a section be made in a given central surface by any plane 
 passing through the centre, the cone described by a constant 
 semidiameter equal to either semiaxis of the section will touch 
 the plane of section ; for if it could cut that plane, a semiaxis 
 would be equal to another radius of the section. Denoting by 
 r, / the semiaxes of the section, conceive two cones to be de- 
 scribed by the revolution of two constant semidiameters equal 
 to r and / respectively. These cones are condirective with the 
 given surface, and have the plane of section for their common 
 tangent plane. Supposing that surface to be expressed by the 
 equation 
 
 and the directive axis to be that of y, the axis of x will be the 
 internal axis of one cone, say of that described by r, and the axis 
 
On the Surfaces of the Second Order. 301 
 
 of z will be the internal axis of the other cone. Let K be the angle 
 made by the two sides of the first cone which lie in the plane 
 xz, and K the angle made by the two sides of the second cone 
 which lie in the same plane ; the former angle being taken so 
 as to contain the axis of x within it, and the latter so as to con-, 
 tain within it the axis of is. Then, considering r, / as radii of 
 the section xz of the surface, we have obviously 
 
 1 _ cos 2 Jic sin 2 K _ j_ /" 1 1\ j /I 1\ 
 
 (3) 
 1 cos 2 |K' sin 2 |K'_., /I IVifi.l 
 
 / 2 ~ $ P ~*\p + ii)~*\p~ii 
 
 observing that when these formulae give a negative value for r 2 
 or r' 2 , in which case the surface expressed by the equation (2) 
 must be a hyperboloid, the direction of r or / meets, not that 
 surface, but the surface of the conjugate hyperboloid expressed 
 by the equation 
 
 Now calling 9 and 0' the angles made by the tangent plane of 
 the cones with the directive planes of the given surface, which 
 are also the directive planes of each cone, the angles /c, K depend 
 on the sum or difference of and 0'. If the latter angles be 
 taken so that their sum may be equal to the supplement of K, 
 their difference will be equal to K', and the f ormulse (3) will 
 become . 
 
 (5) 
 
 by which the semiaxes of any central section are expressed in 
 terms of the non- directive semiaxes of the surface, and of the 
 
3O2 On the Surfaces of the Second Order. 
 
 angles which the plane of section makes with the directive 
 planes.* 
 
 9. From the centre of the surface expressed by equa- 
 tion (2) let a right line OS be drawn cutting perpendicularly in 
 S the plane which touches the surface at S. Let a denote the 
 length of the perpendicular OS, and a, j3, y the angles which it 
 makes with #, y, z. Then 
 
 a 2 = P COS 2 a + Q COS 2 ]3 + R COS 2 y. (6) 
 
 From this formula it is manifest that, if three planes touching 
 the surface be at right angles to each other, the sum of the 
 squares of their perpendicular distances from the centre will be 
 equal to the constant quantity P + Q + R, and therefore the 
 point of intersection of the planes will lie in the surface of a 
 given sphere. If another surface represented by the equation 
 
 be touched by a plane cutting OS perpendicularly in S () , and if 
 (T be the length of OS , then 
 
 do 2 = Po cos 2 a + Qo cos 2 j3 + -Ro cos 2 7 ; 
 
 and therefore when the two surfaces are confocal, that is, when 
 P - Po = Q - Qo = R - R, = k, 
 
 we have a 2 - cr 2 = k, which is a constant quantity. Hence, if 
 three confocal surfaces be touched by three rectangular planes, 
 the sum of the squares of the perpendiculars dropped on these 
 planes from the centre will be constant, and the locus of the 
 intersection of the planes will be a sphere. 
 
 The focal curves of a given surface are the limits of surfaces 
 confocal with it,f when these surfaces are conceived, by the pro- 
 
 * See Transactions of the Royal Irish Academy, VOL. xxi., as before cited. 
 The formulae (5) were first given, for the case of the ellipsoid, by Fresnel, in his 
 Theory of Double Refraction, Memoir es de VInstitut, torn, vii., p. 155. 
 
 f It was by this consideration, arising out of the theorems given in this and the 
 next section about confocal surfaces, that I was led to perceive the nature of the 
 
On the Surfaces of the Second Order. 303 
 
 gressive diminution of their mean or secondary axes, to become 
 flattened, and to approach more and more nearly to *a^ plane 
 passing through the primary axis. And it will appear here- 
 after, that if a bifocal right line, that is, a right line passing 
 through both focal curves, be the intersection of two planes 
 touching these curves, those two planes will be at right angles 
 to each other. Therefore the locus of the .point where a tangent 
 plane of a given central surface is intersected perpendicularly 
 by a bifocal right line is a sphere. The primary axis of the sur- 
 face is evidently the diameter of this sphere. 
 
 Hence we conclude that the locus of the point where a tan- 
 gent plane of a paraboloid is intersected perpendicularly by a 
 bifocal right line is a plane touching the paraboloid at its vertex. 
 For a paraboloid is the limit of a central surface whose primary 
 axis is prolonged indefinitely in one direction, and a plane is 
 the corresponding limit of the sphere described on that axis as 
 diameter. As this consideration is frequently of use in deduc- 
 ing properties of paraboloids from those of central surfaces, it 
 may be well to state it more particularly. It is to be observed, 
 then, that the indefinite extension of the primary axis at one 
 extremity may take place according to any law which leaves the 
 other extremity always at a finite distance from a given point, 
 andj gives a finite limiting parameter to each of the principal 
 sections of the surface which pass through that axis. The 
 simplest supposition is, that one extremity of the axis and the 
 adjacent foci of those two principal sections remain fixed, while 
 the other extremity and the other foci move off, with the centre, 
 to distances which are conceived to increase without limit. Then, 
 at any finite distances from the fixed points, the focal curves 
 approach indefinitely to parabolas, as do also all sections of the 
 surface which pass through the primary axis, while the surface 
 itself approaches indefinitely to a paraboloid ; so that the limit 
 
 focal curves, and the analogy between their points and the foci of conies. And I 
 regarded that analogy as fully established when I found (in March or April, 1832) 
 that the normal at any point of a surface of the second order is an axis of the cone 
 which has that point for its vertex and a focal for its base. 
 
304 On the Surfaces of the Second Order. 
 
 of the central surface is a "paraboloid having parabolas for its 
 focal curves. The limit of an ellipsoid, or of a hyperboloid of 
 two sheets, is an elliptic paraboloid, having one of its focals 
 modular and the other umbilicar, like each of the central sur- 
 faces from which it may be derived ; and the limit of a hyper- 
 boloid of one sheet is a hyperbolic paraboloid, having, like that 
 hyperboloid, both its focals modular. 
 
 10. Let the plane touching at S the surface expressed by 
 equation (2) intersect the axis of x in the point X, and let the 
 normal applied at S intersect the planes yz 9 xz, xy y in the points 
 L, M, N respectively. Since the section made in the surface by 
 a plane passing through OX and the point S has one of its axes 
 in the direction of OX, it appears, by an elementary property of 
 conies, that the rectangle under OX and the co-ordinate x of 
 the point S is equal to the quantity P ; but that co-ordinate is 
 to LS as OS or a is to OX, and therefore the rectangle under er 
 and LS is equal to P. Similarly the rectangle under <r and MS 
 is equal to Q, and the rectangle under cr and NS is equal to R. 
 Thus the parts of the normal intercepted between the point S 
 and each of the principal planes are to each other as the squares 
 of the semiaxes respectively perpendicular to these planes ; the 
 square of an imaginary semiaxis being regarded as negative, 
 and the corresponding intercept being measured from S in a 
 direction opposite to that which corresponds to a real semiaxis. 
 
 The rectangle under <r and the part of the normal inter- 
 cepted between two principal planes is equal to the difference 
 of the squares of the semiaxes which are perpendicular to these 
 planes. This r^tfangle is therefore constant, not only for a given 
 surface, but for all surfaces which are confocal with it. 
 
 Hence the part of the normal intercepted between two prin- 
 cipal planes bears a given ratio to the part of it intercepted 
 between one of these and the third principal plane, whether the 
 normal be applied at any point of a given surface, or at any 
 point of a surface confocal with it. 
 
 If therefore normals to a series of confocal surfaces be all 
 parallel to a given right line, they must all lie in the same plane 
 
On the Surfaces of the Second Order. 305 
 
 passing through the common centre of the surfaces, because 
 otherwise the parts of any such normal, which are intercepted 
 between each pair of principal planes, would not be in a constant 
 ratio to each other. 
 
 The point S being the point at which any of these parallel 
 normals is applied, the plane touching the surface at S is parallel 
 to a given plane ; the perpendicular OS dropped upon it from 
 the centre has a given direction, the plane OSS is fixed, and the 
 directions of the lines OL, OM, ON, in which the plane intersects 
 the principal planes, are also fixed. And as the angle OSS is 
 always a right angle, and the normal at S is always parallel to 
 GS, the distance SS bears a given ratio to each of the distances 
 OL, OM, ON, and therefore also to each of the intercepts MN. 
 Hence, since the rectangle under OS and any one of these in- 
 tercepts is constant, the rectangle under OS and SS is constant. 
 
 Therefore if a series of confocal central surfaces be touched 
 by parallel planes, the points of contact will all lie in one plane, 
 and their locus, in that plane, will be an equilateral hyperbola, 
 having its centre at the centre of the surfaces, and having one 
 of its asymptotes perpendicular to the tangent planes. This 
 hyperbola evidently passes through two points on each of the 
 focal curves, namely the points where the tangent to each curve 
 is parallel to the tangent planes. 
 
 If a series of confocal paraboloids be touched by parallel 
 planes, it will be found that the points of contact all lie in a 
 bifocal right line, and that the normals at these points lie in a 
 plane parallel to the axis of the surfaces ; so that the part of 
 any normal which is intercepted by the two principal planes is 
 constant. This theorem may be proved from the two following 
 properties of the paraboloid : 1. A normal being applied to the 
 surface at the point S, the segments of the normal, measured 
 from S to the points where it intersects the planes of the two 
 principal sections, are to each other inversely as the parameters 
 of these sections. 2. Supposing the axis x to be that of the sur- 
 face, the difference between the co-ordinates x of the point S and 
 of the point where the normal meets the plane of the principal 
 
306 On the Surfaces of the Second Order. 
 
 sections, is equal to the semiparameter of the other principal 
 section. 
 
 11. Let a tangent plane, applied at any point S of a surface 
 of the second order, intersect the plane of one of its focals in the 
 right line 9, and let P be the foot of the perpendicular dropped 
 from S upon the latter plane. The pole of the right line 0, with 
 respect to the principal section lying in this plane, is the point 
 P. Let N be its pole with respect to the focal. Then if T be 
 any point of the right line 0, the polar of this point with respect 
 to the section will pass through P, and its polar with respect to 
 the focal will pass through N ; and if the former polar intersect 
 the diligent curve in A, A', and the latter intersect the focal in 
 F, F', the points F, F' will correspond respectively to the points 
 A, A', and the distances A A' and FF' will be similarly di- 
 vided by the points P and N (see Part I. 8). But since the 
 point S is in the plane of the two directrices which pass through 
 A and A', the lengths AP and A'P, which are the perpendicular 
 distances of S from the directrices, are proportional to the lengths 
 FS and F'S. Therefore FN is to F'N as FS is to FS, and the 
 right line NS bisects one of the angles made by the right lines 
 FS and F'S. And as this holds wherever the point T is taken 
 on the right line 9, that is, in whatever direction the right line 
 FF' passes through the point N, it follows that the right line 
 NS is an axis of the cone which has the point S for its vertex 
 and the focal for its base. Further, if FF' intersect in the 
 point Q, we have FN to F'N as FQ is to F'Q, because N is the 
 pole of with respect to the focal ; therefore FQ, is to F'Q, as 
 FS is to F'S, and hence the right line QS also bisects one of the 
 angles made by FS and F'S. The right lines NS and QS are 
 therefore at right angles to each other ; and as the latter always 
 lies in the tangent plane, the former must be perpendicular to 
 that plane. 
 
 Consequently the normal at any point of a surface of the 
 second order is an axis of the cone which has that point for its 
 vertex and either of the focals for its base. 
 
 It is known that when two confocal surfaces intersect each 
 
On the Surfaces of the Second Order. 307 
 
 other, they intersect everywhere at right angles ; and that 
 through any given point three surfaces may in general be de- 
 scribed, which shall have the same focal curves. If three con- 
 focal surfaces pass through the point S, the normal to each of 
 them at S is an axis of each of the cones which stand on the 
 f ocals and have S for their common vertex. The normals to the 
 three surfaces are therefore the three axes of each cone. 
 
 If the points at which a series of confocal surfaces are touched 
 by parallel planes be the vertices of cones having one of the 
 f ocals for their common base, each of these cones will have one 
 of its axes perpendicular to the tangent planes. Therefore when 
 an axis of a cone which stands on a given base is always parallel 
 to a given right line, the locus of the vertex is an equilateral 
 hyperbola or a right line, according as the base is a central conic 
 or a parabola. 
 
 12. A system of three confocal surfaces intersecting each 
 other consists of an ellipsoid, a hyperboloid of one sheet, and a 
 hyperboloid of two sheets, if the focals be central conies ; but it 
 consists of two elliptic paraboloids and a hyperbolic paraboloid, 
 if the focals be parabolas. In the central system, the ellipsoid 
 has the greatest primary axis, and the hyperboloid of two sheets 
 the least ; and the focal which is modular in one of these sur- 
 faces is umbilicar in the other. The asymptotic cones of the 
 hyperboloids are confocal, the focal lines of each cone being the 
 asymptotes of the focal hyperbola. In the system of parabo- 
 loids, the two elliptic paraboloids are distinguished by the cir- 
 cumstance that the modular focal of the one is the umbilicar 
 focal of the other. 
 
 The curve in which two confocal surfaces intersect each other 
 is a line of curvature of each, as is well known ;* and a series of 
 lines of curvature on a given surface are found by making a 
 series of confocal surfaces intersect it. 
 
 Now if a series of the lines of curvature of a given surface 
 be projected on one of its directive planes by right lines parallel 
 
 * See Dupin's Developpements de Geometrie. 
 
308 On the Surfaces of the Second Order. 
 
 to either of its non-directive axes, the projections will be a series 
 of confocal conies ; and when the surface is umbilicar, the foci 
 of all these conies will be the corresponding projections of the 
 umbilics.* When the surface is not umbilicar, its directive axis 
 will be parallel to the primary axis of the projections. 
 
 The same line of curvature has two projections, according as 
 it is projected by right lines parallel to the one or to the other 
 non-directive axis. In the ellipsoid these projections are always 
 curves of different kinds, the one being an ellipse when the other 
 is a hyperbola ; but in a hyperboloid the projections are either 
 both ellipses or both hyperbolas. In the hyperbolic paraboloid 
 the projections are parabolas. In the elliptic paraboloid one 
 of the projections is always a parabola, and the other is either 
 an ellipse or a hyperbola. 
 
 The corresponding projections of two lines of curvature which 
 pass through a given point of the surface are confocal conies in- 
 tersecting each other in the projection of that point, and of 
 course intersecting at right angles. 
 
 13. A bifocal chord is a bifocal right line terminated both 
 ways by the surf ace. f In a central surface, the length of a 
 bifocal chord is proportional to the square of the diameter which 
 is parallel to it ; the square of the diameter being equal to the 
 rectangle under the chord and the primary axis. 
 
 More generally, if a chord of a given central surface touch 
 two other given surfaces confocal with it, the length of the chord 
 will be proportional to the square of the parallel diameter of the 
 first surface, the square of the diameter being equal to the rect- 
 angle under the chord and a certain right line 21, determined by 
 the formula 
 
 ~ 
 
 wherein it is supposed that the equation (2) represents the first 
 
 * "Exam. Papers," An. 1838, p. xlvi., question 4; p. xcix., question 70. 
 t The theorems in 13 are now stated for the first time. 
 
On the Siirfaces of the Second Order. 309 
 
 surface, and that P', P" are the quantities corresponding to P 
 in the equations of the other two surfaces. 
 
 In any surface of the second order, the lengths of two bifocal 
 chords are proportional to the rectangles under the segments of 
 any two intersecting chords to which they are parallel. 
 
 In the paraboloid expressed by the equation 
 
 y* z* 
 
 + - = X 
 
 P (1 
 
 if x be the length of a bifocal chord making the angles ]3 and y 
 with the axes of y and 2 respectively, we have 
 
 1 = cos 2 /3 + cos 2 7 
 X P 2 
 
 14. At the point S on a given central surface expressed by 
 the equation (2), let a tangent plane be applied, and let k, k' 
 be the squares of the semiaxes of a central section made in the 
 surface by a plane parallel to the tangent plane ; each of the 
 quantities k, k f being positive or negative, according as the 
 corresponding semiaxis of the section is real or imaginary, that 
 is, according as it meets the given surface or not. Then the 
 equations* of two other surfaces confocal with the given one, 
 and passing through the point S, are 
 
 f * 2 _-. ** f * , m 
 
 P~~k Q-k -k~ ' P-k f+ Q-V R-k'~ 
 
 The given surface is intersected by these two surfaces respec- 
 tively in the two lines of curvature which pass through the 
 point S ; the tangent drawn to the first line of curvature at S is 
 parallel to the second semiaxis of the section, and the tangent 
 drawn to the second line of curvature at S is parallel to the first 
 semiaxis of the section. 
 
 When two confocal surfaces intersect, the normal applied to 
 
 * " Exam, Papers," An. 1837, p.c., questions 4, 5, 6 ; An,1838,p. c., questions 71, 72. 
 
3io On the Surfaces of the Second Order. 
 
 one of them at any point S of the line of curvature formed by 
 their intersection lies in the tangent plane of the other, and is 
 parallel to an axis of any section made in the latter by a plane 
 parallel to the tangent plane. Supposing the surfaces to be cen- 
 tral, if two normals be applied at the point S, and a diameter of 
 each 'surface be drawn parallel to the normal of the other, the 
 two diameters so drawn will be equal and of a constant length, 
 wherever the point S is taken on the line of curvature ; the 
 square of that length being equal to the difference of the squares 
 of the primary axes of the surfaces, and the diameter of the sur- 
 face which has the greater primary axis being real, while that of 
 the other surface is imaginary. As the point S moves along the 
 line of curvature, each constant diameter describes a cone condi- 
 rective with the surface to which it belongs ; the two cones so 
 described are reciprocal, and the focal lines of the cone which 
 belongs to one surface are perpendicular to the directive planes 
 of the other surface. 
 
 When two confocal paraboloids intersect, if normals be ap- 
 plied to them at any point S of their intersection, and a bifocal 
 chord of each surface be drawn parallel to the normal of the 
 other, the two chords so drawn will be equal and of a constant 
 length, wherever the point S is taken in the line of intersection 
 of the surfaces ; that constant length being equal to the differ- 
 ence between the parameters of either pair of coincident principal 
 sections. 
 
 15. The point S being the common intersection of a given 
 system of confocal surfaces, of which the equations are 
 
 (10) 
 
 P 77 * Q* + K' = l ' 
 
 suppose that another surface A confocal with these, and ex- 
 pressed by the equation 
 
On the Surfaces of the Second Order. 311 
 
 is circumscribed by a cone having its vertex at S. If the nor- 
 mals applied at S to the given surfaces, taken in the order of the 
 equation (10), be the axes of new rectangular co-ordinates ?, r?, ?, 
 the equation of the cone, referred to these co-ordinates, will be* 
 
 The surfaces of the given system, in the order of their 
 equations, may be supposed to be an ellipsoid, a hyperboloid 
 of one sheet, and a hyperboloid of two sheets; the axes of 
 x, y, z being respectively the primary, the mean, and the 
 secondary axes of each surface. Then P is greater than P', 
 and F greater than P". 
 
 The normals to the given surfaces are the axes of the cone 
 expressed by the equation (12); and if the surface A be 
 changed, but still remain confocal with the given system, it 
 is obvious from that equation that the focal lines of the cir- 
 cumscribing cone will remain unchanged, since the differences 
 of the quantities by which the squares of , TJ, are divided 
 are independent of the surface A. As P' is intermediate in 
 value between P and P", the normal to the hyperboloid of 
 one sheet is always the mean axis of the cone ; the focal lines 
 lie in the plane ?, and their equation is 
 
 * The equation (12) was obtained in the year 1832, and was given at my 
 Lectures in Hilary Term, 1836. The most remarkable properties of cones circum- 
 scribing confocal surfaces are immediate consequences of this equation. That 
 such cones, when they have a common vertex, are confocal, their focal lines 
 being the generatrices of the hyperboloid of one sheet passing through the 
 vertex, was first stated by Professor C. G. J. Jacobi, of Konigsberg, in 1834 
 (see Crelle's Journal, VOL. xn. p. 137). See also the excellent work of M. Chasles, 
 published in 1837, and entitled Aperyu historique sur VOrigine et le Developpe- 
 ment des Methodes en Geometrie, p. 387. The analogy which exists between the 
 focals of surfaces and the foci of curves of the second order was supposed by 
 M. Chasles to have been pointed out in that work for the first time (Gomptes 
 Rendus, torn, xvi., pp. 833, 1106) ; but that analogy had been previously taught and 
 developed in the Lectures just alluded to. 
 
312 On the Surfaces of the Second Order. 
 
 which shows that they are parallel to the asymptotes of a 
 central section made in the hyperboloid of one sheet by a 
 plane parallel to the plane , since the quantities P - P and 
 P -P" are (including the proper signs) the squares of the 
 semiaxes of the section which are parallel to and re- 
 spectively. The focal lines are therefore the generatrices of 
 that hyperboloid at the point S. 
 
 When Bo = 0, the equation (12) becomes 
 
 which is that of the cone standing on the focal ellipse and 
 having its vertex at S. When Q = 0, the same equation 
 becomes 
 
 | + J+| = 0, (15) 
 
 which is that of the cone standing on the focal hyperbola, 
 and having its vertex at S. The normal to the hyperboloid 
 of one sheet at the point S is the mean axis of both cones ; 
 the normal to the ellipsoid is the internal axis of the first 
 cone and the directive axis of the second, while the normal 
 to the hyperboloid of two sheets is the directive axis of the 
 first and the internal axis of the second. 
 
 The three surfaces expressed by the equations 
 
 _- 
 
 P F F f Q Q' Q"""' 
 
 (16) 
 
 * + 5l + *L-l 
 M R' R" ~ 
 
 are a confocal system, having their centres at S, and being re- 
 spectively an ellipsoid, a hyperboloid of one sheet, and a hyper- 
 boloid of two sheets. They intersect each other in the centre 
 of the system expressed by the equations (10), and their normals 
 at that point are the axes of #, /, z respectively. The relations 
 
On the Surfaces of the Second Order. 313 
 
 between the two systems of surfaces are therefore perfectly reci- 
 procal. From the equations (14) and (15) it is manifest that 
 the asymptotic cones of the hyperboloids of one system pass 
 through the focals of the other. 
 
 16. The point S being the intersection of a given system 
 of confocal paraboloids whose equations are 
 
 p q' p' / " 
 
 (17) 
 
 where p-p'=q-q'=4: (h-ti), and p -p" = q-tf' = 4 (h - h") : 
 suppose that another paraboloid A confocal with these, afid 
 expressed by the equation 
 
 V* z 2 
 
 y - + - = x + h Q , (18) 
 
 Po qo 
 
 is circumscribed by a cone having its vertex at S. Then if the 
 normals applied at S to the given system of surfaces, taken in 
 the order of their equations, be the axes of the co-ordinates 
 , j, ? respectively, the equation ef the circumscribing cone will 
 be 
 
 -^- + -^ + -T^ = ; (19) 
 
 p-po p -po p -p 
 
 showing that those normals are the axes of the cone, and that 
 the focal lines of the cone are independent of the surface A, 
 provided it be confocal with the given surfaces. If the hyper- 
 bolic paraboloid be the second surface of the given system, the 
 parameter p' will be intermediate in value between p and p", 
 and the equation of the focal lines of the cone will be 
 
 (20) 
 
 which is the equation of a pair of right lines parallel to the 
 asymptotes of a section made in the hyperbolic paraboloid by 
 
314 On the Surfaces of the Second Order. 
 
 a plane parallel to the plane ?, since the quantities p 1 -p and 
 p' ' -p" are proportional to the squares of the semiaxes of the 
 section which are parallel to ? and respectively. The focal 
 lines are therefore the generatrices of the hyperbolic paraboloid 
 at the point S. 
 
 Putting p and q alternately equal to zero in the equation 
 (19), we get 
 
 ? + &- : i, M+-0, (21) 
 
 p p' p" q q' q" 
 
 the equations of two cones which have a common vertex at S, 
 the first of them standing on the focal which lies in the plane 
 ass, the second on the focal which lies in the plane xy. The 
 mean axis of each of these cones is the normal at S to the 
 hyperbolic paraboloid; the internal axis of either cone is the 
 normal to the elliptic paraboloid which has the base of that 
 cone for its modular focal. 
 
 As the cones which have a common vertex, and stand on 
 the focals of any surface of the second order, are confocal, they 
 intersect at right angles. Therefore when two planes passing 
 through a bifocal right line touch the focals, these planes are 
 at right angles to each other. And as cones which have a 
 common vertex, and circumscribe confocal surfaces, are confo- 
 cal, two such cones, when they intersect each other, intersect at 
 right angles. Therefore when a right line touches two confo- 
 cal surfaces, the tangent planes passing through this right line 
 are at right angles to each other. 
 
 17. When two surfaces are reciprocal polars* with respect 
 to any sphere, and one of them is of the second order, the other 
 is also of the second order. Let the surface B be reciprocal to 
 the surface A before mentioned, with, respect to a sphere of 
 which the centre is S ; and suppose B' and E to be any cor- 
 responding points on these surfaces. Then the plane which 
 
 * Transactions of the Eoyal Irish Academy, VOL. xvn. p. 241 ; "Examination 
 Papers," An. 1841, p. cxxvi., question 4. 
 
On the Surfaces of the Second Order. 315 
 
 touches the surface A at the point R intersects the right line 
 SR' perpendicularly in a point K, such that the rectangle under 
 SR' and SK is constant, being equal to the square of the radius 
 of the sphere. Now if the point K approach indefinitely to S, 
 the distance SR' will increase without limit, the surface B being 
 of course a hyperboloid ; and if through S any plane be drawn 
 touching the surface A, a right line perpendicular to this plane 
 will evidently be parallel to a side of the asymptotic cone of the 
 hyperboloid. The asymptotic cone of B is therefore reciprocal 
 to the cone which, having its vertex at S, circumscribes the 
 surface A. Hence, as the directive planes of a hyperboloid 
 are the same as those of its asymptotic cone, it follows that the 
 directive planes of the surface B are perpendicular to the gene- 
 ratrices of the hyperboloid of one sheet, or the hyperbolic para- 
 boloid, which passes through S, and is confocal with the surface 
 A. And this relation between two reciprocal surfaces ought to 
 be general, whatever be the position of the point S with respect 
 to them ; * for though it has been deduced by the aid of the 
 circumscribing cone aforesaid, it does not, in its enunciation, 
 imply the existence of such a cone. This conclusion may be 
 verified by investigating the equation of the surface B in terms 
 of the co-ordinates , r?, . Suppose p to be the radius of the 
 sphere with respect to which the surfaces A and B are recipro- 
 cal. Then if A be a central surface expressed by the equation 
 (11), and having > ?o, So for the co-ordinates of its centre, the 
 surface B will be represented by the equation 
 
 (P-P )V + (P' - Po) r) 2 + (P" - Po) V 
 
 (22) 
 
 = 2 j o 2 (? ? + ^ + 2oS)- / oS 
 
 but if A be a paraboloid expressed by the equation (18), the 
 equation of B will be 
 
 (p-p Q ) + (p'-p}r? + (p"-p Q }? 
 
 (23) 
 = p4 2 (? cos a + rj cos ]3 + 2 cos y), 
 
3 1 6 On the Surfaces of the Second Order. 
 
 where a, j3, y are the angles which the axis of x makes with the 
 axes of ?, TJ, ? respectively. In the first case, the equation (22) 
 shows that the directive planes of B are perpendicular to the 
 right lines expressed by the equation (13) ; in the second case, 
 the equation (23) shows that the directive planes of B are per- 
 pendicular to the right lines expressed by the equation (20). 
 
 When the surface A is a paraboloid, and the distance of the 
 point E from its vertex is indefinitely increased, the plane 
 touching the surface at E approaches indefinitely to parallel- 
 ism with its axis, and the right line SK, perpendicular to that 
 plane, increases without limit. Therefore the surface B passes 
 through the point S, and is touched in that point by a plane 
 perpendicular to the axis of A. 
 
 When the point S lies upon the surface A, the co-efficient 
 of the square of one of the variables, in the equation (22) or 
 (23), is reduced to zero, and the surface B is a paraboloid 
 having its axis parallel to the normal applied at S to the 
 surface A. This also appears from considering that when S is 
 a point of the surface A, the normal at that point is the only 
 right line passing through S, which meets the surface B at an 
 infinite distance. 
 
 If a series of surf aces -be confocal, their reciprocal surfaces, 
 taken with respect to any given sphere, will be condirective. 
 
 When the equations of any two condirective surfaces are ex- 
 pressed by co-ordinates perpendicular to their principal planes, 
 the constants in the equations may be always so taken that the 
 differences of the co- efficients of the squares of the variables in 
 one equation shall be equal to the corresponding differences in 
 the other. Then, by subtracting the one equation from the 
 other, we get the equation of a sphere. Therefore when two 
 condirective surfaces intersect each other, their intersection is, 
 in general, a spherical curve. But when the surfaces are two 
 paraboloids of the same species, their intersection is a plane 
 curve. 
 
 18. Through any point S of a given surface four bifocal 
 right lines may in general be drawn. Supposing the surface 
 
On the Surfaces of the Second Order. 3 1 7 
 
 to be central, let a plane drawn through the centre, parallel to 
 the plane which touches the surface at S, intersect any one of 
 these right lines. Then the distance of the point of intersec- 
 tion from the point S will always be equal to the primary semi- 
 axis of the surface.* 
 
 If through any point S of a given central surface a right 
 line be drawn touching two other given surfaces confocal with 
 it, and if this right line be intersected by a plane drawn 
 through the centre parallel to the plane which touches the 
 first surface at S, the distance of the point of intersection from 
 the point S will be constant, wherever the point S is taken on 
 the first surface. If this constant distance be called /, and the 
 other denominations be the same as in the formula (7), the 
 wilue of I will be given by that formula, f 
 
 * " Examination Papers," An. 1838, p. xlvii., question 9. 
 
 f In the notes to the last-mentioned work of M. Chasles, on the History of 
 'Methods in Geometry^ will be found many theorems relative to surfaces of the 
 second order. Among them are some of the theorems which are given in the 
 present Paper ; but it is needless to specify these, as M. Chasles's work is so well 
 known. 
 
 
III. JSTOTE RELATIVE TO THE COMPARISON OF ARCS OF 
 CURVES; PARTICULARLY OF PLANE AND SPHERI- 
 CAL CONICS. 
 
 [Proceedings of the Royallrish Academy, VOL. n. p. 446. Read Nov. 30, 1843.] 
 
 THE first Lemma given in my Paper on the rectification of the 
 conic sections* is obviously true for curves described on any 
 given surface, provided the tangents drawn to these curves be 
 shortest lines on the surface. The demonstration remains exactly 
 the same ; and the Lemma, in this general form, may be stated 
 as follows : 
 
 Understanding a tangent to be a shortest line, and sup- 
 posing two given curves E and F to be described on a given 
 surface, let tangents drawn to the first curve at two points T, t, 
 indefinitely near each other, meet the second curve in the points 
 P, p. Then taking a fixed point A on the curve E, if we put s 
 to denote (according to the position of this point with respect 
 to T) the sum (or difference) of the arc AT and the tangent 
 TP, and s + ds to denote the sum (or difference) of the arc A^ 
 and the tangent tp, we shall have ds equal to the projection of 
 the infinitesimal arc P<? upon the tangent ; that is, if a be the 
 angle which the tangent TP makes witji the curve F at the 
 point P, we shall have ds equal to Pp multiplied by the cosine 
 of a. 
 
 Now through the points P, p conceive other tangents TT, 
 
 * Transactions of the Royal Irish Academy, VOL. xvi. p. 79 (supra, p. 20). 
 
Note relative to the comparison of Arcs of Curves. 319 
 
 t'p to be drawn, touching the curve E in the points T', t' ; and 
 let s' and els' have for these tangents the same signification which 
 s and ds have for the former tangents. Supposing the nature of 
 the curve F to be such that it always bisects, either internally or 
 externally, the angle made at the point P by the tangents TP 
 and TT, it is evident that ds = ds' 9 and therefore either s + s f 
 or s - s f is a constant quantity. 
 
 A simple example of this theorem is afforded by the plane 
 and spherical conies. If the curves E and F be two confocal 
 conies, either plane or spherical, and tangents TP, TT be 
 drawn to F from any point P of E (the tangents being of 
 course right lines when the curves are plane, and arcs of great 
 circles when they are spherical; in both cases shortest lines), 
 it is well known that the angle TPT' made by the tangents is 
 always bisected by the conic E. The angle is bisected inter- 
 nally or externally according as the conies intersect or not. 
 Hence we have the two following properties* of confocal 
 conies : 
 
 1. When two confocal conies do not intersect, if one of 
 them be touched in the points T, T' by tangents drawn from 
 any point P of the other, the sum of the tangents TP, TT will 
 exceed the convex arc TT' lying between the points of contact, 
 by a constant quantity. 
 
 2. When two focal conies intersect in the point A, if one 
 of them be touched in the points T, T' by tangents drawn from 
 any point P of the other, the difference between the tangents 
 
 * The first of these properties was originally given for spherical conies by the 
 Rev. Charles Graves, Fellow of Trinity College, in the "notes and additions" to 
 his translation of M. Chasles's Memoirs on Cones and Spherical Conies, p. 77 
 (Dublin, 1841). Mr. Graves obtained it as the reciprocal of the proposition, that 
 when two spherical conies have the same directive circles, any tangent arc of the 
 inner conic divides the outer one into two segments, each of which has a constant 
 area. Both properties, with the general theorem relative to curves described on 
 any surface and touched by shortest lines, were afterwards given in the University 
 Calendar. See " Examination Papers," An. 1841, p. xli., questions 3-6 ; An. 1842, 
 p. Ixxxiii., questions 30-34. These two properties of conies were communicated, in 
 October, 1843, to the Academy of Sciences of Paris, by M. Chasles, who supposed 
 them to be new. See the Comptes Rendus, torn, xvii. p. 838. 
 
32O Note relative to the comparison of Arcs of Curves. 
 
 TP, TT will be equal to the difference between the arcs AT, 
 AT'. 
 
 These properties give the readiest and most elegant solution 
 of problems concerning the comparison of different arcs of a 
 plane or spherical conic. Any arc being given ori a conic, we 
 may find another arc beginning from a given point, which shal] 
 differ from the given arc by a right line if the conic be plane 3 
 or by a circular arc if the conic be spherical. 
 
( 321 ) 
 
 IV. NOTE ON SURFACES OF THE SECOND ORDER. 
 
 [Proceedings of the Royal Irish Academy, VOL. m. p. 429. Read April 12, 1847.] 
 
 LET a surface A of the second order be represented by the 
 equation 
 
 its primary axis being that of x. Through a given point S 
 whose co-ordinates are a?', /', s', conceive three surfaces confocal 
 with A to be described, and let P, P', P", be the squares of 
 their primary semiaxes. Then if normals drawn to these sur- 
 faces respectively at the point S be the axes of a new system of 
 co-ordinates ?, ?j, , and if we put 
 
 P-P,-*, p'-p,"*', p"-p,-r, J + ^ + ^ /, 
 
 jfo ^0 *H> 
 
 the equation of the surface A, referred to the new co-ordinates, 
 will be 
 
 where 5 , no, So are the co-ordinates of its centre. 
 
 From the form of this equation it is evident that, if the sur- 
 face be intersected by the plane whose equation is 
 
322 Note on Surfaces of the Second Order. 
 
 it will be touched along tlie curve of intersection by the cone 
 whose equation is 
 
 This mode of deducing, in its simplest form, the equation of a 
 cone circumscribing a surface of the second order, is much easier 
 than the direct investigation by which the equation (c) was 
 originally obtained. 
 
 Let a right line passing through 8 intersect the plane ex- 
 pressed by the equation (#), in a point whose distance from 8 is 
 equal to or, while it intersects the surface A in two points, P and 
 P', the distance of either of which from 8 is denoted by p. Let 
 the surface It, represented by the equation 
 
 be intersected by the same right line in a point whose distance 
 from 8 is equal to r, the distance r being, of course, a semidia- 
 meter of this surface. Then it is obvious that the equation (a) 
 may be written 
 
 T 1\ 2 
 
 so that, if p and p represent the distances SP and 8P / respec- 
 tively, we have 
 
 1 1 1 j. _ 1 _1 n 
 
 p "w * P p' ~ v r ; 
 and therefore 
 
 This result is useful in questions relating to attraction. For 
 if A be an ellipsoid, every point of which attracts an external 
 point 8 with a force varying inversely as the fourth power of 
 the distance, and if the point 8 be the vertex of a pyramid, one 
 
Note on Surfaces of the Second Order. 323 
 
 of whose sides is the right line SPP', and whose transverse sec- 
 tion, at the distance unity from its vertex, is the indefinitely 
 small area o>, the portion PP of the pyramid will attract the 
 point S, in the direction of its length, with a force expressed by 
 the quantity 
 
 1 IN 2u> 
 
 and putting 6 for the angle which the right line SP makes with 
 the axis of , the attraction in the direction of will be 
 
 cos 
 
 Now, supposing the axis of to be normal to the confocal ellip- 
 soid described through S, it will be the primary axis of the 
 surface B, which will be a hyperboloid of two sheets ; and the 
 surface being symmetrical round this axis, it is easy to see, from 
 the expression for the elementary attraction, that the whole 
 attraction of the ellipsoid will be in the direction of . There- 
 fore when the force is inversely as the fourth power of the dis- 
 tance, the attraction of an ellipsoid on an external point is 
 normal to the confocal ellipsoid passing through that point. 
 
 Hence we infer, that if U be the sum of the quotients found 
 by dividing every element of the volume of an ellipsoid by the 
 cube of its distance from an external point, the value of U will 
 remain the same, wherever that point is taken on the surface of 
 an ellipsoid confocal with the given one. 
 
 The question of the attraction of an ellipsoid, when the law 
 of force is that of the inverse square of the distance, has been 
 treated by Poisson, in an elegant but very elaborate memoir, 
 presented to the Academy of Sciences in 1833.* In the preced- 
 ing year I had obtained the theorems just mentioned, by con- 
 sidering the law of the inverse fourth power ; and, as well as I 
 remember, they were deduced exactly as above, by setting out 
 from the equation (a). But I did not then succeed in applying 
 
 * Memoires de Vlnstitut, torn. xiii. 
 Y2 
 
324 Note on Surfaces of the Second Order. 
 
 the same method to the case where the law of force is that oi 
 nature, probably from not perceiving that, in this case, the ellip- 
 soid ought to be divided (as Poisson has divided it) into concen- 
 tric and similar shells. This application requires the following 
 theorem, which is easily proved : 
 
 Supposing A to be another ellipsoid, concentric, similar, and 
 similarly placed with A\ let the right line SPP' intersect 
 it in the points p and p' ', respectively, adjacent to P and P ; 
 then if the direction of that right line be conceived to vary, the 
 rectangle under Pp and P'p (or under Pp f and P'p') will be to 
 the rectangle under SP and SP* in a constant ratio. 
 
 Denoting the constant ratio by m y and combining this theo- 
 rem with the formula (/), we have 
 
 mr l h \ 
 
 2' 
 
 Now let the two surfaces A and A be supposed to approach in- 
 definitely near each other, so as to form a very thin shell, then 
 ultimately P'p will be equal to PP', and we shall have 
 
 where m is indefinitely small. Therefore if the point S, external 
 to the shell, be the vertex of a pyramid whose side is the right 
 line SP, and whose section, at the unit of distance, from the 
 vertex, is w, the attraction of the two portions Pp and P'p' of 
 this pyramid, which form part of the shell, will be equal to mrw. 
 Hence it appears, as before, on account of the symmetry of the 
 surface B round the axis of ?, that the whole attraction of the 
 shell on the point 8 is in the direction of that axis, and conse- 
 quently (as was found by Poisson) in the direction of the inter- 
 nal axis of the cone whose vertex is S, and which circumscribes 
 the shell. 
 
 To find the whole attraction of the shell, the expression 
 
 mr< cos 9 (0 
 
Note on Surfaces of the Second Order. 325 
 
 must be integrated. Let be the angle which a plane, passing 
 through SP and the axis of , makes with the plane j, then 
 
 to = sin 
 1 //cos 2 sin 2 cos 2 sin 2 8 sin- 
 
 ' -T + ,7 r + ,T, 
 
 When these values are substituted in (t), that expression may be 
 readily integrated, first with respect to 0, and then with respect 
 to 0. 
 
 It is evident that, by the same substitutions, the expression 
 (g) may be twice integrated. 
 
 An investigation similar to the preceding has been given by 
 M. Chasles, for the case in which the force varies inversely as 
 the square of the distance.* He uses a theorem equivalent to 
 the formula (/), but deduces it in a different way. 
 
 From what has been proved it follows that, if V be the sum 
 of the quotients found by dividing every element of the shell by 
 its distance from an external point 8, the value of V will be the 
 same wherever that point is taken on the surface S of an ellip- 
 soid confocal with the surface A of the shell. 
 
 Let S' be another ellipsoid confocal with A, and indefinitely 
 near the surface S. The normal interval between the two sur- 
 faces S and S', at any point 8 on the former, will be inversely 
 as the perpendicular dropped from the common centre of the 
 ellipsoids on the plane which touches S at 8. Hence, supposing 
 the point 8 to move over the surface S, that perpendicular will 
 vary as the attraction exerted by the shell on the point 8 9 when 
 the force is inversely as the square of the distance, or as the at- 
 traction exerted by the whole ellipsoid A on the point 8, when 
 the force is inversely as the fourth power of the distance. 
 
 When the point S is on the focal hyperbola, the integrations, 
 by which the actual attraction is found in either case, are sim- 
 plified, for the surface B is then one of revolution round the 
 axis of ?, and its semidiameter r is independent of the angle 0. 
 
 * Memoires des Savants Etr angers, torn. ix. 
 
326 Note on Surfaces of the Second Order. 
 
 From the expression for the attraction of a shell we can find, 
 by another integration, the attraction of the entire ellipsoid when 
 the law of force is that of nature. And thus the well-known 
 problem of the integral calculus, in which it is proposed to deter- 
 mine directly the attraction of an ellipsoid on an external point, 
 without employing the theorem of Ivory to evade the difficulty, 
 is solved in what appears to be the simplest manner. 
 
PART III. 
 
 ROTATION. 
 
I. ON THE ROTATION OP A SOLID BODY ROUND 
 A FIXED POINT; BEING AN ACCOUNT OP THE 
 LATE PROPESSOR MAC CULLAGH'S LECTURES ON 
 THAT SUBJECT.* COMPILED BY THE REV. SAMUEL 
 HAUGHTON, FELLOW OP TRINITY COLLEGE, DUBLIN. 
 
 [Transactions of the Royal Irish Academy, VOL. xxn. p. 139. Read April 23, 1849.] 
 
 I. COMPOSITION or EOTATIONS. 
 
 LET be the intersection of two axes of rotation, OB,, OR' ; 
 and let the magnitudes of the rotations be represented by w, <*/; 
 then the motion impressed upon the body by these two rotations 
 will be the same as the motion produced by a single rotation 
 round an axis, which is represented in magnitude and position 
 by the diagonal of the parallelogram formed by w, a/. For, 
 draw, through any point I of the body a plane perpendicular to 
 the line 01, and project upon this plane the parallelogram 
 formed by w, a/ ; the sides of this parallelogram will be & sin 
 EOT and a/ sin R'OI. Now the velocities impressed upon the 
 
 * This Essay" On the Rotation of a Solid Body round a Fixed Point," 
 has been compiled from my notes of Professor Mac Cullagh's Lectures, delivered 
 in the Hilary Term of the year 1844, in Trinity College. A short account of some 
 of the results contained in it was published by Professor Mac Cullagh himself, in 
 the Proceedings of the Royal Irish Academy. f As it has appeared to many of Mr. 
 Mac Cullagh's friends desirable that a somewhat more detailed account of his re- 
 searches in this subject should be published, I have, in accordance with this desire, 
 drawn up and presented to the Academy an account of his Lectures on 
 Rotation. I have endeavoured to arrange the subject in a systematic order, and 
 to give the results proved by Mm during the course of the Lectures, carefully ex- 
 cluding all theorems and proofs of theorems, which were not originally given by 
 him, as here stated. SAMUEL HAUGHTON. 
 
 t Vol. 11. pp. 520, 542. 
 
33 Rotation of a Solid Body round a Fixed Point. 
 
 point I by the rotations w and a/ are 01. w sin EOI and 
 01. u/ sin K'OI ; and the directions of these velocities are per- 
 pendicular to the sides of the projected parallelogram. Hence, 
 if this parallelogram be turned in its plane through 90, its 
 sides will represent in magnitude and direction the actual 
 velocities: the resultant of these velocities is perpendicular to 
 the projection of the diagonal of the parallelogram (o>, a/) : 
 this projection, turned round through 90, will represent the 
 actual velocity, which is therefore the same in magnitude and 
 direction as would be produced by a single rotation represented 
 by the diagonal of (w, a/). Hence rotations may be resolved 
 along three rectangular axes by the same laws as couples, and 
 they must be counted positive when the motion produced is 
 from zto x, x to y, y to s, and vice versa. 
 
 II. LINEAR VELOCITIES PRODUCED BY A GIVEN EOTATION. 
 
 Let the origin of co-ordinates be assumed on the axis of 
 rotation, and let the magnitude of the rotation and of its com- 
 ponents be represented by (w, p, q, r) : the velocity of any 
 point (x, y, z) is in a direction perpendicular to the plane con- 
 taining the axis of rotation and the point (a?, y, 2) ; and its 
 magnitude is represented by the area of the triangle whose 
 angles are situated at the origin, the point (a?, y, s), and the 
 point (p, q, r). Hence, the components of the linear velocity 
 are represented by the projections of this triangle on the co- 
 ordinate planes. These projections are 
 
 u = 'qz-ry\ 
 
 v = rx-pz; (1) 
 
 w = py qx. 
 
Rotation of a Solid Body round a Fixed Point. 331 
 
 III. To REPRESENT GEOMETRICALLY THE MOMENTS OF INERTIA 
 OF A BODY WITH RESPECT TO AXES DRAWN THROUGH A 
 FIXED POINT. 
 
 The moment of inertia of a body with respect to any axis 
 (, ft 7) is 
 
 M = A' COS 2 a + B COS 2 /3 + 0' COS 2 y - 2L' COS /3 COS y 
 
 - 2 M' cos a cos y - 2JV" cos a cos ]3 ; 
 where 
 
 A' = / 2 + s 2 dw Z' 
 
 C f = J(a? 2 + y 2 ) dm, N' 
 
 Assume M = ^; ^t being the mass of the body, and r a distance 
 
 measured on the line (a, )3, y), and construct the ellipsoid whose 
 equation is 
 
 C'z z - ZL'yz - 2M'xz - 2N'xy = n ; (2) 
 
 then it is evident that the moments of inertia of the body with 
 respect to axes passing through the fixed point are represented 
 by the squares of the reciprocals of the radii vectores of this 
 ellipsoid. Assume A = ^u 2 , B = ^ C = pc 2 , and let the axes of 
 co-ordinates be the axes of the ellipsoid ; its equation will thus 
 become 
 
 <rV + &y + cV = l; (3) 
 
 and the equation of the reciprocal ellipsoid will be 
 
 This latter ellipsoid may be called the ellipsoid of gyration, as 
 the perpendiculars on its tangent planes represent the radii of 
 
33 2 Rotation of a Solid Body round a Fixed Point. 
 
 gyration ; this is evident from the consideration, that these per- 
 pendiculars are reciprocal to the radii vectores of the ellipsoid 
 (3). In fact, the moment of inertia with respect to any axis 
 will be represented by the formula 
 
 M = (a z cos 2 a + tf cos 2 13 + c 2 cos 2 7) M = ^uP 2 = 2 ; (5) 
 
 [R)P] denoting the radius vector and perpendicular on tangent 
 plane of the ellipsoid (4) ; and (Iff] the corresponding lines in 
 the ellipsoid (3). 
 
 IV. To FIND THE MAGNITUDE, POSITION, AND DIRECTION OF 
 THE STATICAL COUPLE PRODUCED BY THE CENTRIFUGAL 
 FORCES. 
 
 If from any point (a?, y, z) of the body, a perpendicular be 
 let fall on the axis of rotation (a, /3, 7), the centrifugal force 
 will be represented by the product of the square of the angular 
 velocity and this perpendicular : the corresponding elementary 
 statical couple will be found by multiplying the centrifugal 
 force by the distance from the foot of the perpendicular to the 
 origin, which is represented by the quantity (x cos a + y cos /3 
 + is cos 7). The components of the elementary couple will be 
 proportional to the projections of the triangle formed by the 
 lines before mentioned. The components of the elementary 
 couples must be integrated for the entire extent of the body, 
 and the integrals thus found will be the components of the 
 couple produced by centrifugal force: the expressions are as 
 
 follows : 
 
 co 2 (x cos a + y cos ]3 + z cos y) (z cos /3 - y cos 7) dm ; 
 
 w 2 (x cos a + y cos j3 + z eos 7) (x cos 7 - z cos a) dm ; 
 eo 2 (x cos a + y cos /3 + z cos 7) (y cos a - x cos j3) dm. 
 
 If the axes of co-ordinates be principal axes, these expressions, 
 
 when integrated, will become 
 
 w 2 cos /3 cos y(B-C) = qr (B-C}\ 
 
 a; 2 cos a cos y(C-A) =pr (C - A) ; (6) 
 
 w 2 cos a cos )3 (A - B] =pq (A - B) : 
 
Rotation of a Solid Body round a Fixed Point. 333 
 
 p, q, r being the components of the angular velocity w. The 
 position of the resultant couple may be expressed by means of 
 the ellipsoid (4). If a tangent plane be drawn to this ellipsoid 
 at the point (#, y, 2), and perpendicular to the line (a, |3, 7), it 
 may be easily shown that the projections of the triangle formed 
 by the radius vector and perpendicular are represented by the 
 quantities 
 
 cos |3 cos 7 (I* - c 2 ), cos a cos 7 (c 2 - # 2 ), cos a cos J3 (a z - b*) : 
 
 these three expressions multiplied by juw 2 will produce the quan- 
 tities used in (6). Hence it appears that the couple produced 
 by the centrifugal forces lies in the plane of the radius vector 
 and perpendicular to a tangent plane of the ellipsoid (4) ; the 
 tangent plane being perpendicular to the axis of rotation. Also, 
 the magnitude of the resultant couple is proportional to the 
 triangle formed by the radius vector and perpendicular. 
 
 The differential equations of motion commonly used in the 
 solution of this problem may be deduced immediately from 
 equations (6). In fact, as the axes of co-ordinates are axes of 
 permanent rotation, the increment of angular velocity round 
 each axis will be equal to the statical couple of the applied forces 
 (including centrifugal forces), divided by the moment of inertia 
 round that axis. The statement of this fact, in analytical lan- 
 guage, will give the equations of motion : 
 
 (7) 
 
 (L, M, N) being the components of the applied statical couple. 
 The position and magnitude of the couple produced by the 
 centrifugal forces are easily found by the method which has been 
 just given ; but the direction will be found more readily by 
 
334 Rotation of a Solid Body round a Fixed Point. 
 
 taking more particular axes of co-ordinates. Let 'the axis of 
 
 rotation be the axis OZ, and the plane of radius vector and 
 
 perpendicular be the co-ordinate plane 
 
 XOZ. In the accompanying figure OB' 
 
 and OP' are the radius vector and per- 
 
 pendicular of the ellipsoid (2), and OR, 
 
 OP the radius vector and perpendicular 
 
 of the ellipsoid (4), which is reciprocal 
 
 to the former; the rotation is positive, 
 
 in the direction indicated by the arrow. 
 
 As the rotation is round the axis of z, it is Fig. i. 
 
 easy to see that the statical couple produced by centrifugal force 
 
 will have for components, round the axes of x and y respectively, 
 
 the quantities ^\yzdm^ w z jxzdm taken with their proper sign; 
 
 i. e. the components are w 2 Z', o> 2 M' ; L ', M' being coefficients 
 
 in the equation of the ellipsoid 
 
 A'# + BY + C'z z - ZL'yz - ZM'xz - Wxy = p. 
 
 The tangent plane to this ellipsoid, applied at the point (x, y, z), 
 will be 
 
 (Ax- M'z - N'y] x f + (B'y - N'x - L'z) y' + (C'z - L'y - M'x] z' = p. 
 
 At the point B' the tangent plane will be perpendicular to the 
 plane XOZ, and will be found by making x = 0, y = 0, and de- 
 stroying the coefficient of y' in the preceding equation. These 
 conditions give us L' = 0, which proves that the statical couple 
 produced by centrifugal force lies altogether in the plane XOZ. 
 The equation of the tangent plane is the same as the equation 
 of the line B'P', and is 
 
 Hence we obtain 
 
 M' 
 
 tan = - _. 
 
 The value of the centrifugal couple is u> W, which is found 
 
Rotation of a Solid Body round a Fixed Point. 335 
 
 from the preceding equation by replacing C' and tan by their 
 values /iP 2 , and -= ; Q being the line EP. 
 
 We thus obtain finally the centrifugal couple lying in the 
 plane XOZ, and expressed by the equation 
 
 w*lxzdm = - ^PQ. (8) 
 
 It thus appears that the centrifugal couple lies in the plane of 
 radius vector and perpendicular, is proportional to the area of 
 the triangle EOF, and has a direction opposite to the direction 
 of rotation. 
 
 V. To FIND THE EELATION BETWEEN THE PLANE OF PRINCIPAL 
 MOMENTS AND THE Axis OF EOTATION AT ANY INSTANT. 
 
 The motion of the body at any instant consists of a rotation 
 of a certain magnitude round a certain axis ; this rotation might 
 be produced by an impulsive couple of a determinate magnitude 
 and direction. The ^statical impulsive couple thus conceived is 
 the couple of principal moments. Let this couple be represented 
 by 6r, and act round the axis OE (fig. 1, p. 334) ; then the cor- 
 responding axis of rotation will be the perpendicular OP, and 
 the relation between 6r and w may be thus found : Let the axes 
 of co-ordinates be the axes of the ellipsoid (4), the radius vector 
 being determined by the angles (A, n, v), and the axes of rota- 
 tion by the angles (a, |3, 7). From mechanical considerations 
 we obtain the equations 
 
 G cos X = Ap = juw a z cos a ; 
 G cos fj. = Bq = jua; b z cos/3 ; 
 G cos v = Cr = IULW c 2 cos 7. 
 Hence we obtain 
 
 cos A _ a? cos a cos.^i _ b* cos/3 
 cos v c 2 cos 7' cos v c 2 cos 7* 
 
 (9) 
 G 
 
3 3 6 Rotation of a Solid Body round a Fixed Point. 
 
 The first two of these equations prove that the axis of rota- 
 tion is the perpendicular on tangent plane of the ellipsoid, and 
 the last equation gives the magnitude of the rotation in terms 
 of the impressed couple and quantities determined by the nature 
 of the body itself. Equations (9) are true, whatever be the 
 forces acting on the body ; if no forces act, G will be fixed in 
 magnitude and position in space, by the principle of conserva- 
 tion of areas, but will change its position in the body, the axis 
 of rotation accompanying it, and changing its position both in 
 the body and in space. 
 
 VI. BOTATION PRODUCED BY CENTRIFUGAL FORCE ; PARTICU- 
 LAR PROPERTIES OF THE MOTION WHEN NO FORCES ACT. 
 
 The axis of rotation produced by the centrifugal couple always 
 lies in the plane of principal moments. This theorem may be 
 thus proved : Let the radius vector and perpendicular be drawn, 
 which coincide with the axis of principal moment and axis of ro- 
 tation at any instant ; a line perpendicular to the plane of radius 
 vector and perpendicular is the axis of centrifugal couple ; 
 this line and the original radius vector are axes of the section of 
 the ellipsoid made by their plane : at the point where the axis 
 of the centrifugal couple pierces the ellipsoid let a tangent plane 
 be applied ; the perpendicular let fall on this tangent plane is 
 the axis of rotation produced by centrifugal forces. From the 
 construction it is evident that the plane of the second radius 
 vector and perpendicular is perpendicular to the axis of G ; hence 
 the axis of the centrifugal couple and the axis of rotation pro- 
 duced by it always lie in the plane of principal moment. Two 
 important corollaries follow from the theorem just demonstrated, 
 in the case where no forces act : First, the component of an- 
 gular velocity round the axis of primitive impulse is constant 
 during the motion. Secondly, the radius vector which coincides 
 with the axis of G is of constant length during the motion. The 
 first theorem is obvious ; for as the axis of rotation produced by 
 centrifugal force is always perpendicular to the axis of 6r, it 
 
Rotation of a Solid Body round a Fixed Point. 337 
 
 cannot alter the rotation round that axis. The second theorem 
 follows from equation (9), from which we deduce 
 
 w cos = . (10) 
 
 The left-hand member of this equation is constant by 'the pre- 
 ceding theorem ; and G is constant, since there is no external 
 force ; therefore R is constant. 
 
 As the axis of G is fixed in space, and the line R is constant, 
 it is evident that the axis of G will describe in the body the cone 
 of the second degree, determined by the intersection of the ellip- 
 soid (4) with the sphere whose radius is R. The equation of 
 this cone is 
 
 R*-a* , R*-V . R*-c* , 
 - - x* + - y 2 + - 2 2 = 0, (11) 
 
 2 z * 
 
 As the axis of principal moments describes this cone in the body, 
 it is accompanied by the axis of rotation, which is always the 
 corresponding perpendicular on tangent plane of the ellipsoid. 
 The cone described by the axis of rotation might be found thus. 
 Let tangent planes be applied to the ellipsoid along the sphe- 
 rical conic in which the cone (11) cuts the ellipsoid. From the 
 centre let fall perpendiculars on these tangent planes ; the locus 
 of these perpendiculars is the required cone. 
 
 VII. THE Axis OF PRINCIPAL MOMENTS is FIXED IN 
 
 SPACE. 
 
 This is evident from D'Alembert's principle, but may be 
 shown by geometrical considerations in the particular case under 
 consideration. The axis varies in position in the body, in con- 
 sequence of the centrifugal couple, which must be compounded 
 with the impressed couple at each instant. Eeferring to equa- 
 tion (8), the value of the centrifugal couple is - fivfPQdt, the 
 principal moment being G = pwPjR, (vid. (9)). Hence the angle 
 
338 Rotation of a Solid Body round a Fixed Point. 
 through which the axis of principal moments shifts in an element 
 
 of time is - C : this angle, multiplied by the constant radius 
 
 R 
 
 vector, will give the elementary motion on the spherical conic 
 traced by the axis of principal moment on the surface of the 
 ellipsoid : this motion is therefore - wQdt ; but in the same time 
 the point of the body which coincides with the point where the 
 axis of moments pierces the spherical conic will describe the 
 angle + wQdt in consequence of the angular rotation. Hence 
 the axis of moments will remain fixed in space, and will move 
 in the body with a velocity proportional to the tangent of the 
 angle between the radius vector . and perpendicular, the motion 
 being in a direction opposite to the direction of the rotation. 
 
 This is evident from the consideration that Qw = Pw tan <, Po> 
 
 f3 
 being constant and equal to = (rid. (9)). 
 
 x 
 
 VIII. TO FIND THE MOTION OF THE PRINCIPAL AXIS IN THE 
 
 BODY. 
 
 First Method. 
 
 
 
 The point of the principal axis of moments, which is situated 
 at the distance R from the centre, moves on the spherical conic 
 which has been determined. Let this point be projected on the 
 three co-ordinate planes ; then, since the spherical conic is pro- 
 jected into a conic section, the movement of the axis of moments 
 is reduced to the movement of a point on a conic section, accord- 
 ing to a law which must be determined. The radius vector 
 describes an elementary triangle in the surface of the cone (11) : 
 let the projections of this triangle on the co-ordinate planes be 
 (dA l9 dAi, dAj) ; we obtain easily 
 
 dA l dz dy dA z _ dx d* dA, = dy d_x 
 
 ~dr= y jr*di' ~dT dt x dt dt ~ dt' J df 
 
 Substituting in these equations the values of the velocities given 
 by (1), we obtain 
 
Rotation of a Solid Body round a Fixed Point. 339 
 
 
 (12) 
 
 dt 
 
 p 
 
 = Pw 
 
 These equations prove that the areolar velocity of the projection 
 on a co-ordinate plane varies as the ordinate to that plane. By 
 means of the method of quadratures, we may determine from 
 equations (12) the position of the projections of the principal 
 axis at any instant, and hence deduce the position of the axis 
 itself. 
 
 / Second Method. 
 
 If the spherical conic be projected on a cyclic plane of the 
 ellipsoid of gyration, by lines parallel to x and s, the projections 
 will be two concentric circles, and the 
 corresponding projections will lie on 
 the same ordinate SII' (fig. 2). The 
 inner circle will belong to the projec- 
 tion parallel to a?, if R be greater than - 
 b, and will belong to the projection 
 parallel to z if R be less than b ; and 
 if R be equal to b, the two circles will 
 coincide with each other and with the Fi g- 2 - 
 
 spherical conic, which in this case becomes the circular section of 
 the ellipsoid. The projected point will revolve round the cir- 
 cumference of the inner circle, and will vibrate on the circum- 
 ference of the outer circle between the dotted lines. It is evi- 
 dent that the mean axis of the ellipsoid OY lies in the plane of 
 the figure. Let SI and SI' be equal to />, p, and let (7, <?' denote 
 the radii of the two circles : the velocities of the projections in 
 the circles will evidently be 
 
 F=^, 
 P dt' 
 
 _ 
 dt ' 
 
34 o Rotation oj a Solid Body round a Fixed Point. 
 C and C' having the values 
 
 The value of -j- deduced from (1) is 
 tit 
 
 - z - - P p' sin cos 9. 
 dt \c 2 ay \ c V 
 
 being the angle made by the plane of the circular section with 
 the plane (#, ?/), 
 
 a c //fl 2 -& 2 \ a 
 
 smO = - / - , cos 9 = T 
 
 z 
 
 Introducing these values of -^, sin 6 and cos 0, and for Pw its 
 
 P 
 
 value =r, we obtain finally for the velocities 
 
 G 
 
 "' 
 
 (13) 
 
 The velocity of each projection, therefore, varies as the ordinate 
 of the other. This theorem enables us to find a simple expression 
 for the time. Using the angle (0) marked in fig. 2, we obtain 
 
 dt 
 
 (^, (7, K] belonging to the projection parallel to axis of x. If 
 (i//, (7', K') be the corresponding quantities for the other projec- 
 tion, we obtain also 
 
Rotation of a Solid Body round a Fixed Point. 341 
 
 K C 
 or, since it is easily seen that > = -^>, we obtain finally 
 
 _TL t/ 
 
 (14) 
 dil, 
 Kdt 
 
 The motion of the principal axis of moments is, therefore, ex- 
 pressed by an elliptic function of the first kind. 
 
 The motion of the axis of moments is determined by the 
 magnitude of the radius vector of the ellipsoid, which is the axis 
 of the original couple impressed upon the body ; if this radius 
 vector be greater than the mean axis of the ellipsoid, the cor- 
 responding spherical conic will have the axis of x for its internal 
 axis ; and if the radius be less than the mean axis, the axis of 
 z will be the internal axis of the conic ; in no case will the mean 
 axis be the internal axis of the spherical conic. If the radius R 
 be nearly equal to either the greatest or least semi-axis, the ex- 
 pression (14) for the time may be integrated. Let R be nearly 
 equal to the greatest semi-axis. The first of the equations (14) 
 belongs to the interior circle, which is of small dimensions in 
 the case supposed ; the second equation expresses the vibratory 
 motion of the projection, through a small arc of the outer circle, 
 which will have a radius much greater than the inner circle ; 
 we may, therefore, suppose the angle $ to be equal to its sine. 
 
 C' 
 
 Multiplying both sides of the equation by -^ we obtain. 
 
 C' 
 ri f if -77 d\L 
 
 -K* 1/V s- 
 
 Hence 
 
 -7T=sin (K't + A). (15) 
 
 If To denote the time of a complete oscillation or revolution of 
 
34 2 Rotation of a Solid Body round a Fixed Point. 
 
 the axis of moments about the axis of x, and T r the time of a re- 
 volution of the body round the axis of x, the following relation 
 between these two periods may be readily deduced from (15) : 
 
 be 
 
 r ' r V.( (*-*) (-*)' 
 
 If the axis of moments, and consequently the axis of revolution, 
 be situated near the axis of greatest or least inertia, it will 
 always continue near this axis ; if, however, it be situated near 
 the mean axis, the movement of the body will be determined by 
 the following construction : Let the two cyclic planes of the 
 ellipsoid be drawn through the mean axis*; they will divide the 
 ellipsoid into two regions, in one of which is situated the axis of 
 maximum inertia, and in the other the axis of minimum inertia. 
 The spherical conic described by the axis of principal moments 
 will have the first or second of these axes for its internal axis, 
 according as R is greater or less than the mean axis. If the 
 axis of principal moments lie in one of the cyclic planes, the 
 spherical conic becomes a circle, and its two projections become 
 identical with itself (fig. 2, p. 339) ; the expressions (14) -are 
 reduced to the form 
 
 COS 
 
 which when integrated gives 
 
 or, 
 
 . 7T , 
 
 cot - ~ 2 
 
 $o being the value of corresponding to t = 0, and K being ex 
 pressed by the following quantity : 
 
 t 
 
 ILL b*ac 
 
 It is evident from the equation (17) that the axis of moments 
 will coincide with the mean axis of inertia at the end of an in- 
 finite time. 
 
Rotation of a Solid Body round a Fixed Point. 343 
 
 IX. To FIND THE POSITION OF THE BODY IN SPACE AT THE 
 
 END OF ANY GIVEN TlME. 
 
 First Method. 
 
 The radius vector of the ellipsoid of gyration, which is per- 
 pendicular to the plane containing the axes of principal moment 
 and of rotation, always lies in the plane of principal moment, 
 and describes in that plane areas proportional to the time. 
 
 Let OGr, Oil be the axes of principal moment and of rota- 
 
 tion ; OR', OO', the axes of centrifugal 
 couple and of corresponding rotation ; 
 the plane CtOQ' will contain the two 
 successive positions of the axis of rota- 
 tion. Let 01 be the position of the 
 axis of rotation at the end of the time 
 $t ; then $u will be equal to the angle 
 described in the fixed plane by the 
 line OR'. Let R' and P f be the radius 
 vector and perpendicular correspond- 
 ing to the centrifugal couple and its 
 
 Fig. 3. 
 
 axis of rotation. The following relations are evident from the 
 figure : 
 
 w sin Q'OI cos 0' 
 
 because 
 
 sinliOI 
 
 ^,r\-r 
 sin O 01 = 
 
 , 
 
 sm0 
 
 sin 
 
 sn 
 
 but, from mechanical considerations, 
 
 PR' G 
 
 PRu sin 
 
 ; because 
 
 sn 
 
 tP'# 
 
 Hence, by equating the geometrical and mechanical expression, 
 we obtain 
 
 8*. (18) 
 
344 Rotation of a Solid Body round a Fixed Point. 
 
 The position of the body in space is thus reduced to quadra- 
 tures ; but the problem may be solved more readily in the fol- 
 lowing manner. 
 
 Second Method. 
 
 The axis of principal moments, appearing to move in a 
 direction opposite to the rotation, describes in the body the cone 
 whose equation has been given (11). If the cone reciprocal to 
 this cone be described, one of its sides will lie in the fixed 
 plane, and the whole motion of the body in space will be the 
 same as the motion of this cone, which partly slides and partly 
 rolls on the fixed plane, the sliding motion being uniform. This 
 theorem is evident by resolving the angular velocity o> into two 
 components, one round the axis of principal moments, and the 
 other in a direction perpendicular to this, round the side of the 
 reciprocal cone, which is in contact with the fixed plane. These 
 components are o> cos and w sin ; w cos $ being constant and 
 producing the sliding motion, while u> sin represents the an- 
 gular velocity round the side of the cone in contact with the 
 fixed plane. The angle described by the side of the reciprocal 
 cone in the fixed plane at the end of a given time is, therefore, 
 the algebraic sum of two angles, one of which is proportional to 
 the time, and the other is the angle described in the cone in con- 
 sequence of the rotation w sin 0, and is, therefore, measured by 
 the arc of a spherical conic. The position of the body at the 
 end of the time t is thus found : determine by equation (14) 
 the position of the axis of principal moments in the cone (11) ; 
 the corresponding position of the component axis of rotation in 
 the reciprocal cone is therefore known. Hence the angle de- 
 scribed in the time t in the fixed plane is 
 
 9 = J(u COS0^ = w C08(f).t . (19) 
 
 The equation of the reciprocal cone is 
 
 _**_ + jv_ + _<?*_ . 
 
 > + + * 
 
Rotation of a Solid Body round a Fixed Point. 345 
 
 In (19) the positive or negative sign must be used according 
 as E is less or greater than the mean axis of the ellipsoid ; this 
 is evident from the composition of rotations, and from the con- 
 sideration that in the former case the axis of rotation falls inside 
 the cone (11), while in the latter case it falls outside. 
 
 X. To FIND A POINT, IF ANY, IN A GIVEN Axis OF EOTATION, 
 
 WHICH BEING FIXED, THE AxiS WILL BE PERMANENT. 
 
 Let B'B" (fig. 4) be the given axis, round which the body 
 revolves with a rotation expressed by w ; describe the ellipsoid 
 of gyration round the centre of gra- 
 vity 0, and draw OP' parallel to B'B". 
 The centrifugal force uPrdm at any 
 point (#, y, z) may be resolved into two 
 components, u?pdm and <i> 2 . B'P' . dm ; 
 r and p denoting the distances of the 
 point from the axes B'B" and OP' re- 
 spectively ; the effect of the rotation 
 round B'B" is therefore the same as an 
 equal rotation round OP', together with 
 a number of parallel and equal forces applied to each point of 
 the body. The rotation round OP' produces a centrifugal couple 
 represented by - juw 2 . OP . PB (md. (8)); or, determining the point 
 B' by the condition OP . PB = OP' . P'B', the centrifugal couple is 
 - /W . OP' . P'B'. The resultant of the parallel forces is a force 
 applied at the centre of gravity, acting in the direction parallel 
 to B'P', and equal to juo> 2 . B'P'. Comparing this with the cen- 
 trifugal couple, it is evident that the forces at destroy each 
 other, and, therefore, the total result of the rotation round B'B" 
 is to produce a force acting at the point B', which has been just 
 determined. If this point be fixed, the axis B'B" will be a per- 
 manent axis of rotation. The condition by which the point B' 
 is found is, that the triangle OB'P' is equal to and in the same 
 plane with the triangle OBP : hence, if an ellipsoid confocal to 
 
346 Rotation of a Solid Body round a Fixed Point. 
 
 the ellipsoid of gyration be described through the point B', it 
 will be perpendicular to the line R'B". The general construc- 
 tion for permanent axes is, therefore, the following : Let the 
 ellipsoid of gyration be described, and confocal ellipsoids : any 
 line which pierces one of these ellipsoids at right angles is a per- 
 manent axis of rotation for the point of intersection. 
 
PART IV. 
 
 A T T R A C T.I O N 
 
 
( 349 ) 
 
 I. ON A DIFFICULTY IN THE THEORY OF THE ATTRAC- 
 TION OF SPHEROIDS. 
 
 [Transactions of the Royal Irish Academy, VOL. xvi. p. 237. Read May 28, 1832.] 
 
 
 AN approximate theorem, discovered by Laplace, and relating 
 to the attraction of a solid slightly differing from a sphere, on 
 a point placed at its surface, has given rise to many disputes 
 among mathematicians.* I hope the question will be set in a 
 clear light by the following remarks. 
 
 Let us consider the function which expresses the sum of 
 every element of a solid divided by its distance from a fixed 
 point, and let us denote it, as Laplace has done, by the letter Y. 
 It is necessary to find the value of Y for a pyramid of inde- 
 finitely small angle, the fixed point being at its vertex. Calling 
 the small solid angle of the pyramid (or the area which it 
 intercepts on the surface of a sphere whose radius is unity and 
 centre at the vertex), it is manifest that the element of the 
 pyramid at the distance r from the vertex is 0rVr; dividing 
 therefore by r, and integrating, we have -J^r 2 , or multiplied 
 into half the square of the length, for the value of Y. 
 
 Again, supposing the force to vary inversely as the square 
 of the distance the only hypothesis that can be of use in the 
 present inquiry the attraction of the same pyramid on a point 
 at its vertex, and in the direction of its length, is manifestly 
 equal to $r. 
 
 Let us now consider a solid of any shape, regular or irregu- 
 lar, terminated at one end by a plane to which the straight 
 line PQ is perpendicular at the point P; and let there be a 
 sphere of any magnitude, whose diameter P'Q' is parallel to 
 PQ. Let P" be a fixed point, and from the points P, P / , P", 
 draw three parallel straight lines Pp, P'p, P"p" 9 the first two 
 
 * See Pontecoulant, Theorie analytique du systeme du monde, tome ii. p. 380 ; with 
 the references there given. 
 
350 Difficulty in Theory of Attraction of Spheroids. 
 
 terminated by the surfaces of the solid and of the sphere, the 
 third, Py, i n ^ ne same direction with them and equal to their 
 difference, without regarding which of them is the greater, and 
 
 suppose all the points p" 9 taken according to the same law, to 
 trace the surface of a third solid. Let Pp, P'p'> P"p" > be edges 
 of three small pyramids with their other edges proceeding from 
 P, P', P", parallel, and having of course the same solid angle, 
 which we shall call $, and denote by r, r', r", their respective 
 lengths, and by F, F', F", the values of the function Y for each 
 of them. Drawing pR perpendicular to PQ, the attraction of the 
 pyramid Pp in the direction of PQ will be equal to x PR. 
 Call this attraction A, and let a be the radius of the sphere. 
 
 Since /' is the difference of r and r', we have r 2 + r' 2 - r" 2 
 = 2rr' = 2PM x P'Q', and multiplying by 1 </> we find 0r 
 + r ' _ !0 r "2 = 2a<j> x PR, that is F+ V - V" = 2aA. The 
 same thing is true for any other three pyramids similarly related 
 to each other, throughout the whole extent of the three solids 
 which are exhausted by them at the same time ; and hence, if 
 we now denote by F, F', F", the whole values of the function Y 
 for the three solids, and by A the whole attraction, of the first of 
 them parallel to PQ on a point at P, we shall still have F+ V 
 - V" = 2aA. 
 
 To express this general theorem in the notation of Laplace, 
 we have merely to observe that the attraction A is synonymous 
 
 with - ( ), and that the quantity V for the sphere is equal 
 
 \U')" J 
 
 4 
 
 to ira z . Substituting these values, we find 
 
 an exact equation, differing from the approximate one of Laplace 
 only in containing the quantity V ', and totally independent of 
 
Difficulty in Theory of Attraction of Spheroids. 351 
 
 the nature of the surface or of the magnitude of the sphere ; the 
 only things supposed being that all the lines drawn from P meet 
 the surface again but once, and that no part of it passes beyond 
 a plane through P at right angles to PQ. 
 
 With respect to the limit of the quantity V", it is obvious 
 that if a hemisphere be described from P" as a centre, with a 
 radius equal to the greatest difference 8 between the lines Pp, 
 jPjp', the solid P"p" will lie wholly within this hemisphere, and 
 consequently V" will be less than the value of V for the hemi- 
 sphere, that is, less than ?rS 2 ; for here all the little pyramids 
 from the centre have the same length 8, and their bases are 
 spread over the hemispherical surface ; wherefore V" = 2ir 
 x ig 2 = TrS 2 . All this is independent of anything but the suppo- 
 sition just mentioned. 
 
 If now PQ be supposed to be a spheroid of any sort, 
 slightly differing from the sphere P'Q', and such that the 
 line PQ, perpendicular to the surface at P, passes nearly 
 through the centre, then all the differences, of which 8 is the 
 greatest, being of the first order, the quantity V' ', which is less 
 than TrS 2 , will be of the second order ; and therefore neglecting, 
 as Laplace has done, the quantities of that order, we get the 
 theorem in question. 
 
 It may be well to apply the general theorem to the simple 
 case in which the first solid is a sphere of the radius of, because 
 both Lagrange and Ivory have used this case to show that the 
 reasonings of Laplace are incorrect. In this instance, then, the 
 surface described by the point p" is that of a sphere whose 
 
 radius is the difference between a and a' ; and the values of 
 
 444 4 
 
 F, F', F", and A, are ~ ?ra' 2 , ^7ra 2 } - ?r (a' - of and -iraf respec- 
 o o o o 
 
 tively. 
 
 Substituting these values in the equation F+ V - V" = 2aA, 
 
 4 
 and omitting the common factor ^TT, the resulting equation 
 
 a' + # - (a? - a)* = 2aa 
 ought to be identical ; and so it manifestly is. 
 
 November, 1831. 
 
( 352 ) 
 
 II. ON THE ATTRACTION OF ELLIPSOIDS, WITH A NEW 
 DEMONSTRATION OF CLAIRAUT'S THEOREM, 
 BEING AN ACCOUNT OF THE LATE PROFESSOR 
 MAC CULLAGH'S LECTURES ON THOSE SUBJECTS. 
 COMPILED EY GEORGE JOHNSTON ALLMAN, LL.D., 
 OF TRINITY COLLEGE, DUBLIN. 
 
 [Transactions of the Royallrish Academy, VOL. xxn. p. 379. Read June 13, 1853.] 
 
 PROPOSITION I. 
 
 If P be any point on the surface of an ellipsoid, and PCi be 
 drawn perpendicular to an axis 00, and an ellipsoid be described 
 through Ci concentric, similar, and similarly placed to the given 
 ellipsoid ; then the component of the attraction of the given ellipsoid 
 on P in a direction parallel to 00 is equal to the attraction of the 
 inner ellipsoid on the point Ci. 
 
 This theorem is an extension of that given by Mac Laurin* 
 relating to the attraction of a spheroid on a point placed on its 
 surface. It may, moreover, be established by means of the same 
 geometrical proposition from which Mac Laurin deduced his 
 theorem. 
 
 Through the point P let a chord PP' of the given ellipsoid 
 be drawn parallel to the axis 00. Now, suppose both ellipsoids 
 to be divided into wedges by planes parallel to each . other, and 
 passing respectively through this chord and the parallel axis of 
 the inner ; and suppose the wedges to be divided into pyramids, 
 the common vertex of one set "being at P, and of the other at d. 
 
 * De caus. Phys. Flux, et Eefl. Mar is, sect. 3 ; or see Airy's Tract on the 
 Figure of the Earth, Prop. 8. 
 
On the Attraction of Ellipsoids. 
 
 353 
 
 Observing that any two of these parallel planes cut the two 
 surfaces in similar ellipses, such that the semi-axis of one is 
 equal to the parallel ordinate of the other, it is easy to see that 
 
 Fig. 1. 
 
 the reasoning employed by Mac Laurin may be used to esta- 
 blish the truth of the theorem stated above. 
 
 PROPOSITION II. 
 
 To calculate the Attraction of an Ellipsoid on a Point placed at 
 the extremity of an A.XIS* 
 
 Let the semi-axes of the ellipsoid be a, b, c, where a > b > c, 
 and let the point on which it is required to find the attraction 
 be (Fig. 1), the extremity of the least axis. 
 
 Suppose the ellipsoid to be divided by a series of cones of revo- 
 lution which have a common vertex C and a common axis CO', C' 
 being the vertex of the ellipsoid opposite to ; it will be sufficient 
 to find an expression for the attraction of the part of the ellipsoid 
 contained between two consecutive conical surfaces, whose semi- 
 angles are and -f dO respectively. Suppose now the part of 
 the ellipsoid between two consecutive cones to be divided into 
 
 * Proceedings of the Royal Irish Academy, YOL. in. p. 367. 
 2A 
 
354 
 
 On the Attraction of Ellipsoids. 
 
 elementary pyramids with a common vertex C, Let CP be one 
 of these elementary pyramids, whose solid angle is w ; let PQ, 
 be drawn perpendicular to CO' ; from the centre draw a radius 
 
 Fig. 2. 
 
 vector OE parallel to CP, and from the extremity E let fall a 
 perpendicular ES on the axis CO'. 
 
 Now the attraction of the elementary pyramid CP on the 
 material point ju, placed at its vertex = ^t/pw.CP ; and the com- 
 ponent of this attraction in the direction of the axis is 
 
 /U/JOO).CQ, = 2///J30). 
 
 OE 2 cos 2 
 
 Now suppose the radius vector OE to revolve around the axis 
 OC', then the attraction on the point C of the portion of the 
 ellipsoid bounded by the two cones of revolution, whose semi- 
 
On the Attraction of Ellipsoids. 355 
 
 angles are 9 and 9 + d9 respectively, since it is made up of the 
 components in the direction CC' of the attractions of all the 
 elementary pyramids CP, is 
 
 .S(OE 2 w) = - ^cos 2 0t/0.S(OE 2 6fy), 
 
 i/ C 
 
 dQ being the angle between two consecutive sides of the cone 
 generated by the revolution of OE. 
 
 But S (OE 2 d(f>) is equal to twice the superficial area of the 
 part of this cone which is enclosed within the ellipsoid. More- 
 over, the projection on the plane ab of this portion of the sur- 
 face of the cone is an ellipse, whose semi-axes are n sin 0, r z sin 0, 
 and whose area is TTT^TZ sin 2 0, r l and r 2 being the maximum and 
 minimum values of OE : the superficial area o4ke portion of 
 the cone within the ellipsoid is therefore 7rnj<sm& L 
 
 Hence it follows that 4*A 
 
 **?**& 
 
 The attraction on the point C of the part of the 
 tained between the two cones of revolution, whose common 
 vertex is at C, and whose semi-angles are and 9 + d9 respec- 
 spectively, is therefore 
 
 c 
 where 
 
 2 sin 2 
 
 On substituting these values, the expression given above 
 
 becomes 
 
 abc cos 2 sin 06/0 
 
 ^ 
 
 (a 2 cos 2 + c 2 sin 2 0) ^/ (6 2 cos 2 + c 2 sin 2 0) ' 
 
 Hence the attraction of the solid ellipsoid on the point C at the 
 extremity of the least axis is 
 
 7T 
 
 abc cos 2 sin 
 
 (a 2 cos 2 .+ c 2 sin 2 0) v/(6 2 cos 2 + c 2 sin 2 0)' 
 2 A2 
 
356 On the Attraction of Ellipsoids. 
 
 Let cos 6 = u, this expression becomes 
 
 , 
 
 """ 
 
 abcifdu 
 
 (1) 
 
 In the same way it may be shown that the attraction of the 
 ellipsoid on a point p placed at the extremity of the mean 
 axis is 
 
 dbctfdu 
 
 f 1 
 
 Jo 
 
 and on a point at the extremity of the greatest axis, 
 
 . . f 1 abc u 2 du 
 
 " p } v/jtf 2 + n* (b* - O) V {^ 2 + w 2 (c 2 - a 2 )}' 
 
 It will be seen in a subsequent proposition, that these three 
 expressions are not independent of each other, the values of the 
 three attractions in question being connected by an equation. 
 
 PROPOSITION III. 
 
 To give Geometrical Representations of the ^Attraction of an 
 Ellipsoid on Points placed at the extremities of its least and mean 
 Axes* 
 
 On the greater axis OAo of the focal ellipse assume a point 
 
 such that OKi = - OA ; from the point KI draw a tangent 
 c 
 
 * Proceedings of the Royal Irish Academy, VOL. HI. p. 367. 
 
On the Attraction of Ellipsoids. 357 
 
 KiQi to the focal ellipse, and let T = tan KiQ! - arc A Qi ; then 
 the attraction of the ellipsoid on the particle /* placed at the ex- 
 tremity C of the least axis is 
 
 4tr fjifp abc* 
 
 (a 2 - c 2 ) (b* - c*) 
 
 T. (2) 
 
 For let a point K (see fig. 3) be assumed on the greater axis OA 
 of the focal ellipse, such that 
 
 OA 
 
 from K let a tangent KQ, be drawn to the focal ellipse, and let 
 OP be the perpendicular let fall from and KQ, ; then ^ denot- 
 ing the angle A OP, 
 
 OK 2 , cos 2 1/, = ^- {c 2 + w 2 (V - c 2 ) J . cos 2 </,. 
 c 
 
 Moreover, 
 
 OK 2 , cos 2 ^ = OP 2 = (<# - c 2 ) cos 2 ^ + (b z - c 2 ) sin 2 ^. 
 Equating these values, and solving for sin 2 *//, we get 
 
 (a* - ") u z 
 
 V 
 
 Sin I/ = 
 
 . . . -- rr-. 
 
 c 2 + u? (a 2 - c 2 ) 
 Now 
 
 d. (tan KQ - arc A Q) = sin $d. OK 
 
 (a* - c 2 ) (b* - c 2 ) u 2 du 
 
 By comparing this expression with (1), given in the last propo- 
 sition, it appears that the attraction on the point C of the portion 
 of the ellipsoid contained between the two conical surfaces whose 
 semi- angles are and 9 + dO, respectively, is 
 
 * Transactions of the Royal Irish Academy, VOL. xvi. p. 79. Proceedings of 
 the Royal Irish Academy, VOL. n. p. 507 (supra, p. 255). 
 
358 On the Attraction of Ellipsoids. 
 
 Now, in order to obtain the attraction of the whole ellipsoid 
 on the point C, we have to integrate the expression given above 
 between the limits u - and u = 1, or OK = OA and OK = OK X ; 
 from which it appears that its value is 
 
 It is easy to see that the attraction of the part of the ellipsoid 
 contained within the conical surface, whose semi-angle is equal 
 to the angle cos^w, is 
 
 where t = tan KQ, - arc A Q. 
 
 To represent the attraction on a point ^ placed at the extre- 
 mity of the mean axis, assume on the transverse OA of the focal 
 
 /i 
 
 hyperbola a point KI such that OKi = OA 7, and from K! 
 
 draw a tangent K^ to the hyperbola, and let T = tan K^ 
 - arc A Qi ; then the attraction of the ellipsoid on the point fi is 
 
 - ^"fr (5- - ?) I? - v) T - (4) 
 
 To prove this, assume a point K such that 
 
 from K draw a tangent KQ to the hyperbola, and from let 
 fall a perpendicular OP on this tangent; then if i// = angle 
 A.OP, 
 
 _ 
 
 1 * 
 
 Hence, by following a method similar to that used in finding the 
 representation of the attraction on a point at the extremity of 
 the least axis, the expression given above may be easily obtained. 
 The attractions (7, B of the ellipsoid on points placed at the 
 extremity of the least and mean axes are thus represented by 
 
On the Attraction of Ellipsoids. 359 
 
 means of arcs of the focal ellipse and hyperbola respectively. In 
 consequence of the third focal conic of the ellipsoid being ima- 
 ginary, no direct geometrical representation can be given for the 
 attraction A on a point placed at the extremity of its greatest 
 axis. It will, however, be found, as was intimated above, that 
 a simple relation exists between the three attractions, which 
 enables us to represent this last by means of arcs of both focal 
 conies. 
 
 The relation alluded to is 
 
 A B C 
 
 - + - + -- 4^* (5) 
 
 This can be easily proved by the help of the following geo- 
 metrical theorem : 
 
 If from the extremities A, B, C of the three axes of an ellip- 
 soid three parallel chords A.p, B<?, Cr, be drawn, and if these 
 chords be projected each on the axis from whose extremity it is 
 drawn, then the sum of these three projections, Aa, B/3, Cy, 
 divided respectively by the lengths of the axes AA', BB', CO', 
 on which they are measured, will be equal to unity. 
 
 Now conceive three chords Ap, A>', A^", to be drawn from 
 A, making each with the other two very small angles, and so 
 forming a pyramid with a very small vertical solid angle o> ; 
 and from B and C let two systems of chords Bg, B/, B<?", and 
 Cr, Cr', Cr", be drawn, each system forming a very small pyra- 
 mid whose three edges are parallel to the three edges AJO, A//, 
 Ap", of the pyramid which has its vertex at A. 
 
 The attractions of the three pyramids, reduced each to the 
 direction of the axis passing through its vertex, will be equal to 
 /u/jow.Aa, ju/pw.B/3, JU//QW . Cy respectively ; and, therefore, the 
 sum of those attractions divided respectively by the lengths of 
 the axes will be 
 
 Aa B/3 C 7 
 
 Proceedings of the Royal Irish Academy, VOL. u. p. 525. 
 
360 On the Attraction of Ellipsoids. 
 
 Let pyramids thus related be indefinitely multiplied, and 
 the ellipsoid will be simultaneously exhausted from the three 
 points A, B, C. 
 
 Hence the sum of the whole attractions at A, B, C, divided 
 respectively by the lengths of the corresponding axes, will be 
 2irfifp, or, 
 
 ABC 
 
 PROPOSITION IV. 
 
 To find an expression for the potential V of a system of par- 
 tides at a point M, ivhose distance from the centre of gravity of the 
 system is very great compared with the mutual distances of the 
 particles. 
 
 It is proved by Poisson,* that if the origin of co-ordinates 
 be at the centre of gravity of the system 
 
 x ', /', z being the co-ordinates of the distant point, and / its 
 distance from the origin. Let now the principal axes at that 
 centre be taken as axes of co-ordinates ; then, since 
 
 m = 0, 2#s dm = 0, ^yz dm = ; 
 
 Hence, if A, B, C be the three principal moments of inertia, 
 and / the moment of inertia of the system round OM, 
 
 (6) 
 
 * Mecamque, torn .i. p. 178. 
 
On the Attraction of Ellipsoids. 361 
 
 PROPOSITION V. 
 
 A system of material particles attracts a point M, whose distance 
 from the centre of gravity of the attracting system is very great 
 compared with the mutual distances of the particles ; then if a tan- 
 gent plane be drawn to the " ellipsoid of 'gyration"* perpendicular 
 to OM, the ivhole attraction lies in the plane OST, where S is the 
 point in which this tangent plane intersects OM, and T its point 
 of contact tvith the ellipsoid. 
 
 Let a, ]3, 7 be the direction angles of OT ; a', )3', y' of OM ; 
 and ai, |3i, 71 of TS ; and a , j3 , y of the normal to the 
 
 Fig. 4. 
 
 plane OST ; and let OS, OT and the angle SOT, be denoted by 
 p, r, and respectively. It will be sufficient to prove that the 
 component Q of the attraction in the direction of the normal to 
 the plane OST is zero. 
 
 We shall first find the components X, F, Z, of the attraction 
 in the directions of the axes, and thence deduce the value of Q. 
 
 * The centre of this ellipsoid is at the centre of gravity ; its axes are in the 
 directions of the principal axes, and their lengths are determined by the equations 
 
 Ma* = A, Mb* = B, M & = C. 
 
 This ellipsoid is used by Professor Mac Cullagh in his u Theory of Kotation": see 
 Eev. S. Haughton's Account of Professor Mac Cullagh's Lectures on that subject, 
 Transactions R. I. A., VOL. xxn. p. 139 (supra, p. 329). 
 
362 On the Attraction of Ellipsoids. 
 
 Now, 
 
 dV M , 3 3 dl 
 
 but, 
 
 dl ^(A-I)cosa t dl 
 dx'~ / ' 4/ 
 
 Hence we have 
 
 M , 3 .. , 
 
 X = 2 GOSa + -^ (A + B + C- 51) COS a + -- 
 
 (7) 
 
 M , 3 , 3(7 cos 7 r 
 
 ' 
 
 r 
 
 Now, 
 
 Q = X cos a 4 Y cos j3 + Z cos 70 ; 
 but, 
 
 sin cos a = cos (3 cos 7' - cos 7 cos $ , 
 
 sin cos j3 = cos 7 cos a - cos a cos 7', 
 sin cos 70 = cos a cos j3' - cos /3 cos a' ; 
 The following relations, moreover, exist : 
 
 a z cos a = rp cos a, b z cos /3' = r/? cos |3, c z cos 7' = rp cos 7 : (8) 
 
 hence, by substitution, we have 
 
 b z c z c z - a z 
 
 COS = : COS /3' COS 7', COS j3 = : COS 7' COS a', 
 
 pr sin jw- sin < 
 
 COS 70 = : COS a' COS/3'. 
 
 pr sin 4 
 
On the Attraction of Ellipsoids. 363 
 
 Substituting these values for cos a , cos j3 , cos 70, in the expres- 
 sion for Q, and observing that 
 
 COS a' COS a + COS |3' COS )3o + COS y' COS 70 = 0, 
 
 we get 
 
 Q = T. - : cos a' cos Q' cos y'= o. (9) 
 
 r 4 pr sm 
 
 PROPOSITION YI. 
 
 The same things being supposed, to find the other Components 
 of the Attraction, namely It in the direction of the centre of gravity 
 MO, and P in the transverse direction TS. 
 
 To find ^; , 
 
 JR = X cos a + Y cos jS' + Z cos y' ; 
 
 = +~(A + + C-3I). (10) 
 
 To find P ; 
 
 P = X cos ai + Y cos )3i + Z cos 71 ; 
 but, 
 
 sin cos ai = cos a cos - cos a, 
 
 sin0 cos]3i= cos|3'cos0 - cos/3, 
 
 sin cos 71 = cos 7' cos - cos 7. 
 
 Substituting for cos a, cos|3, 0087 their values from (8), we get 
 
 cosai = -- r^- cos a', cos/3i = -- : cosjS', 
 pr sm ^>r sm 
 
 c 2 -^ 2 
 
 COS 71 = -- r COS 7 . 
 
 jt?r sm 
 
 Substituting these values of cosai, cosjS,, 0087!, and observing 
 
 that 
 
 cos a cos d! + cos ]3' cos ]3i + cos 7' cos 71 = 0, 
 
364 On the Attraction of Ellipsoids. 
 
 we have 
 
 P = -- TT 
 
 r 4 pr sin 
 
 or, 
 
 SxST. (11) 
 
 The negative sign indicates* that the force P acts in the direction 
 TS, i. e. from the radius vector towards the perpendicular of the 
 ellipsoid of gyration. If the force P be resolved into three 
 others in the direction of the axes, it is evident from the values 
 given in Proposition V. for X, Y, Z> that these components are 
 
 3 (A -I) , 3(5-7) Q/ 3(0-7) 
 
 4 cos a', ~7 - cos -I cos 7 -t (12) 
 
 * The direction of the force P, which Professor Mac Cullagh determines by the 
 interpretation of the negative sign, may be very clearly seen from the following 
 considerations. This force exists in every case where the three principal moments 
 of inertia of the system at are not all equal, that is, when the ellipsoid of gyra- 
 tion is not a sphere. The greatest axis of that ellipsoid is manifestly towards that 
 part of the body in which there is a deficiency of attracting matter. If we now con- 
 sider the position of a perpendicular on a tangent plane of an ellipsoid with rela- 
 tion to the corresponding radius vector, we shall find that it always lies away 
 from the greatest axis. But the transverse force has been shown to be in the 
 plane of radius vector and perpendicular. Therefore, the direction of the transverse 
 force, being towards the preponderating matter, must be from T to S. 
 
 f The results given by Professor Mac Cullagh in Propositions V. and VI. may 
 be otherwise obtained, and, perhaps, with greater facility, by introducing the con- 
 sideration of the statical moment of the attracting force. * 
 
 If the three principal moments of inertia were equal to each other, then the 
 whole attraction would be in the direction of the centre of gravity, and its mag- 
 
 nitude would be 
 
 M 
 rft 
 
 In general, however, the attracting mass will be of an irregular shape ; there will 
 exist then, in addition to the principal part of the attraction, which will be central, 
 a transverse force which will tend to cause a motion of rotation about the centre of 
 gravity. 
 
 The components of the moment of this transverse force in the three principal 
 
 planes are 
 
 x'Y-y'X, y'Z-z'Y, z'X-x'2; 
 
 * See Rev. R. Townsend, in the " Dublin University Examination Papers," 1849, p. 51. 
 
On the Attraction of Ellipsoids. 365 
 
 PROPOSITION YIL 
 
 An Ellipsoid is composed of ellipsoidal strata of different densities 
 and of variable but small ellipticities ; to find the Components, cen- 
 tral and transverse, of its Attraction on an external point. 
 
 The values found in the last Proposition for the components 
 of the attraction of any mass on a very distant point will be 
 found to hold in the present case, whatever be the position of 
 the attracted point. In order to show this, we shall first prove 
 it for a homogeneous ellipsoid of small ellipticities. Such an 
 ellipsoid being given, another confocal with it can be con- 
 structed so small, that the distance to the attracted point may be 
 
 but from (7), 
 
 x'Y- y'X = -- - cos a cos ft' = -- - (a 2 - i 4 ) cos a" cos ft', 
 
 q / T> _ f~1\ "\M 
 
 y' Z z'Y = -- - - - cos ft' cos 7' = -- - (b* - c 2 ) cos ft' cos 7', 
 
 3(0 -A) 3M.. 
 
 z X - x Z = -- - cos 7 cos a = -- TJJ- (o 2 a 2 ) cos 7 cos a . 
 
 Now it is well known, that 
 
 ( 2 - 2 ) cos o' cos ft', | (i 2 - c 2 ) cos ft' cos 7', i (<? 2 - a 2 ) cos 7' cos a, 
 
 are the areas of the projections of the triangle OST on the principal planes. Hence 
 it follows that the resultant moment lies in the plane of the radius vector OT and 
 the perpendicular OS to the corresopnding tangent plane of the ellipsoid of gyra- 
 tion ; the tangent plane being perpendicular to OM. It appears, also, that the 
 magnitude of the resultant moment is 
 
 and therefore that the transverse component of the attraction 
 
 . 
 
 Or, the values of the central force and the moment of the transverse force may 
 be obtained directly from the expression (6) for the potential T. This function is 
 of such a nature, that its* differential coefficient with relation to any line (the sign 
 being changed) is equal to the resolved part of the attraction in that direction ; and 
 the differential coefficient with relation to any angle (the sign being changed as 
 
366 On the Attraction of Ellipsoids. 
 
 regarded as very great, compared with the axes of this ellipsoid. 
 The components of the attraction of this small ellipsoid on the 
 distant point are given by the expressions (10) and (11) : now as 
 the attractions of two confocal ellipsoids on an external point are 
 in the same direction, and proportional to their masses : the 
 components of the attraction of the proposed ellipsoid will, 
 therefore, be 
 
 the letters with suffixes referring to the small ellipsoid. 
 
 The attracting ellipsoids being confocal, their ellipsoids of 
 gyration are confocal also ; hence it follows that 
 
 7l/" 
 
 =r (A, + B, + d - 3/0 = A + B + C - 37, 
 JxL\ 
 
 and 
 
 x 
 
 xTi = OS x ST. 
 
 before) gives the component in the plane of that angle of the moment of the 
 attractive force. 
 Hence, 
 
 
 
 dr 
 Again, if N be the component of the moment of the attractive force round OZ, 
 
 ' 
 
 but 
 
 (xf --y' ) F(x' z + y' 2 ) = where Fis any function ; 
 \ ay ax I 
 
 Ax'* + '2+ Cz'* 
 
 (C08'COS0'). 
 
 The two other components of the moment may be similarly obtained. The 
 remainder of the proof is the same as in the former part of this note. 
 
On the Attraction of Ellipsoids. 367 
 
 It appears, from this, that the central and transverse components 
 of the attraction of a solid ellipsoid of uniform density, and 
 whose ellipticities are small, on any external point whatever, 
 are given by the same formulae as the corresponding components 
 of the action of any mass on a distant point. 
 
 Now it is a property of moments of inertia, that they are 
 sultractive, that is, the difference of the moments of inertia of 
 two masses with relation to any axis is equal to the moment of 
 inertia of the difference of those masses with relation to the same 
 axis. And the values at which we have arrived for the central 
 force, and for the three components of the transverse force, con- 
 tain in each term either a mass or a moment of inertia in the 
 first power, and therefore, these values also are subtractive. 
 Hence the two components of the attraction of a homogeneous 
 mass contained between two concentric and coaxal ellipsoids of 
 small ellipticities, are given by formulae (10) and (11). Now 
 suppose an ellipsoidal mass to be composed of strata bounded 
 by ellipsoids of different but small ellipticities, each stratum 
 being homogeneous throughout its extent, while the density 
 varies from one stratum to another according to any law ; then, 
 since those formulae hold for the action of each stratum sepa- 
 rately, and since the terms of which they are made up are in 
 their nature additive, they hold for the entire mass.* 
 
 PROPOSITION VIII. 
 
 An oblate Spheroid is composed of spheroidal strata of different 
 densities and of variable but small ellipticities ; to find the Com- 
 of its Attraction on any external point. 
 
 The expressions given in the last Proposition for R and P 
 become simplified in this case. Let OZ be the axis of revolu- 
 tion, and let A denote the angle which OM makes with the plane 
 
 * See Professor Mac Cullagh, in the " Dublin University Examination Papers," 
 1833, p. 268. 
 
368 On the Attraction of Ellipsoids. 
 
 XY ; then since A and B are equal, we have 
 
 I = A cos 2 \ + C sin 2 A ; 
 and therefore 
 
 A + E + C-3I = (C-A) (l-3sin 2 A); 
 also 
 
 M OS x OT = (A - C) sinA cos A. 
 
 Substituting these values in the expressions for R and P, we 
 have 
 
 ^ = + a -3 sin* A), 
 
 C A 
 P = 3 7r - cos A sin A. (14) 
 
 The direction of the force P is towards the plane of the 
 equator ; this appears from the shape of the " ellipsoid of gyra- 
 tion," which in this case is a prolate surface of revolution. 
 
 PROPOSITION IX. CLAIRAUT'S THEOREM. 
 
 Whatever be the law of variation of the Earth's density at 
 different distances from the centre, if the ellipticity of the surface be 
 added to the ratio which the excess of the polar above the equatorial 
 
 K 
 
 gravity bears to the equatorial gravity, their sum will be ^ q, where 
 
 4/ 
 
 q is the ratio of the centrifugal force at Equator to equatorial gravity. 
 
 For suppose the attracted point M to be on the surface of 
 the earth, which is known to be an oblate spheroid of small 
 ellipticity. Then, from the principles of Hydrostatics, since the 
 tangential force is zero, we have 
 
 R cos B - P sin 9 - w*r cos A cos (0 - A) = 0, (15) 
 
 where to denotes"the angular velocity, and the angle which 
 the tangent to the meridian through the attracted point makes 
 with the radius vector. Developing cos (B - A) and arranging, 
 we obtain 
 
 (R - o> 2 r cos 2 A) cos 9 = (P + w 2 r cos A sin A) sin 9. (16) 
 
On the Attraction of Ellipsoids. 369 
 
 But, from the property of the elliptic section made by the plane 
 of the meridian, we have 
 
 , Q e* sin X cos A rt . x 
 cot = - = 2e sm A cos A, q. p.. 
 
 1 - e 2 cos 2 A 
 
 where e is the excentricity and f the ellipticity of this ellipse. 
 
 Substituting in (16) this value of cot 9, and the values of R 
 and P from (13) and (14), the equation of equilibrium becomes 
 
 \M 3C-A 
 
 + ~ 
 
 (1-3 sin 2 A) - w z r cos 2 A J 2e sin A cos A 
 
 C- A 
 
 3 - + w 2 r ) sin A cos A, 
 
 r* 2 r 4 
 
 or, approximately, 
 
 j + o ^r^ (1-3 sin 2 A) -rfa cos 2 AJ 2 = 3^^+^M. 
 
 \ Ct> & (I (i 
 
 If we neglect quantities of the second order, this equation 
 becomes 
 
 &M 0-A , 
 
 r- = o + or#. (17) 
 
 a 2 a 4 
 
 We have thus arrived at a relation which enables us to ex- 
 press the unknown quantity C-A, in terms of quantities which 
 are all known, and, therefore, to eliminate the former from any 
 other equation in which it may occur. 
 
 Now let R e and R p denote the equatorial and polar attrac- 
 tions respectively ; we have from the general value of R (13), 
 
 M 3 C-A 
 
 but 
 
 M 
 
 "' 
 
 _M ~ 
 
 ~~*~~' 
 
370 On the Attraction of Ellipsoids. 
 
 But 
 
 G p = E p and G e R e - <*> 2 ; 
 
 Eliminating ^ by means of equation (17), we get 
 
 G p - G e _ 5 o^ 
 
 ~6T~ f 2"^" ; 
 
 or 
 
SUPPLEMENT. 
 
 EGYPTIAN CHRONOLOGY 
 
( 373 ) 
 
 I. ON THE CHKONOLOGY OF EGYPT. 
 
 {Proceedings of the Royal Irish Academy, VOL. i., p. 66. Read April 24, 1837.] 
 
 IN this Paper the author endeavours to ascertain the names 
 of the Egyptian sovereigns who were contemporary with Moses. 
 For this purpose he finds it necessary to determine the interval 
 between two celebrated epochs the reign of Menes and the 
 Exodus of the Israelites. He conceives that the former epoch 
 is fixed by the " old chronicle " at the distance of 443 years 
 from the beginning of a cynic (or canicular) cycle ; and he thinks 
 it strange that this simple meaning should not have occurred to 
 chronologists, who have universally supposed the " cynic cycle " 
 of the old chronicle to be a series of demi-god kings who derived 
 that appellation from the dog-headed Anubis. The canicular 
 cycle is a well-known period of 1460 years, which the Egyptians 
 seemed to have used for computing time, as we sometimes use 
 the Julian period. One of these cycles commenced in the year 
 2782 before the Christian era ; and if we reckon 443 years in 
 advance, we shall have the year B. c. 2339 for the commence- 
 ment of the reign of Menes. This date agrees well with the com- 
 putation of Josephus, who says that the interval from Menes 
 to Solomon was upwards of 1300 years. Again, we are told by 
 Clemens of Alexandria, that the Exodus of the Israelites took 
 place 345 years before the beginning of a canicular cycle. This 
 is evidently the cycle which commenced B. c. 1322 ; and hence 
 we have B. c. 1667 for the date of the Exodus. The interval 
 between Menes and the Exodus was, therefore, about 670 years. 
 
374 On the Chronology of Egypt. 
 
 If, now, we take the catalogue of Eratosthenes, which com- 
 mences with Menes, we shall find, at the distance of 670 years 
 from Menes, a king named Achescus Ocaras, who reigned only 
 one year ; preceded by a king named Apappus, who reigned a 
 hundred years, and succeeded by queen Nitocris, who reigned 
 six years. Mr. Mac Cullagh thinks that Apappus is the king 
 in whose reign Moses was born ; that Ocaras is he who pursued 
 the Israelites to the Red Sea ; and that Nitocris is the famous 
 queen mentioned by Herodotus. It may be objected that 
 Eratosthenes gives us the succession of Theban kings, whereas 
 the Pharaohs of the Mosaic history reigned in Lower Egypt ; 
 but it is remarkable that the three sovereigns mentioned above 
 are found in Manetho's dynasties among those who reigned at 
 Memphis ; and it is singular that these are the only sovereigns 
 (except Menes and his immediate successor) in which the dynas- 
 ties of Manetho and the catalogue of Eratosthenes agree. All 
 the other names are different. Of course the predecessors of 
 Apappus, at Thebes and at Memphis, were different ; and thus 
 we can easily understand how there arose up at Memphis " a 
 new king who knew not Joseph." It would appear, in fact, 
 that Apappus was of a Theban family, and that he succeeded, for 
 some reason or another, to the throne of Lower Egypt. He was 
 only six years old (as we learn from Manetho) when he came to 
 the throne ; and it is natural to suppose that his chief advisers, 
 as he grew up, were the courtiers who accompanied the young 
 king from his own country to Memphis, and who knew nothing 
 of Joseph, and cared nothing for his people. Accordingly, when 
 Apappus arrived at manhood he issued an order that every 
 male child of the Hebrews should be destroyed, lest they should 
 grow too numerous for the Egyptians ; and, under these circum- 
 stances, Moses was born in the twenty-first year of his reign, 
 and was saved by the king's young daughter, a girl about ten 
 years old. About the sixtieth year of Apappus, Moses was 
 obliged to fly to the land of Midian, for having killed an 
 Egyptian ; and when at length the king of Egypt died 
 " after many days," as it is in the original Moses returned in 
 
On the Chronology of Egypt. 375 
 
 the beginning of the reign of Ocaras, before whom were per- 
 formed those signs and wonders which prepared the way for the 
 departure of the Israelites. On the night of the Passover, the 
 king lost his first-born, perhaps his only son ; and this may be 
 the reason that he was succeeded by his sister Nitocris. The 
 short reign of Ocaras (a single year) might be explained by 
 supposing he was drowned in the Eed Sea ; but as there is 
 nothing in the sacred narrative which obliges us to admit that 
 the king perished in this manner, we may adopt the account of 
 Herodotus, that he was murdered by his subjects. "We may 
 imagine that some of his nobles remained with Pharaoh on the 
 shore ; and that when they saw the sea return and swallow up all 
 that had gone in after the Israelites, they murdered the king, 
 whose obstinacy had brought such calamities on his people, and 
 then placed his sister Nitocris on the throne. As Nitocris was the 
 daughter of Apappus, there is nothing to prevent us from sup- 
 posing that the queen, now ninety years old, was the princess who 
 had saved the infant Moses. Weary of her life, she lived only 
 to avenge her brother. For this purpose, says Herodotus, she 
 constructed a large subterranean chamber, to which, when it was 
 finished, she invited the principal agents in her brother's death ; 
 and there, by the waters of the Nile admitted through a secret 
 canal, they were drowned in the midst of the banquet. The 
 queen then threw herself into a room filled with ashes, where 
 she perished. 
 
 
( 376 ) 
 
 II. ON THE CATALOGUE OP EGYPTIAN KINGS, WHICH 
 IS USUALLY KNOWN BY THE NAME OF THE 
 LATERCTTLUM OF ERATOSTHENES. 
 
 [Proceedings of the Royal Irish Academy, VOL. n. p. 366. Read January 9, 1843.] 
 
 THIS Catalogue, which the distinguished mathematician and 
 philosopher whose name it bears drew up by command of 
 Ptolemy Euergetes, contains a long series of kings who reigned 
 at Thebes in Upper Egypt, and has been preserved to us in 
 the Chronographia of Greorgius Syncellus, a Greek monk of the 
 eighth century. It is a document which has been made much 
 use of by chronologers ; by some of whom (as by Sir John Mar- 
 sham for example, who calls it " venerandimmum antiquitatis 
 monumentum") , it has been reckoned of the very highest autho- 
 rity ; but it is extremely corrupt in the latter part, owing to the 
 carelessness with which it was transcribed either by Syncellus 
 himself or his immediate copyists. The writers on Egyptian 
 antiquities have in consequence been much perplexed in settling 
 the chronology of the reigns in which the errors exist, and the 
 attempts that have been made to remove the confusion have 
 only served to increase it. It was the object of the author to 
 restore the document to its original state, and he showed that 
 this might be effected, with complete certainty, by a proper at- 
 tention to the manuscripts of Syncellus. Of these only two are 
 
On the Catalogue of Egyptian Kings. 377 
 
 known : one has been used by Father Q-oar, the first editor of 
 the Chronogr aphia (Paris, 1652) ; the other, which is a much 
 better one, has been collated by Dindorf, the second and latest 
 editor. Dindorf 's edition was published at Bonn, in the year 
 1829, as part of the Corpus Scriptorum Historic^ Byzantince, and 
 on its first appearance Mr. Mac Cullagh had satisfied himself as 
 to the original readings of the Catalogue, and had seen how to 
 account for the errors which, probably from Syncellus's own 
 negligence, had crept into it ; but he did not publish his conclu- 
 sions at the time, thinking that similar considerations could not 
 fail to occur to some of the numerous writers who were then 
 giving their especial attention to such subjects. This, however, 
 has not been the case. Chronologers have continued to follow in 
 the footsteps of Groar, a man of little learning, and of no critical 
 sagacity, who corrected the Catalogue most injudiciously, and 
 whose corrections, strange to say, are left without any remark 
 by Dindorf. Thus Mr. Cory, in his Ancient Fragments, a work 
 much referred to, merely transcribes Groar's list ; and Mr. Culli- 
 more, in attempting to reconcile ancient authors with each other 
 and with the monuments, has adopted an hypothesis respecting 
 the identity of two sovereigns which is not tenable when the 
 true version of the Catalogue is known. Even in Groar's edition, 
 however, there was quite enough to have led a person of ordi- 
 nary judgment to the correct readings of the Catalogue, though 
 perhaps they could not be said to be absolutely certain without 
 the additional light obtained from that of Dindorf. 
 
 The Catalogue in question professes to contain the names of 
 thirty-eight sovereigns, with the years of their reigns ; the whole 
 succession occupying, as is stated, a period of 1076 years ; but 
 it is only in the last eight reigns that the errors and inconsist- 
 encies occur. The thirty-second prince is called Stamenemes |3, 
 that is, Stamenemes the Second, though there is, at present, no 
 other of that name in the list ; and the beginning of his reign 
 as appears from the years of the world, which Syncellus has 
 annexed according to the Constantinopolitan reckoning follows 
 
37$ On the Catalogue of Egyptian Kings. 
 
 the termination of the preceding one by an interval of twenty- 
 six years. Jackson, in his Chronological Antiquities y is positive that 
 this prince is called the Second by a mistake, and adds the years 
 that are wanting to the reign of his predecessor, as Groar had 
 previously done. In the first part of this view all authors, with- 
 out exception, are agreed, though they do not explain how a 
 mistake, so very odd, could have originated ; but the learned 
 Marsham who, having adopted the short chronology of the 
 Hebrew Bible, is so hard pressed to find room for the Egyptian 
 dynasties that he is obliged to begin the reign of Menes the 
 very year after the Deluge is glad to omit the twenty-six 
 years altogether, thus reducing the sum of all the reigns to 
 1050 years, contrary to what is expressly stated by Syncellus. 
 The natural inference from the state of the MSS. is, however, 
 simply this : that the thirty-second king was Stamenemes I., 
 that he reigned twenty-six years, and was succeeded by Stame- 
 nemes II. "We may easily conceive that the eye of the tran- 
 scriber, deceived by the identity of names, passed over the first, 
 and rested on the second, thus occasioning the error. Indeed 
 there can now be no doubt that this was the fact ; because, in 
 the MS. marked (B) by Dindorf, the next king is numbered as 
 the thirty-fourth, the next but one as the thirty-fifth, and so on ; 
 which shows that a name had dropped out, and this name could 
 be no other than that of Stamenemes I., who must have filled 
 the vacant interval, and must consequently have reigned the 
 number of years that has been assigned to him. 
 
 As neither Gk>ar nor any other writer perceived this omission, 
 the successor of Stamenemes II. has always been reckoned as 
 the thirty-third in the list, and the next following as the thirty- 
 fourth, &c. But as one error begets another, the omission was 
 compensated by the insertion of an anonymous king, who is 
 placed thirty-sixth in the list, with a reign of fourteen years ; 
 the insertion being necessary to complete the number (thirty- 
 eight) which the Catalogue ought to contain. And, by a fur- 
 ther error, these fourteen years are taken out of the reign of the 
 
On the Catalogue of Egyptian Kings. 379 
 
 thirty-seventh sovereign, who ought to have nineteen years 
 instead of the five that have been hitherto assigned to him. This 
 last error was occasioned by an ignorant correction of a mistake 
 which is found in both the MSS., and which therefore probably 
 arose from the carelessness of Syncellus himself. The thirty- 
 seventh king and his predecessor are stated to have begun to 
 reign in the same year of the world, and to have reigned the 
 same number of years (five). Now from what goes before it is 
 plain that both these numbers belong to the thirty-sixth king ; 
 and from the year of the world in which the thirty-eighth and last 
 king began to reign, it is clear that the thirty-seventh reigned 
 nineteen years. The mistake in the MSS. is one which might 
 easily be made by a thoughtless writer ; for the Catalogue is 
 given in detached portions a few reigns at a time separated 
 by a great quantity of other matter, and the name of the thirty- 
 sixth king ends one of these portions, while that of the thirty- 
 seventh begins another ; so that, not having both before his eyes 
 at the same moment, a person so careless as Syncellus might, 
 without being conscious of it, attach the same reign and date to 
 the two names, by transcribing twice over the same line of num- 
 bers in the Catalogue which he was copying ; the whole of which 
 Catalogue, in all likelihood, he had previously drawn up in a 
 tabular form, with the years of the world annexed according to 
 his own chronology, that it might be ready, as any portion of it 
 was wanted, for immediate transference to his pages. Such 
 seems to be the natural account of the matter ; but, as usual, it 
 does not occur to Groar, who takes the opportunity, which the 
 confusion affords him, of foisting in his supplementary king 
 between the two last mentioned, giving each of these five years, 
 as in the MS., by which means he obtains room for him ; while 
 on the other hand he alters the year of the world attached to the 
 thirty-seventh king, so as to make it suit his hypothesis. 
 
 The following is a view of the last eight reigns, as they ap- 
 pear to have stood in the original document, compared with the 
 erroneous list of Groar. The years of the world are omitted, as 
 
380 On the Catalogue of Egyptian Kings. 
 
 being of no importance, except so far as they are useful in the 
 preceding argument. 
 
 I. ,GOAE'S LIST. 
 
 Years. 
 
 31. Peteathyres reigned 42 
 
 32. Stamenemes ,, 23 
 
 33. Sistosichermes ,, 55 
 
 34. Maris 43 
 
 35. Siphoas 5 
 
 36. Anonymous 14 
 
 37. Phruoro 5 
 
 38. Amuthartcsus 63 
 
 II. CORRECTED LIST. 
 
 Years. 
 
 31. Peteathyres reigned 16 
 
 32. Stamenemes I. 26 
 
 33. Stamenemes II. ,, 23 
 
 34. Sistosichermes ,, 55 
 
 35. Maris 43 
 
 36. Siphoas ,, 5 
 
 37. Phruoro 19 
 
 38. Amuthartceus 63 
 
 The interval of time which has been shown to belong to the 
 first Stamenenes, and which was added by Groar to the reign of 
 Peteathyres, is differently disposed of by Mr. Cullimore, in a 
 chronological table which he has given in the second volume of 
 the Transactions of the Eoyal Society of Literature. His object 
 being to compare the lists of Eratosthenes, Manetho, &c., with the 
 supposed hieroglyphical series, he makes Saophis, the fifteenth 
 in Eratosthenes' Catalogue, the same as a king whose name 
 is read Phrathek Osirtesen ; but the forty-third year of the 
 latter is mentioned on the monuments, whereas Saophis has 
 only twenty-nine years in the Catalogue. To escape from this 
 difficulty, therefore, Mr. Cullimore adds the unappropriated in- 
 terval to the reign of Saophis, thus giving him fifty-five years 
 instead of twenty-nine. But it now appears that such a suppo- 
 sition is altogether inadmissible, and consequently the two per- 
 sonages in question cannot be identified a circumstance which 
 proves that there is some fault in Mr. Cullimore's assumptions, 
 and that his other conclusions, at least in this part of his Table, 
 cannot be relied on. 
 
 The corrections here given do not interfere with the infe- 
 rences drawn by Professor Mac Cullagh from the Catalogue of 
 Eratosthenes in a former Paper on Egyptian Chronology (Pro- 
 ceedings of the Koyal Irish Academy, vol. i., p. 66),* because the 
 
 * Supra, p. 373. 
 
On the Catalogue of Egyptian Kings. 381 
 
 portion of the Catalogue with which he was there concerned ter- 
 minates with the reign of Queen Nitocris, the twenty-second in 
 the list. The corrections, indeed, though not hitherto published, 
 were made long before the date (April, 1837) of that Paper, 
 but not before he had adopted the hypothesis therein proposed, 
 as an answer to the old and ever-recurring question Who were 
 the Egyptian sovereigns that were contemporary with Moses ? 
 For it was in consequence of this hypothesis, which had sug- 
 gested itself to him at a very early period, that he was led to 
 examine the Catalogue minutely, in order to discover whether 
 his chronology was affected by its errors. 
 
 Having been led to refer to his hypothesis, Mr. Mac Cullagh 
 took occasion to observe that, in the interval which had elapsed 
 since it was published, he had not met with any facts that were 
 opposed to it : on the contrary, the more he considered it, the 
 more he was inclined to believe in its reality ; though it was 
 entirely different from every other that had been proposed, 
 either by modern chronologers or by the early Fathers of the 
 Church, in their manifold attempts to connect the narrative of 
 Moses with the remaining fragments of Egyptian history. The 
 hypothesis, indeed, is the only one which, while it gives a pro- 
 bable date for the Exodus, also satisfies what Mr. Mac Cullagh 
 conceives to be the necessary conditions of the question ; namely, 
 a very long reign of at least eighty years during which the 
 Israelites were persecuted, succeeded by a very short one 
 apparently not more than a year during which their deliver- 
 ance was wrought ; and it is interesting in itself, on account of 
 the remarkable connexion which it establishes between sacred 
 and profane history, and the highly dramatic character of the 
 events which are thus, for the first time, brought into view. 
 
 THE END. 
 
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