UC-NRLF 
 
 B 14 ESE 7fc,3 
 
 
FRESNEL AND HIS FOLLOWERS, 
 
 A CRITICISM. 
 
 TO WHICH AEE APPENDED 
 
 OUTLINES OF THEORIES OF DIFFRACTION AND TRANSVERSAL 
 VIBRATION. 
 
 BY ROBERT MOON, M.A. 
 
 TELLOW OF QUEENS' COLLEGE, CAMBRIDGE, 
 
 CAMBRIDGE: 
 
 MACMILLAN, BARCLAY, AND MACMILLAN; 
 
 LONDON: GEORGE BELL. 
 
 1849 
 

 CAMBRIDGE : 
 Metcalfe and Palmer, Printers, Trinity-street. 
 
VICE-CHANCELLOR AND HEADS OF HOUSES 
 IN THE UNIVERSITY OF CAMBRIDGE, 
 
 THE FOLLOWING APPEAL 
 
 AGAINST THE CONTINUANCE OF A GREAT BRANCH OF THE 
 
 MATHEMATICAL COURSE PURSUED IN THAT UNIVERSITY, 
 
 IS RESPECTFULLY INSCRIBED 
 
 BY THEIR 1GJ3EDIENT SERVANT, 
 
 THE AUTHOR. 
 
 248218 
 
CONTENTS. 
 
 Page 
 
 On Fresnel's Theory of Diffraction ... 1 
 
 A New Theory of Diffraction . . . .7 
 
 On Fresnel's Theory of Double Refraction . . 14 
 
 Note by Mr. Archibald Smith, with Remarks thereupon . 23 
 
 Remarks by 'Jesuiticus' .... 26 
 
 Reply to ' Jesuiticus' . . . . .29 
 
 Note by Professor Potter 39 
 
 On Double Refraction Continued . . . .40 
 
 On Fresnel's Theory of Polarization where there is no Double Refraction 48 
 On Mr. Kelland's Theory of Heat ... 60 
 
 On Professor Powell's Account of the Theory of Finite Intervals 73 
 On Mr. Green's Theory of Reflection and Refraction . 87 
 
 On Professor MacCullagh's Theory . . . .92 
 
 Reply to some Remarks of Mr. Airy ... 97 
 
 Professor Challis's Theory of Luminous Rays . . . 103 
 
 Mr. Stokes's Theory . . . . .114 
 
 Mr. O'Brien's Theory . . . . .127 
 
 A New Theory of Transversal Vibration . . . 132 
 
 On the Supposed Incompatibility of Transversal Vibration with the 
 Sonorous Medium. Fallacy of Poisson's Proof of the Uniform 
 Velocity of Sound * . . . .145 
 
 On Professor Challis's mo*e Recent Speculations . . 153 
 
EKRATA. 
 
 Page 44, line 2, for xy read XYZ. 
 Page 56, line 5, for evident read incident. 
 Page 81, line 20, for $ read 8. 
 
PREFACE. 
 
 THE circumstances under which the following pages are 
 presented to the public are probably familiar to many of my 
 readers. It fell to my lot to denounce a theory not merely 
 all but universally received, but which had been made the 
 subject of the most extravagant eulogy. In conducting such 
 a discussion it was hardly to be expected that I should speak 
 with very profound respect of persons who, while possessing 
 the highest reputation for sagacity and judgment, had been 
 seduced into patronising the most extraordinary errors. It 
 happened however that the tone of animadversion which I 
 felt called upon to assume did not meet with the approbation 
 of the conductor of the periodical which I had made the 
 medium of my communications, who, with a practical respect 
 for authority which is sometimes found to accompany very 
 opposite theoretical views, refused to continue the publication 
 of my papers. 
 
 Undoubtedly, under the circumstances in which I was 
 placed, I might have pursued a different course. I might 
 have contented myself in the progress of my criticism with 
 suggesting that this or that principle was doubtful, that such 
 and such processes were unsatisfactory, that such and such 
 conclusions were not warranted by the premises from which 
 they were deduced. Had I done no more than this it is 
 possible that my refutation might have been admitted to an 
 honorable niche in the Optical Pantheon, side by side with 
 the theories which it professed to overthrow. It is not im- 
 possible that the easy faith which could tolerate a dozen con- 
 flicting theories on the same subject at one and the same 
 
PREFACE. 
 
 time, might have regarded with indulgence a mild and merely 
 speculative scepticism, which, although suggesting doubts as 
 to every existing theory, made no very urgent call upon the 
 adherents of any particular fallacy to renounce the worship 
 of the false object of their admiration. This however was not 
 the object I had in view. FresneFs Theory appeared to me to 
 be something moie than erroneous, it appeared to me to be 
 absurd ; and that not merely deductively, like a position in 
 Euclid, but primd facie, so absurd that it ought never to have 
 been entertained. It appeared to me that, besides the injury 
 which ordinarily results from the admission of a false theory, 
 a grievous wrong had in this case been done to the cause 
 of science, by the introduction of principles of the truth of 
 which there was not a shadow of evidence, and by the 
 allowance of processes the fallacy of which, if unsanctioned 
 by authority, the merest tyro might have detected. It ap- 
 peared to me that under these circumstances there was a call 
 for the strongest animadversion. It was not so much my 
 object to refute Fresflel's Theory with its train of subsidiaries, 
 as to stigmatise the whole system of investigation which had 
 been adopted; to shew that the methods which had been 
 employed, not only did not but that they could not lead to 
 a satisfactory result. I felt it to be desirable that there should 
 be no mistake as to the nature of my views. The delusion 
 had been too long entertained and too extensively diffused to 
 admit of a doubtful assertion of opinion. There was too 
 much of a disposition to put down opposition with a high 
 hand, to allow of anything approaching to irresolution. It was 
 necessary to give the lie direct to the Theory. 
 
 It is possible that to some of my readers Fresnel's Theory 
 may seem less objectionable than it has done to me; it is 
 possible that his hypotheses may appear to them less extra- 
 vagant, his contradictions les palpable, his errors less gross, 
 his contempt of the ordinary principles of reasoning less 
 complete than I have considered them. It may be that, 
 
PREFACE. IX 
 
 although not differing from me materially in these respects, 
 they may think the fact of having overlooked such defects 
 less inexcusable than it has appeared to me : but holding 
 the opinions I entertained, it was not possible but that I 
 should express myself strongly with regard to them. It was 
 not possible that I should comment upon principles which 
 go to subvert the foundation of philosophical enquiry with the 
 same composure as I might correct a mistake in arithmetic; 
 it was not possible that I should speak of errors which I believe 
 to be without a parallel in the history of science, as if they were 
 of every-day occurrence and such as any one is liable to : 
 others might have acted differently under the same circum- 
 stances for myself I felt it would be misrepresentation to 
 have done so. 
 
 If any one should be disposed to doubt the reality of the 
 evils I have ascribed to the reception of Fresnel's Theory, 
 I would recommend to his perusal the account given by Sir 
 John Herschel (Encyc. Met. Art. LIGHT) of the Theory of 
 Double Refraction ; and I would suggest to him whether some 
 singular cause must not have been in operation, when one so 
 accomplished, so able, could be betrayed into so complete a 
 mystification as is there exhibited. 
 
 If we look at the different theories which have been proposed 
 as to the constitution of the ethereal medium Fresnel's ; 
 M. Cauchy's (two) ; Mr. Kelland's (two) ; Mr. Tovey's (two at 
 least); Sir John Lubbock's; Mr. O'Brien's (two); Professor 
 Challis's ; to say nothing of Mr. Green's, Mr. MacCullagh's, 
 and Mr. Stokes's, every one of which possesses at this moment 
 some more or less bewildered votaries ; if I say we look at these 
 several theories, the reflection cannot but suggest itself that 
 those must be very odd principles of investigation which from 
 the same results lead to such opposite causes. From the 
 powerful machinery they have brought to bear upon the sub- 
 ject, one might suppose it was the object of these gentlemen 
 to invent light rather than to discover the means already 
 
x PREFACE. 
 
 devised and put in action for the accomplishment of that great 
 phenomenon : nor can we be surprised if with such an aim 
 the production of each optical Frankenstein should be some 
 frightful monster. 
 
 Much has at various times been urged in favour of the 
 system of trial and error, and the example of Kepler has been 
 sufficiently enlarged upon; but I never heard it contended by 
 the stan chest advocates of the haphazard system of philosophy, 
 that Kepler proposed twelve curves, at one and the same time, 
 as candidates for the honour of representing the earth's orbit. 
 The followers of Fresnel certainly require some exponent of 
 their system of logic. If the spirit of Newton were again to 
 walk the earth, and could be made cognizant of the Theory of 
 Double Refraction as originally propounded, it would surely 
 appear to that illustrious shade that some extraordinary revo- 
 lution had taken place in the laws of evidence since his hand 
 balanced the facts of creation. The praises of Bacon have 
 always been a favourite theme with his countrymen, occasion- 
 ally I should imagine rather* to the amusement of our conti- 
 nental neighbours, and never more conspicuously than at the 
 present day; yet it would be difficult to conceive a more 
 thorough repudiation of the principles of Bacon than is exhibited 
 by the disciples of the Fresnelian Theory of Optics. The 
 general equations of fluid motion, and the equations representing 
 the motions of a system of attracting particles, are the magazines 
 from which the optical theories of the present day are con- 
 structed. To devise a system of assumptions which will reduce 
 one or other of these systems of equations to a form representing 
 undulatory motion is the triumph of optical genius. In selecting 
 such assumptions little attention is paid to their antecedent 
 probability. At first, it may be, some circumspection is evinced. 
 The first admission required of us may have some fantastical 
 vraisemblance it may be true or may be not, the chances may 
 be about equal ; but very soon this caution is lost, and in the end, 
 provided we cannot distinctly shew that an hypothesis is untrue, 
 
PREFACE. XI 
 
 we arc bound to receive it ; for however profoundly convinced 
 we may be of its having no foundation in nature, if it can be 
 shewn " that it is riot absurd nor contradictory to sound me- 
 chanical principles," (and in this respect tolerably ample lati- 
 tude is allowed,) " that such may be true" according to Sir John 
 Herschel, we have done something. The effect of all this is 
 what might have been anticipated. We have after all an 
 intellectual as well as a moral conscience ; and the mental, like 
 the moral faculty, once abused will cease to give true indications. 
 When that is assumed as a plain fact which the mind of an 
 angel could not compass, can we wonder if intellectual blind- 
 ness should be the retribution for the intellectual sin ? After 
 all the extraordinary principles which the writers above enume- 
 rated have either invented or assented to, can we be surprised 
 to find that each one has fallen into one or more signal mathe- 
 matical errors?* 
 
 The methods adopted by Newton in the development of 
 his great discoveries, when applied on an extended scale, ceased 
 to be commodious, and the blind admiration which long refused 
 to admit the defects of the machinery employed, became a 
 serious obstacle to the progress of science in this country. It 
 may yet be a question whether a similar consequence may not 
 result from the indiscriminate use of the ponderous machinery 
 of the great analysts of the last age. After all, there are other 
 ways of apprehending truth besides through the medium of 
 
 * I am not sure that I am justified in including in the above censure 
 M. Cauchy, as he seems rather to have failed in producing a result than in 
 giving one which is erroneous. Of the different theories commented on in the 
 following pages, I would here particularly call attention to that of Mr. Stokes. 
 If the method of that gentleman is henceforth to be the standard of philosophical 
 investigation, it will remain to us but to weep over the grave of the Inductive 
 Philosophy. Time was when the mathematician was thought to be possessed 
 of methods of peculiar force and efficacy, and the most commanding intellects 
 were apt to shrink from contest with one entitled to that honourable name as 
 from a man possessed of a keen and unerring weapon ; but if the processes of 
 Mr. Stokes are those that are to be adopted, mathematics will not only cease 
 to be demonstrative (a favourite crotchet with some), they will cease to be 
 reasoning. It is strange that the mystical taint which threatens religious 
 opinion in this country should extend itself into natural philosophy: but 
 when we find the Historian of the Inductive Sciences expressing his belief in 
 the inspirations of genius, what can surprise us ? 
 
Xil PREFACE. 
 
 a differential equation. The principle of interferences was not 
 discovered from the consideration of the equation of sound. 
 We are still, notwithstanding all the efforts of M. Poisson and 
 Mr. Stokes, profoundly ignorant of the general principles of 
 molecular action, and yet we are able to solve many practical 
 questions depending upon those principles with perfect pre- 
 cision. The equations of motion were not intended to super- 
 sede thought, and he must have a strange idea of the principles 
 of philosophical inquiry who supposes that a solution of the 
 problem of light will be most readily obtained by attempting 
 that of a problem infinitely more general; or that the easiest 
 mode of ascertaining a particular motion of a particular medium 
 can be by means of an investigation involving every conceivable 
 motion in an inconceivable number of media. It may be 
 possible to trace in the flights of birds the vagaries of human 
 action to discern in the entrails of animals that which is 
 hidden in the bosom of time to interpret the whisperings of 
 the Dodonsean grove into the language of articulately speaking 
 man : but as certain as it is that fio man has ever yet appeared 
 of eye sufficiently keen or understanding sufficiently en- 
 lightened to derive any solid information from these imaginary 
 sources of knowledge, so sure is it that no theory of light will 
 ever be extricated from the tangled web of the general equa- 
 tions of motion. 
 
 In the course of the following pages I have endeavoured to 
 point out some simple circumstances connected with undu- 
 latory motion which have not hitherto been adverted to. I 
 have endeavoured to shew that the transversal motion which 
 has been made the subject of such convulsive efforts, when 
 we have divested ourselves of our ponderous harness, may not 
 be altogether impregnable to our attacks. It had long at- 
 tracted my attention, that unless some modification were made 
 in the definition of a wave, a spherical wave consisting of trans- 
 versal vibrations was a simple impossibility. If a wave be 
 defined to be a system of similar and similarly situated laminae 
 
PREFACE. Xlll 
 
 or surfaces each member of which consists of particles in the same 
 state of motion, it is geometrically impossible to devise a standard 
 according to which the particles in any spherical lamina or 
 surface (the vibrations being transversal) can be said to be in 
 the same phase. But although it was thus obvious that we 
 could not describe a wave consisting of transversal vibrations 
 as made up of elements the particles in which were in the same 
 state of motion, there was no objection to our considering 
 it as made up of elements the motion of the particles in which, 
 although not identical, still possessed such a distinct and recog- 
 nizable character as would enable us to speak of them as being 
 in the same phase. It was not difficult to suggest such a modi- 
 fication of the ordinary definition of identity of phase as satis- 
 fied all the geometrical requirements of the problem, and the 
 geometrical character of the motion being once distinctly appre- 
 hended, it was easy to see how it might be maintained mecha- 
 nically. The mechanism I have suggested for this purpose 
 may not be the actual mechanism, but at any rate it is one 
 which is feasible ; and without having recourse to such far- 
 fetched illustrations as the waving of corn, the undulations 
 on the surface of water, or the vibrations of a stretched cord, 
 it will, unless I am. greatly mistaken, render the conception 
 of transversal vibration as familiar to the mind as that of the 
 vibration which occurs in sound. 
 
 In applying my scheme to the explanation of polarization 
 as it occurs in crystallized media, I was led to observe, that to 
 whatever else might be attributable the variation of velocity in 
 crystalline waves, it certainly was not owing to the fact of waves 
 consisting of vibrations executed in one plane being more readily 
 transmissible than those in which the direction of vibration is 
 different; but, on the contrary, that the velocity is a charac- 
 teristic imposed upon the wave immediately upon its entering 
 the crystal, and that having once passed a certain narrow 
 boundary the velocity will be maintained exactly in the same 
 manner as the velocities of different coloured rays are main- 
 
xiv PREFACE. 
 
 tained in non- crystallized media, i. e. irrespective of the direction 
 of the vibration.* 
 
 Another result to which I have been conducted in the pro- 
 gress of these inquiries, and which may be thought of some 
 interest, is the possibility of the occurrence of transversal undula- 
 tions contemporaneously with and depending upon direct waves. 
 
 The theory of diffraction is presented to the reader as exhibiting 
 results which assuming the existence of ethereal waves must 
 certainly occur, and as being competent, whatever be the 
 opinion arrived at as to its ultimate truth, to give an intelligible 
 general explanation of the more prominent phenomena. When 
 first studying this department of Optics as a student at Cam- 
 bridge, I was led to ask myself, what became, after the diffraction, 
 of the system of general waves which might be incident upon 
 an aperture ? That each wave must extend itself laterally was 
 obvious, and it was not difficult to point out how this might 
 be effected mechanically. But what would be the effect of the 
 spreading ? Would it affect the continuity of each wave ? and 
 what would be its effect on the Relative positions of the waves 
 inter se ? It appeared to me that the continuity of each wave 
 considered individually would not be affected, but that succes- 
 sive waves would vary their forms from that which they ori- 
 ginally possessed, and that in different degrees, so that two 
 consecutive waves intersecting at a small angle might give 
 rise to maxima and minima of intensity. In the case of a 
 series of plane or spherical waves advancing perpendicularly 
 towards a circular aperture, it was easy to picture to the mind 
 each diffracted wave converted, after passing the aperture, into 
 a paraboloid of revolution and intersecting other waves con- 
 secutive to it in lines of annular form. 
 
 It is remarkable that a theory which explains the phenomena 
 of diffraction with such admirable exactness as that of Dr. Young 
 is known to do, should have completely fallen into oblivion 
 
 * As to the cause of a wave of given type being propagated uniformly in 
 a uniform medium, and the uniform velocity of all waves in the free ether, 
 I have arrived at certain conclusions to which I shall probably take some 
 other occasion of giving publicity. 
 
PREFACE. XV 
 
 in this country, having been entirely superseded by the wild 
 and erroneous scheme of Fresnel. This circumstance I conceive 
 to be attributable to the overwhelming sense of deficiency 
 which the early introducers of the continental mathematics 
 into this country appear to have experienced when com- 
 paring their own attainments and capabilities with those of their 
 neighbours; and which seems to have deprived them to a 
 certain extent of that power of discrimination which could 
 alone give value to their labours, and without which, in fact, the 
 study of mathematics would be of little more utility than read- 
 ing the newspapers. Sir John Herschel tells us that " Fresnel 
 has shewn a minute though decided difference between" the 
 places of the fringes " as given by this theory and by direct 
 measurement" ; and has moreover remarked, that were this 
 the "true explanation, they could hardly be supposed abso- 
 lutely independent of the figure of the edge of the opaque 
 body, which experience shows they are." This may be true, 
 but it seems to have been somewhat hastily taken for granted 
 on the faith of a rival, and the circumstance appears the more 
 remarkable if we compare the rejected theory with the extra- 
 ordinary and unintelligible alternative which Fresnel had to 
 propose. I have indulged in these remarks for the purpose 
 of suggesting that had Dr. Young's theory of Diffraction been 
 originally brought before me with due prominence, it might 
 have been doubtful whether I should have made an attempt to 
 devise another. As my theory however has been invented, 
 and, as I conceive, is competent to explain the phenomena, and 
 as the truth of the statement as to the indifference of the 
 phenomena to the form of the edge may be true, I have thought' 
 proper to introduce the theory here, the rather as, whether 
 true or not, as a theory of diffraction, it at any rate affords an 
 explanation of the motion of diffracted waves a subject to 
 which, so far as I am aware, no attention whatever has hitherto 
 been paid. 
 
 And now I must commend alike my theories and my criticism 
 to the attention of the mathematical world. Many, I am aware, 
 
XVI PREFACE. 
 
 more competent than myself have regarded with the same disap- 
 proval the recent theories of optics ; but either from want of suf- 
 ficient interest in the subject, or from the circumstances of their 
 position or temper, have shrunk from undertaking the thankless 
 and dangerous duty which I have endeavoured to fulfil. I feel 
 some love for truth and some contempt for absurdity, and I 
 have beheld with some indignation a system of inquiry so 
 monstrous as that of Fresnel introduced to the attention of the 
 youthful student not merely without a warning voice as to 
 its totally inadmissible character, but with the highest encomiums 
 upon the success and genius of its author. The evils which 
 have resulted from the reception of this theory are already 
 of a sufficiently aggravated character. It has had the effect 
 of divorcing for the time physical from mathematical science ; 
 and great as are the advances which have of late years been 
 made by the former in this country, they are in no degree 
 attributable to the aid of the mathematician, but must be wholly 
 ascribed to less pretending but more adroit and more discri- 
 minating hands. It is time that we should begin to retrace 
 our course, and it has appeared to me highly desirable that the 
 first step towards the reduction of the evil should be taken 
 by that great university which has always been looked upon 
 as the bulwark of mathematical knowledge in this country, 
 by the interdiction for the future of a branch of study, the con- 
 tinuance of which can have no other effect than that of corrupting 
 the judgment or disgusting the taste of precisely the most able 
 and the most promising of her sons. 
 
 P.S. I cannot omit this opportunity of expressing my sense 
 of the obligations I am under to my friend Professor Potter for 
 the valuable assistance rendered to me upon many occasions 
 during the preparation of the following pages, and which, from 
 the extent and accuracy of his acquaintance with the phe- 
 nomena of physical optics, is such as few persons could have 
 afforded. 
 
 8 Mount Street, Berkeley Square, 
 May, 1849. 
 
ON FRESNEL'S THEORY OF DIFFRACTION. 
 
 THE principle upon which Fresnel professes to explain the 
 phenomena of diffraction is thus enunciated by Mr. Airy : 
 
 " The effect of any wave in disturbing any .given point 
 may be found by taking the front of the wave at any given 
 time, dividing it into an indefinite number of small parts, 
 considering the agitation of each of these small parts as the 
 cause of a small wave which will disturb the given point, 
 and finding by summation or integration the aggregate of all 
 the disturbances of the given point produced by the email 
 waves coming from all parts of the great wave." 
 
 Mr, Airy offers a few words "in demonstration of this 
 principle," but I cannot think that he himself would consider 
 his reasoning upon the subject to be of any peculiar force : 
 and were the case otherwise, it would not be difficult to 
 suggest d priori objections at least as cogent as Mr. Airy's 
 d priori arguments. But it is not with the apparent reason- 
 ableness of this hypothesis that I have now to do, but simply 
 with its truth, and upon this enquiry I shall at once enter. 
 
 The hypothesis in question is thus applied by Sir John 
 Herschel to the investigation of the leading experiment in 
 the theory of Diffraction, viz. that in which spherical waves 
 are diffracted at a single edge. (Encyc. Met. Art. LIGHT, 7 IB.) 
 
 " Let us consider a wave AMF(fi.g. 1) propagated from 0, and 
 of which all that part to the right of A is intercepted by an opake 
 body A G ; and let us consider a point P in a screen at the 
 distance AS behind A as illuminated by the undulations ema- 
 nating simultaneously from every point of the portion AMF 
 according to the theory laid down in Art. 628, et seq. For 
 simplicity, let us consider only the propagation of undulations 
 
2 ON FRESNEI/S THEORY OF DIFFRACTION. 
 
 in one plane. Put AO = a, AB = b, and suppose \ = the length 
 of an undulation ; and drawing PN any line from P to a point 
 near M, put PN=f, NM= s, PB = x; then supposing P very 
 near to B, and with centre P, radius PM describing the circle 
 QM y we shall have 
 /= PQ + QN= V{O + b) z + x*}-a + QN = b + 2(a * +&) + QJV1 
 
 Now QN is the sum of the versed sines of the arc s to radii OM 
 and PM, and is therefore equal to 
 
 I 
 
 = - 1 - + v = 
 
 so that finally 
 
 Now if we recur to the general expression in Art. G32, for the 
 motion propagated to P from any limited portion of a wave, we 
 shall have in this case #0(0)= 1, because we may regard the 
 obliquity of all the undulations from the whole of the efficacious 
 part of the surface AMN as very trifling, when P is very dis- 
 tant from A in comparison of the length of an undulation ; and 
 as we are now considering undulations propagated in one plane, 
 that expression becomes merely 
 
 and the corresponding expression for the excursions of a 
 vibrating molecule at P will be 
 
 If then we put for f its value, and take 
 
 and consider that in those expressions t and x remain constant 
 while s only varies, the latter will take the form 
 
 cos efdr cos r + sin v* sin 
 
which shews that the total wave on arriving at P may be 
 regarded as the resultant of two waves, x cos and x" sin 0, 
 differing in their origin by a quarter undulation, and whose 
 amplitudes x and x' are given by the expression 
 
 (f 4 
 
 the integrals being taken between the limits of y corresponding 
 to s = - A M and s = + oo ". 
 
 It is obvious that if the principle of secondary waves can 
 be applied to a problem like the above, it may also be applied 
 to find the illumination at any point in front of a wave when 
 no diffraction takes place. In this case, taking the integrals 
 X', X" between the limits + oo and - oo , we obtain 
 
 ab\ t b 
 
 substituting for 6. It is evident from this result that the phase 
 
 # 2 
 
 of vibration will be the same for all points at which b + ; - =- 
 
 2 (a + b) 
 
 is constant, where x and a 4 b are the coordinates measured 
 in the plane of the paper from of the point whose vibration 
 is considered. But it is clear that where there is no diffraction 
 the phase must be the same for all points at which (a + bj + x z 
 is constant, since the waves are by hypothesis spherical. Hence 
 it appears that the hypothesis of secondary waves gives us a 
 false result in this case. 
 
 But it may possibly be urged that in the case of diffraction 
 at a single edge there are peculiar circumstances operating, 
 such as the obliquity of emanation of the secondary waves, 
 which we cannot take accurately into account; and that to 
 these the failure of the method in the above instance may be 
 attributable, I shall not stop to enlarge upon the imperfection 
 
 B2 
 
4 ON FRESNELS THEORY OF DIFFRACTION. 
 
 of a- theory of diffraction which fails to explain a phenomenon 
 which has always been regarded as the touchstone of the subject, 
 but shall proceed to examine whether in the case of narrow 
 apertures any greater reliance is to be placed in the principle 
 of secondary waves than in that which we have been con- 
 sidering. 
 
 Let AB (fig. 1') be the aperture, and let OP bisect it. With 
 centre describe the circles AEB, aeb indefinitely near to 
 each other. AB will then represent the front from which the 
 small waves are supposed to emanate in the ordinary theory. 
 It is obvious that if we draw Ac, Bd touching the inner circle 
 in b and d, we may consider the small waves to emanate from 
 cabdj and the illumination at P obtained upon this hypothesis 
 ought, if the theory were true, to be identical with that obtained 
 upon the hypothesis of AB being the origin of emanation. 
 
 In estimating the effect at P of the small waves from cdbd, 
 it will be necessary to ascertain the effect upon the latter of the 
 diffraction which they will necessarily undergo at A and B. 
 For this purpose, produce each side of the aperture to meet the 
 inner wave in a and b respectively, then it is evident that the 
 effect upon P of the small wave from any point of cabd which 
 lies between a and b, will be the same as if no opaque body 
 intervened, since in the case of any such wave the angle between 
 the lines which join the centre of emanation with the diffracting 
 edge and the point P, whose illumination is being considered 
 respectively, will be greater than a right angle. 
 
 I shall now shew that the illumination at P arising from 
 the small waves from ac, bd, will be of the order S 8 , where 
 % = OE- Oe. 
 
 Referring to fig. 1, it will be seen that if a be the radius 
 of a wave emanating from a luminous point 0, and which 
 touches the diffracting edge, f the distance of any point N 
 in the wave from the point P in front of it, s the distance 
 measured along the front of N from the point where the 
 wave is intersected by OP, the illumination / at P due to 
 the wave from N will be 
 
 (1) _/&>?? (*-/). 
 
 A. 
 
ON FRESNEL'S THEORY OF DIFFRACTION. 5 
 
 between the proper limits ; where 
 
 (1) / = /|( a + rj + a 3 - 2a (a + r) cos S -\ 
 
 = r + a I 1 - cos - ) , 
 
 V / 
 
 if a be indefinitely small ; 
 
 O ty TT 9 ty TT 
 
 .'. I = f{ds cos (vt - r - a) a cos - sin (vt - r - a)} 
 
 A A a A 
 
 2?r x . s 
 
 = s cos ~Y" (vt - r - a), 
 A 
 
 neglecting squares of a : or if s' be the difference between the 
 limiting values of s, 
 
 I = s' cos (vt - /), 
 
 where s' is of the same order as 8. Hence if we now take s 
 to represent the arc cdbd in fig. 1, the illumination at P due to 
 the small waves from ac may be represented by 
 
 S (7) = s'fds cos ( v t - r} ; 
 A 
 
 and since ac can be diminished indefinitely by diminishing S, 
 it follows that fds cos (vt - r) can be diminished indefinitely, 
 
 and S (/) will be of the order S 2 . 
 
 If we now apply the formulae (1) to find the illumination x 
 at P, produced by the small waves from aeb, it is evident that 
 we shall have 
 
 X = fds cos - (vt -fyf 
 where /) = J( : + r$ + a, z - 2 a (a l 4- rj cos I , 
 
 the limits of p being ea and - el (e l and - e t suppose) ; where 
 a, = a - S and r x = a + 8. 
 
 If s be small /J = r l + -^ J s*, 
 
 and X = I </s cos ( vt- r^ - 
 
 J -e l A V 
 
 But if we had considered the emanation to take place from 
 
6 ON FRESNEL'S THEORY OF DIFFRACTIONS. 
 
 AB instead of db, we should in the same manner have found 
 
 [ +e , 2?r / , a + r z \ 
 
 X = ds cos -- (vt - r s 8 , 
 
 J-e X V 2ar J 
 
 where e = AE\ and comparing these two values of X it is 
 obvious that we must have 
 
 sin /27r a + r 2 
 
 P*i 7 sm /27r a. + r. 2 \ / , sn 
 ds . -* - 1 . s 2 = ds 
 
 J -^ COS \ X 20/j J J -e CO 
 
 cos \ X 2#r 
 
 and therefore I ds sin - 1 l s z - I ds sin s 2 , 
 
 Jo X a/j J A, ar 
 
 an equation which holds accurately so long as we confine 
 ourselves to the first powers of 8. 
 
 Putting < 0) for the second member of the last equation, 
 it may easily be shewn that 
 
 f'i . TT a, + r, 2 //<VA , f / ar \ A . 
 
 J.^x-^r'^Vl-JnVfeJ '}' 
 
 therefore 
 
 Now 
 and 
 
 Substituting these values in (2) we get 
 
 whence, expanding the right-hand member of the equation, and 
 equating to zero the coefficient of 3, we obtain 
 
 an equation which is not satisfied by 
 
 a 
 
 Hence in the case of narrow apertures equally as in the 
 case of diffraction at a single edge, the hypothesis of secondary 
 waves leads to inconsistent results. It is obvious therefore that 
 the theory depending upon that hypothesis must be entirely 
 rejected. 
 
A NEW THEORY OF DIFFRACTION. 
 
 (Altered from the Philosophical Magazine, Vol. xxvu. p, 46.) 
 
 IN a former paper I endeavoured to point out the fallacy of 
 the celebrated hypothesis devised by Presnel for the ex- 
 plication of the phenomena of Diffraction. I endeavoured 
 to shew that when that principle is applied to the investigation 
 of facts which are simple and ascertainable, it gives results 
 which are inconsistent with truth ; and that when it is applied 
 to the explanation of those phenomena which it is the object of 
 every theory of diffraction to elucidate, it leads to conclusions 
 which are inconsistent with each other. I did not attempt to 
 refute the evidence which may have been brought forward in 
 support of this principle, simply because I found none. Those 
 who are desirous of ascertaining Mr. Airy's very vague senti- 
 ments with regard to this subject, will find them embodied 
 in five lines of p. 267 of his Mathematical Tracts. Sir John 
 Herschel appears to have considered that the principle of 
 secondary waves was "either self-evident or followed imme- 
 diately from the elementary principles of dynamics;" but he 
 nowhere shews us how it follows from those principles, nor 
 have I heard of any of his successors having ventured upon 
 so desperate an attempt.* 
 
 But though entertaining a strong opinion as to the absurdity 
 of the principle of secondary waves, I am desirous that the 
 scope of these observations should not be misunderstood. The 
 principle in question has been represented by some leading 
 writers on this subject as so completely part and parcel of the 
 undulatory theory, that the student who is little conversant 
 
 * The hypothesis of secondary waves seems to bear considerable resemblance 
 to the ancient astronomical theory of solid epicycles as wild, and as fallacious, 
 
8 A NEW THEORY OF DIFFRACTION. 
 
 with these subjects might suppose that by rejecting it we reject 
 the theory in toto, but such is by no means the case. The prin- 
 ciple is quite collateral to the theory. With the investigations 
 into the phenomena of polarized light, it has absolutely no con- 
 nexion whatever. Rejecting it as I do, I still admit and profess 
 my belief in the truth of the undulatory theory of reflexion and 
 refraction, of the explanations of the interesting experiments of 
 the two mirrors and the prism of a small refracting angle, and 
 of that admirable portion of the theory which relates to the 
 phenomena observed in the shadows of thin fibres and the 
 colours of Newton's rings. But if there be any soundness in 
 the principles I have laid down, the remaining portion of the 
 theory of diffraction, the whole theory of apertures and of the 
 shadows of extended bodies must be entirely rejected. 
 
 Although, however, I thus feel constrained to shew the 
 futility of the received theory of diffraction, I do not consider 
 that the elucidation of this class of phenomena upon undulatory 
 principles is by any means to be despaired of. With regard to 
 Dr. Young's hypothesis, which attributes the fringes about the 
 edge of the shadow of an extended body to the interference 
 of the direct light with the small portion reflected from 
 the edge of the diffracting body, I have indeed little to 
 say. It has been very positively asserted that the breadth 
 of the fringes is independent of the nature of the diffract- 
 ing body and of the form of the edge, a fact which if es- 
 tablished would form an insuperable objection to the reception 
 of Dr. Young's hypothesis. It may be shewn, however, that 
 the phenomena of diffraction may be accounted for on the 
 simplest principles, and in a manner entirely free from this 
 objection. 
 
 Suppose a spherical wave, diverging from a point, to consist 
 of one condensation and one rarefraction. ( Vide fig. 2.) 
 
 Let O be the point of divergence, A'BA'B' the condensed, 
 AS AB' the rarefied portion of the wave after it has been dif- 
 fracted by the opake body A C. It is obvious that .the portion 
 AB'A'B", in which the density of the ether is greater than that 
 of rest, will tend to relieve itself by flowing in the direction 
 B"A" into the geometrical shadow, (in which the ether has the 
 
A NEW THEORY OF DIFFRACTION. 9 
 
 mean density); while on the other hand, as the density in 
 AB A'B' is less than that of rest, the ether will tend to flow 
 in the direction A'B' out of the geometrical shadow into the 
 portion AB AB'. Hence the condensation A'B 1 A"B" will be 
 prolonged into the geometrical shadow and likewise the rare- 
 faction ABA'B', or in other words the wave itself will be 
 prolonged within the shadow. 
 
 If we now examine into the effect of this result (which it will 
 be observed is the inevitable consequence of the diffraction) upon 
 the constitution of the wave, it will be obvious in the first place 
 that there will be no discontinuity, i. e. the wave will be con- 
 tinuous up to the point where it finally terminates. We have 
 here considered light as consisting of waves of which the vi- 
 brations are executed in the direction of transmission, but the 
 same would be the case whatever were the nature of the 
 vibration; for it is certain that the waves have the power of 
 diffusing themselves within the geometrical shadow ; and if we 
 lay aside the consideration of any disturbing effect produced by 
 the forces inherent in the diffracting body, the non-existence 
 of which we are now assuming to be established upon experi- 
 mental grounds, it seems inconceivable that this effect should be 
 operated otherwise than continuously. 
 
 This consideration, if just, is of the highest importance, for 
 it shews us that if the phenomena of diffraction are to- be ac- 
 counted for at all upon undulatory principles, and if it is at the 
 same time to be considered as established, that those phenomena 
 are entirely independent of the specific action of the diffracting 
 body, there is but one way left in which to look for an expla- 
 nation. If each wave taken singly tends to preserve its con- 
 tinuity, interference is impossible except through the intersection 
 of consecutive waves nearly parallel to each other. 
 
 Now, in order to the intersection of which we are here speak- 
 ing, it is obvious that the waves which before diffraction are 
 concentric and spherical, must, after diffraction, vary from the 
 spherical form, and that too in different degrees, in order to 
 enable a second w T ave to overtake so as to intersect its immediate 
 predecessor. That such a change of form should take place is 
 not at all improbable. We are accustomed indeed to consider 
 
10 A NEW THEORY OF DIFFRACTION. 
 
 the velocity of waves traversing a uniform medium as invariable ; 
 but the waves to which attention has hitherto been directed, 
 have always been either continuous or quasi continuous, i. e. 
 they have either been supposed to be spherical or to be propa- 
 gated along the axis of a cone or cylinder. But the case of 
 broken waves is strikingly different. 
 
 Suppose a plane wave to traverse a cylindrical tube having 
 its front perpendicular to the axis. If a portion of the side 
 of the tube towards its extremity were cut away, it is obvious 
 that when the wave arrives at this point it will no longer 
 retain its form of a plane surface, but the part of it which is 
 no longer supported by the side of the tube will be weakened 
 by diffusion, and will fall behind the position it would have 
 held had the tube been preserved intact. This is precisely 
 the case of the experiment in diffraction which we are now 
 considering. 
 
 We are told indeed that in the free ether all disturbances are 
 propagated with the same velocity, but, as I have before sug- 
 gested, it is impossible to reason strictly from the case of con- 
 tinuous waves to that of the waves which we are now considering, 
 which, besides being subjected to all the circumstances which 
 affect the former, have this further peculiarity of being subject 
 to a lateral infusion and effusion of particles. In the one case 
 the effect of the wave is exerted only on the particles in its 
 front; in the other the motion is not'only communicated forwards, 
 but also laterally. It is certain that the part of the wave which 
 has extended itself within the geometrical shadow is not circular ; 
 for if it were it would obviously be impossible that any inter- 
 ference should take place within the geometrical shadow of 
 a narrow fibre : and if the part of the wave within the geome- 
 trical shadow deviate from the spherical form, is it very incre- 
 dible that the part without the shadow, immediately contiguous 
 to the former, should suffer a similar deviation ? 
 
 Assuming then that a wave, after diffraction, may undergo 
 such change of form as I have described, and to such an extent 
 as to allow of its overtaking its immediate predecessor, it is easy 
 to see how the intersection of the two, when nearly parallel 
 to each other, should produce interference. 
 
A NEW THEORY OF DIFFRACTION. 11 
 
 Let ABB Z A Z , CDD 2 C 3 (Fig. S) represent portions of two waves 
 intersecting each other at a small angle ABB^Aj CDD 1 C 1 
 respectively representing the condensed portions of the waves 
 A^B^B^A^ C^D^Dflr, the rarefied portions. A very little con- 
 sideration will serve to shew that the result of the intersection 
 must be to give us a bright point in the centre with a dark one 
 on each side. As the waves move on, since their relative 
 change of form will be very slow, the locus of their intersection 
 will describe a curve differing slightly from a straight line ; and 
 as the relative change of form goes on diminishing as the wave 
 advances, it is obvious that the path described will bear a gene- 
 ral resemblance to an hyperbola which lies very near to its 
 asymptote. The interference I have described will, it is true, 
 take place at a finite distance from the edge of the diffracting 
 body, but it is easily conceivable that this distance may be so 
 small as to be inappreciable to the senses. If we suppose again 
 that the original wave is intersected in a similar manner by its 
 second consecutive wave, it is obvious that a second line of inter- 
 ference will be formed which will lie without the former ; for 
 they will both have the same sensible origin, and the relative 
 variation of form in the last case will obviously be greater than 
 in the former, so that the locus of the intersection will tend 
 to recede further from the geometrical shadow. The existence 
 of additional lines of interference may be accounted for in 
 a similar manner, and we are thus led to a new and further 
 analysis of the composition of light, which is this, that in ordi- 
 nary cases, i.e. when the light is steady, what is commonly 
 called homogeneous light (meaning thereby light of one colour) 
 really consists of a number of different waves which occur in 
 the same invariable cycle, the members of the cycle being so 
 adjusted with regard to each other as to produce the effects 
 I have above endeavoured to describe. 
 
 It may seem strange that waves which, when subjected to the 
 test of refraction, evince no difference of velocity, should, when 
 tried by another standard exhibit so remarkable a diversity; 
 but when we consider the vast variety of waves which are 
 supposed to traverse the free ether with the same velocity, it is 
 not unnatural to imagine that even those which traverse a 
 
12 A NEW THEORY OF DIFFRACTION. 
 
 given medium with the same velocity may in reality comprise 
 a variety of types. 
 
 That such a systematic adjustment as that which I have sup- 
 posed to exist among the waves composing homogeneous light 
 should occur, need not be thought incredible when we reflect 
 that, the emission of the various coloured waves which produce 
 the solar spectrum must to a certain extent occur in an invari- 
 able order, or otherwise we should have the relative brightness 
 of the different parts of the spectrum varying from one moment 
 to another, which there is no reason to suppose is the case. It 
 is true that the eye is not a very critical judge in such matters ; 
 and therefore the uniformity of emission which the apparent 
 absence of change in the intensity of the different parts of the 
 solar spectrum appears to indicate, may not after all be so accu- 
 rate as one might at first suppose. But whether this uniformity 
 be, in point of fact, accurate, or only approximate, it is just as 
 easy to conceive it the one as the other; in other words, it 
 is not more inconceivable that the order of emission of the 
 different coloured rays should be invariable, than that its devia- 
 tions should be so small as not to be discoverable by the senses. 
 Hence the hypothesis as to the accurate adjustment of the 
 waves composing light of one colour, which I have supposed, 
 hardly presents more difficulty than the theory of colours. 
 
 But whatever be thought of the probability of this hypothesis, 
 it is obvious that, if we are to consider the objections to Dr. 
 Young's theory as established, our choice lies almost between 
 this and none. Before we allow ourselves to be driven to the 
 conclusion that the phenomena of diffraction are to be attributed 
 to the action of the diffracting body on the waves taken sepa- 
 rately, rather than to the mutual action of the waves upon one 
 another, we shall do well to reflect upon the extreme improba- 
 bility, which upon a popular view of the matter there seems, of 
 such being the case, and of the refined nature of the considera- 
 tions which would be requisite in order to expose by accurate 
 methods the fallacy of such popular impressions, and let us 
 then judge of the likelihood of such an investigation being 
 conducted to a satisfactory issue. Moreover, if the waves in- 
 dividually possess the faculty, when diffracted, of breaking up 
 
A NEW THEORY OF DIFFRACTION. 13 
 
 into parts, how does it happen that this only occurs without 
 the geometrical shadow ? If any portion of the wave were 
 broken into fragments, we should surely expect that it would 
 be that which has been newly created, namely that within the 
 geometrical shadow ; yet we know that in this direction nothing 
 of the kind takes place. 
 
 But it is time that I should commend my theory, such as it is, 
 to the consideration of my readers. If I have dealt severely with 
 the theories of another, it has not been simply because I be- 
 lieved them to be erroneous, but because I considered them to 
 involve a total disregard of all sound principles of argument. 
 Of the extraordinary vagaries in which Fresnel was in the habit 
 of indulging, we have a sufficient specimen in his theory of 
 diffraction, which I have considered in my first paper ; and of 
 which other examples equally remarkable will be given in 
 the subsequent portion of this work. The mere fact of such 
 views having been published and not contradicted might not 
 have induced me to take, with regard to them, the course I 
 have adopted ; but when they are not merely passed over in 
 silence, or with approval, by the highest authorities, but have 
 likewise obtained for their author one of the greatest reputations 
 of modern times, I thought it was time to speak out. The 
 theory which I have proposed may not be true, but, unless 
 self-love deceives me, it contains nothing that is very impro- 
 bable or which cannot be readily understood. When I say 
 that Fresnel's theory both of polarized and unpolarized light is 
 liable to the severest censure upon both of these grounds, I 
 hope I shall be considered to have offered a sufficient apology 
 for claiming a leniency of consideration in my own case, which 
 I have not thought proper to extend to him. 
 
 Berne, August 17, 1846. 
 
ON FRESNEL'S THEORY * OF DOUBLE 
 REFRACTION. 
 
 (From the Philosophical Magazine, Vol. XXVII. p. 553.) 
 
 HAVING on several previous occasions treated of the theory of 
 unpolarized light, and having as I trust effectually exposed the 
 futility of the celebrated hypothesis devised by Fresnel for the 
 elucidation of many of the principal phenomena in that depart- 
 ment of Optics, I now come to the consideration of the subject 
 of polarized light ; upon this treatment of which, Fresnel's great 
 fame now principally rests, and to whose views in regard to 
 which many of his adherents, who have felt themselves com- 
 pelled to give up his theory of diffraction, still cling with un- 
 shaken fidelity. What my own faith in this respect may be, 
 it is unnecessary at present to disclose further than this, that 
 whether the original idea of transversal vibrations is due to 
 Young only or not, I can pledge myself to prove that the 
 researches of Fresnel have not advanced us one step beyond it. 
 
 FresnePs first step in the mathematical part of his theory 
 is to prove the existence of the axes of elasticity, " a proposition," 
 says Mr. Smith (Cambridge Mathematical Journal, Vol. i. p. 1), 
 " on which the whole theory of double refraction depends, and 
 which Fresnel has proved by a method which has the advantage 
 of geometrical distinctness, but which is long and rather diffi- 
 cult to follow out on that account." On the same account I 
 shall prefer to give Mr. Smith's elegant analysis, instead of the 
 cumbrous processes of Fresnel, trusting that my readers will 
 take my word for it, that whatever it may want in ( ' geometrical 
 distinctness" it gains in logical clearness. 
 
 "The proposition is thus stated: In every system of par- 
 ticles acting on each other with forces which are functions of 
 their mutual distances, there are three directions at right angles 
 to each other, along which if a particle be displaced, the forces 
 of restitution will act in the same direction. 
 
15 
 
 "Let xyz be the coordinates of the attracted, xyp^xjy^... 
 those of the attracting points, ?V 2 r 3 ... the distances between the 
 attracted and attracting points, ^(rj . 2 (r 2 ) . ^> 3 (r 3 )... the attrac- 
 tions ; X YZ the total resolved forces along the axes ; then we 
 shall have 
 
 + 
 
 w tv* 
 
 i ' s> ' 2 
 
 and similarly for Y and Z. Now let 
 R = - S/0 (r) 
 
 then X = ~ = 0, 
 
 when the particle is in equilibrio. 
 
 " Let the particle receive a small displacement, the projections 
 of which on the coordinate axes are &c, &/, z. Then, supposing 
 the displacement to be very small, the force of restitution may 
 be taken as proportional to it, so that we have 
 
 " Now the force of restitution will be in the direction of the 
 displacement if X YZ be proportional to &, y, 2. 
 
 x Y z 
 
 Let then p = = -7- = -p ; 
 
 dx dy dz 
 
 d*R d*R e/ 2 ^ ^ 
 
 then putting - = A, - - B, - = C, 
 
 ~i - T > J J > 7 T 
 
 dzdy dzdx dxdy 
 
16 ON FRESNEL'S THEORY OF DOUBLE DIFFRACTION. 
 
 and, substituting in the former equations, they become 
 (-4 - p) &P + CBy + B'z = 0, 
 C'Sx + (B - p} y + A'Sz = 0, 
 B'Sx + A'Sy + (C 1 - p) $z = 0." 
 
 From which it is easy to prove, supposing the above process 
 correct, " that there are three directions at right angles to each 
 other, along which if a particle be displaced, the force of resti- 
 tution acts in the same direction." 
 
 But the fact is, the above process is entirely fallacious, if it 
 is meant to apply to the case of the motion of a particle of the 
 ethereal medium by which light is produced. What is meant 
 by the mysterious principle that, supposing the displacement to 
 be very small, the force of restitution may be taken as propor- 
 tional to it," I confess myself unable to* comprehend : but this 
 I do understand, that when the coordinates of the particle whose 
 motion is being considered vary from xyzto x + $z, y -f- Sy, z + $z, 
 the coordinates of the other particles of the medium will vary 
 from ztffr, x$ z z z , &c. to x l + S^, ^ + Sy,, z l + z l3 x 2 + &c a , 
 y z -t- 8y a , z 2 + $z z , &c., and that therefore the above values (A) 
 for the resolved parts of the force on the particle whose motion 
 is being considered are entirely fallacious. 
 
 The true value of X in this case is 
 
 + f?~ **, t -Z-T- fa + ^- *** + &c - > 
 
 c&ufc, efo</y a ' e&ok, 
 
 and similarly for y and ^: from which it is evident, that to 
 talk of the existence of axes of elasticity in every system of 
 particles acting on each other with forces varying according to 
 their mutual distances is mere absurdity. And hence it appears 
 that the "proposition on which the whole theory of double 
 refraction depends" is entirely untrue. 
 
 Will it be urged, however, that although the general pro- 
 position does not hold, there still may be particular systems of 
 
17 
 
 particles for which it does hold ? I do not hesitate to state my 
 belief that the existence of such a system is impossible, and at 
 any rate would challenge any analyst whatever to suggest any 
 such. 
 
 The case then stands thus : a writer states a proposition as 
 the basis of a theory ; he offers a proof of the proposition which 
 turns out to be fallacious ; and not only is the proof itself erro- 
 neous, but during the investigation there appears a degree of 
 evidence approaching to certainty, that the proposition itself, 
 after modifying it in every conceivable way consistent with the 
 circumstances of the case to which it is meant to apply, is 
 untrue ; and there is moreover a perfect certainty that it is in- 
 capable of proof. Thus we have a fundamental proposition of 
 which a false proof is given, a certainty that if true it must 
 always remain a mere assumption incapable of independent 
 proof ; and this in the face of the fact that there is every reason 
 to suppose it untrue. Such a combination of circumstances 
 would have decided the fate of any other theory ; why is this 
 to be made an exception to the rule ? But to return. 
 
 Assuming the existence of the axes of elasticity, we are next 
 introduced to the surface of elasticity. Referring the co-ordi- 
 nates to the axes of elasticity, we have 
 X - ax = ar cos en 
 
 Y = by = br cos |3 1 where a, b, c are constants. 
 Z = c$z = cr cos 7 J 
 
 The rest I shall give in the words of Sir John Herschel, (vide 
 Encyc. Met. Art. LIGHT, 1004). 
 
 " M. Fresnel next conceives a surface, which he terms the 
 * surface of elasticity,' constructed according to the following 
 law : On each of the axes of elasticity, and on every radius 
 r drawn in all directions, take a length proportional to the 
 square root of the elasticity exerted on a displaced molecule 
 by the medium in the direction of the radius or to VJP: then if 
 we call R this length, or the radius vector of the surface of 
 elasticity, we shall have 
 
 1C 1 = {ar . (cos a) 2 + br . (cos ]3) 2 + cr . (cos y) 2 } x const., 
 
 the values of R parallel to the axes are then had by the equations 
 
 R* = const, x ar, R* = const, x br, R z = const, x cr, 
 
 c 
 
18 ON FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 which we shall express simply as a z , b*> c z , so that the equation 
 of the surface of elasticity will be of the form 
 
 R* = a 2 cos 2 X + b* cos 2 Y + c* cos 2 Z, 
 
 where X, Y, Z now stand for a, /3, y, the angles made by R with 
 the axes of co-ordinates. 
 
 " Let us now imagine a molecule displaced and allowed to 
 vibrate in the direction of the radius R, and retained in that 
 direction, or at least let us neglect all that portion of its motion 
 which takes place at right angles to the radius vector. Then 
 the force of elasticity by which its vibrations are governed 
 will be proportional to jR 2 , and the velocity of the luminous 
 wave propagated by means of them, in a direction transverse to 
 them (or at right angles to R\ will be proportional to R" 
 
 Of this extraordinary proposition the accomplished author 
 does not offer one syllable of proof or explanation. Whether 
 Fresnel's writings are equally deficient I am not aware ; but 
 another eminent mathematical writer, the present Astrono- 
 mer Royal, after bestowing, as we may reasonably suppose, 
 some degree of care on the study of Fresnel's papers, appears 
 to have found nothing better in the way of demonstration than 
 the following, (vide Airy's Tracts, 2nd edit., p. 341): 
 
 " To explain on mechanical principles the transmission of 
 a wave in which the vibrations are transverse to the direction 
 of its motion. 
 
 " In fig. (4) let the faint dots represent the original situations 
 of the particles of a medium, arranged regularly in square order, 
 each line being at the distance h from the next. Suppose all 
 the particles in each vertical line disturbed vertically by the 
 same quantity, the disturbance of different vertical lines being 
 different. Let x be the horizontal abscissa of the second row, 
 x - h that of the first, and x + h that of the third ; let u, u lt and 
 u be the corresponding disturbances. The motions will depend 
 upon the extent to which we suppose the forces are sensible. 
 Suppose the only particles, whose forces on A are sensible, to 
 be BCDEFG (omitting those in the same line as their attrac- 
 tions are equal and in opposite directions) ; and suppose them 
 to be attractive, and as the inverse square of the distance ; and 
 
ON FRESNEL'S THEORY OF DOUBLE REFRACTION. 19 
 
 the absolute force of each = m. The whole force, to pull A 
 downwards is 
 
 m (h + u - Uj) m (u - u^) 
 
 m (h - u + u^ m (h + u - u') 
 
 m (u - u') m (h - u + u 1 ) 
 
 {tf + ( - ujf {h*+(h-u + uj } l 
 
 Expanding these fractions and rejecting powers of u - u^ and 
 u - u above the first, the force tending to dimmish u is 
 
 ^ . f du , d z u tf 
 
 Putting for lf M -_.A + __, 
 
 du , d'u h 2 
 and for u , u + h + 2 ; 
 
 dx dx z 1.2 
 
 l\md z u 
 we find 
 
 an equation of exactly the same form as that for the transmission of 
 sound. The solution therefore has the same form ; and therefore 
 the transversal motion of particles supposed here follows the same 
 law, that is, it follows the law of undulation." And, moreover, if 
 the above were correct, the velocity of the luminous wave would 
 be proportional to the square root of the force of elasticity in 
 a direction transverse to the direction of the course of the wave. 
 Whether the above illustration, if indeed it may be called 
 such, is due to Fresnel, or to Mr. Airy himself, I am not aware ; 
 but if to the latter, it is plain that Fresnel's mantle has fallen 
 upon him.. The whole is erroneous from beginning to end. Not 
 only do the mathematics fail to meet the case under consider- 
 ation, but there is a palpable mathematical error in the process, 
 which, even admitting the data, completely vitiates the result 
 
 02 
 
80 ON FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 I need but to advert to the circumstance, that in the approxi- 
 mate values substituted for u v and u, it is assumed that h is 
 small with respect to u, or that the distances between the par- 
 ticles are small compared with their actual motions, a sup- 
 position entirely at variance with the assumed data of the 
 problem. Hence it is plain that this supposed illustration is 
 for every purpose entirely worthless. 
 
 Thus as we were compelled to assume the existence of the axes 
 of elasticity, not only in default, but in the face of evidence, so we 
 are compelled to assume the rule as to, the mode of calculating the 
 velocity on a bare analogy to a case presenting the most striking 
 difference from that under consideration, namely that of the 
 transmission of an undulation when the vibrations are in the 
 direction of transmission. We are not only compelled to assume 
 the existence of undulations, consisting of vibrations executed in 
 directions perpendicular to the course of the wave, respecting 
 which it is not too much to say, that it is impossible for the 
 mind to conceive the possibility of their existence, but we are 
 to suppose ourselves* acquainted with an exact law to which 
 
 * It is easy to conceive of transversal as the consequence of direct vibration, 
 but I confess myself unable to conceive the possibility of there being a surface 
 of transversal vibrations in the same phase : take a sphere for example. At all 
 events, if the hypothesis of transversal vibration is to hold its ground, it must 
 have much more thought bestowed upon it than it has yet received. The most 
 painful circumstance connected with the late history of the undulatory theory, 
 is the manner in which ideas, in themselves perhaps valuable as hints, have 
 been dressed up into a settled theory. A truly philosophical mind, to which the 
 idea of transversal vibrations had once suggested itself, would have set itself to 
 work to discover, if possible, some method by which such motion could be con- 
 ceived, and would not have rested satisfied so long as a doubt existed as to the 
 perfect feasibility of the scheme, Thus it is that we may account for Dr. Young's 
 not having attempted to carry his first notion any further. He saw, no doubt, the 
 difficulties by which the idea of transversal vibrations was beset ; and was well 
 aware that till these were overcome, it was hopeless to attempt to enter into the 
 discussion of their nature and consequences. Fresnel, on the contrary, was 
 satisfied with a series of possibilities, upon which he has built a theory, not 
 only of no value in itself, as having nothing solid to rest upon, but from its 
 crudity and manifold errors, discreditable to himself and to the age by which 
 it has been received. 
 
 It may be observed, that in denouncing, as above, undulations consisting of 
 transversal vibrations I had an eye chiefly to their original emission. I still 
 hold it to be impossible that a wave could be emitted consisting of transversal 
 vibrations in the same phase ; but at the time of writing the above it had not 
 occurred to me, that although not originally belonging to it, such a character 
 
ON FRESNEL'S THEORY OF DOUBLE REFRACTION. 1 
 
 they are subject. Of the worth of such a theory I leave my 
 readers to judge. The discussion of the remaining portion of 
 it I must defer to another opportunity. 
 
 Liverpool, Novembers, 1845. 
 
 might be impressed on the wave after its emission. My views on this subject 
 will be expressed hereafter. 
 
 October 3, 1846. 
 
REMARKS OF MR. ARCHIBALD SMITH 
 
 RELATING TO PART OF THE LAST PAPER. 
 
 (From the Philosophical Magazine, Vol. xxvm. p. 48.) 
 
 To the Editor of tlie Philosophical Magazine and Journal. 
 
 GENTLEMEN,~III an article on Fresnel's Theory of Double 
 Refraction, in the Supplement to the December No. of the 
 Philosophical Magazine, Mr. Moon has quoted part of an article 
 of mine, in the first volume of the Cambridge Mathematical 
 Journal, in which the following passage occurs : " Let the 
 particle receive a small displacement, the projections of which 
 on the coordinate axes are $x, Sy, $z. Then supposing the dis- 
 placement to be very small, the force of restitution may be taken 
 as proportional to it, so that we have, &c." I am not surprised 
 that Mr. Moon should remark on this passage, " what is meant 
 by the mysterious principle ( supposing the displacement to be 
 very small, the force of restitution may be taken as proportional 
 to it,' I profess myself unable to understand." 
 
 The clause in italics, which was added to my manuscript 
 when it was sent to the press, to remove, I believe, what was 
 thought an abruptness in the reasoning, is certainly incorrect 
 when applied to a doubly refracting medium. What was in- 
 tended to be expressed, no doubt was, that in the case supposed 
 terms involving powers of &c, fy, $z, higher than the first might 
 be neglected. But this expression is only equivalent to the 
 other in the case of a singly refracting medium. I may mention 
 that in the middle of page 7, of the article in the Cambridge 
 Mathematical Journal, the word " rays" was, by mistake, sub- 
 stituted for waves. These mistakes are corrected in the second 
 
MR. A. SMITH'S REMARKS, ETC. 23 
 
 edition of the first volume of the Cambridge Mathematical Jour- 
 nal, which is now printing. As they have been noticed in your 
 Journal, I shall feel much obliged by your inserting this expla- 
 nation, when you can afford space for it. 
 
 Your obedient Servant, 
 
 ARCHIBALD SMITH. 
 
 25, Old Square, Lincoln's Inn* 
 
 Dec. 20. 
 
 REMARKS UPON THE FOREGOING. 
 
 Mr. Smith is " not surprised" that I should remark, " what is 
 meant by the mysterious principle ' supposing the displacement to 
 be very small the force of restitution may be taken as proportional 
 to it) I profess myself unable to understand ;" and yet, for any- 
 thing which appears in Mr. Smith's letter, he might well have 
 been so. He says, " the clause in italics is certainly incorrect 
 when applied to a doubly refracting medium." This remark is 
 itself incorrect. If we have 
 
 v d*R d z R s d z R . 
 X = - Ex + - Sy + 8z, 
 dx z dxdy dxdz 
 
 and if $r be the radius vector of the particle measured from its 
 point of rest, a, /3, y the inclinations of the radius to the coordi- 
 nate axes, since 
 
 $x = $r cos a, $y = $r cos )3, Sz = $r cos y, 
 it is obvious that it is literally true that the force of displace- 
 ment is proportional to Sr, i. e. to the displacement itself. 
 
 When I animadverted upon the clause in question, I did not 
 consider it at all in the point of view in which Mr. Smith appears 
 to look upon it. It appeared to me that it was meant to be the 
 statement of a distinct physical principle. The principle is in- 
 comprehensible enough, it is true, and it is utterly incapable of 
 proof; but Mr. Smith must remember that such objections 
 have, in general, very trifling weight with the followers of 
 Fresnel ; and though Mr. Smith may choose to repudiate the 
 
MR. A. SMITH'S REMARKS, ETC. 
 
 idea of having meant to assume anything of the kind, yet by 
 doing so he renders himself liable to equal animadversion upon 
 another ground. For if Mr. Smith does not mean to assume 
 as a self-evident principle that the force of displacement on 
 any particle depends on the displacement of that particle 
 alone, and is independent of the displacements of the particles 
 surrounding it, he has either committed a gross error in 
 his demonstration, or else he has mistaken the point he had 
 undertaken to prove: i.e. either he has mistaken the partial 
 differentials of X, Y, Z for the complete differentials, or else he 
 has considered that he had to prove something short of "the 
 existence of the axes of elasticity." Mr. Smith may have 
 thought with my antagonist, Jesuiticus, that it was sufficient 
 to prove that in a system of particles acting upon each other 
 by their mutual attractions, there are three directions in which 
 if a particle be suddenly disturbed, the force of restitution will 
 in the beginning of the motion act in the direction of the dis- 
 placement; but I must beg to assure him that this is not enough. 
 Mr. Smith has himself described the proposition he had taken 
 in hand to prove as one " on which the whole theory of 
 double refraction depends;" and he must know that the pro- 
 position so adverted to is tantamount to nothing short of this, 
 that if Xy y, z be the coordinates of any particle measured from 
 the point of rest, the equations which govern its motion will be 
 of the form 
 
 equations which are not to hold for an instant, in the BEGINNING 
 OF THE MOTION, but throughout all time. Does Mr. Smith 
 think the general equation follows from the particular one ? 
 Does he think that if the above equations hold " in the begin- 
 ning of the motion," that they must therefore, as a matter of 
 course, hold throughout the motion? Does he believe this to 
 be a self-evident fact, or one of which we have strong evidence ? 
 If he does, I would beg to assure him, on the contrary, that the 
 demonstration of the particular proposition, of itself, affords 
 the most convincing evidence that the general proposition is 
 untrue. 
 
MR. A. SMITH'S REMARKS, ETC. 25 
 
 Mr. Smith appears to have a very humble opinion of the 
 judgment of the unfortunate editor, who added the clause to 
 his manuscript: but it appears to me that that individual, 
 whoever he may be, has displayed rather more sagacity in 
 this instance than Mr. Smith himself. What he inserted 
 at any rate bore the semblance of an argument ; and though 
 altogether inadmissible, it shewed his consciousness of the 
 defect in the reasoning, which seems entirely to have escaped 
 the notice of Mr. Smith. Indeed that gentleman's blindness 
 appears to have been of so enduring a character, that in taking 
 public notice of a paper in which the futility of his investiga- 
 tion was asserted and proved, he has been insensible to the 
 absurdity of leaving entirely without comment the vital objec- 
 tions to his argument, and confining his efforts to an ill con- 
 sidered correction of a trifling and imaginary error. A cer- 
 tain character in a well known modern novel did not scruple 
 to chouriner y who would have scorned to steal. So Mr. Smith 
 is indifferent to the consequences of an error in the higher 
 branches of analysis or physical science, but immediately takes, 
 the field in defence of his trigonometry. 
 
 Hamburg, Sept. 16, 1846. 
 
REMARKS ON A PAPER BY MR. MOON 
 
 ON FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 By JESUITICUS. 
 (From the Philosophical Magazine, No. 185.) 
 
 THE hypothesis on which Fresnel's Theory of Double Refrac- 
 tion is based is the following : 
 
 " That the displacement of a molecule of the vibrating 
 medium in a crystallized body is resisted by different elastic 
 forces, according to the different directions in which the dis- 
 placement takes place." 
 
 This is not a mere speculative hypothesis, but is based on 
 experiment. It is found that glass, possessing only the power 
 of single or ordinary refraction, may be made by the application 
 of heat, or by mechanical pressure, to possess that of doubfe 
 refraction. 
 
 It is further supposed that the medium is symmetrical with 
 respect to three rectangular axes in space, but in general not 
 symmetrical with respect to any other axis through the same 
 origin. These axes are called the axes of elasticity. It is then 
 proved, that if any particle of the ether be suddenly disturbed, 
 the other particles remaining quiescent; the force of restitution 
 developed by such disturbance will not in general be in the 
 direction of the displacement, but only when such displacement 
 is in the direction of the aforesaid axes of elasticity. The elegant 
 demonstration of Smith, quoted by Mr. Moon, is, by Mr. Moon's 
 own shewing, fully adequate to establish the theorem as I have 
 enunciated it, which is doubtless the sense in which Fresnel 
 (the illustrious Fresnel, " whose name is enrolled amongst those 
 which pass not away") doubtlessly conceived it. 
 
 Any one who understands the subject must at once acknow- 
 ledge that any theory of light must be, to a considerable extent, 
 
MR. MOON'S REMARKS, ETC. 7 
 
 imaginative ; and that theory which can explain the greatest 
 number of facts ought to claim the attention of the philosopher 
 more than any other. It is to this that the undulatory theory 
 owes its great celebrity, and of all parts of the undulatory 
 theory that of double refraction is the most extraordinary. It 
 ought to be regarded as a stupendous monument of human 
 ingenuity. It must not be forgotten how admirably the pro- 
 perties of uniaxal crystals follow from the general investigation 
 of the biaxal class ; but above all, how from this same investi- 
 gation conical and cylindrical refraction were discovered by 
 Sir William Hamilton. Such an unexpected refinement as 
 this, which probably would never have been recognised by the 
 mere experimentalist, undirected by the skill of so great an 
 analyst, is surely no slight recommendation of the theory. 
 
 Mr. Moon subsequently gives a quotation from Airy's Tracts, 
 concerning which he is by no means sparing in arrogant and 
 supercilious criticism. But is it likely that Airy would make 
 such a fool of himself as Mr. Moon earnestly endeavours to 
 represent ? It must be remembered that at the time Airy's 
 Tracts were published, very little of the undulatory theory was 
 studied or known in Cambridge. 
 
 It was the part of this philosopher, therefore, to put every- 
 thing as much as possible in the clearest and most simple point 
 of view. That there are, and will perhaps long continue to be, diffi- 
 culties in the undulatory theory, none of its supporters will deny. 
 None of these difficulties are shirked or glossed over in the Tract 
 of Airy : he plainly acknowledges each as it arrives. He, no 
 doubt, himself considered the part quoted by Mr. Moon more as 
 an illustration than anything else. Those who wish to see the 
 matter treated with all the generality of which it is capable, 
 are referred to a tract on the Reflection and Refraction of Light 
 at the surface of two contiguous media, by the late famous George 
 Green, in the Cambridge Philosophical Transactions. 
 
 I have one word more with Mr. Moon. He says that on 
 substituting for u l9 
 
 du , d z u h 2 Q 
 u - h + - - - , &c., 
 dx dx z 1.2 
 
8 MR. MOON'S REMARKS, ETC. 
 
 and for u, 
 
 du , d z u h* 
 u + T h + ~r~2 r^ &c - 
 
 dx dtf 1.2 
 
 that h is considered small with respect to u. 
 
 Does Mr. Moon know anything of analysis ? He was eighth 
 wrangler in 1838, and therefore ought to know something. 
 His knowledge, however, has served him miserably on this 
 occasion. The substitutions, stopping at h?, merely require 
 that h should in comparison with the length of a wave, not with 
 respect to u. 
 
REPLY TO JESUITICUS. 
 
 (Altered from the Philosophical Magazine, for March, 1846.) 
 
 AN anonymous writer, who subscribes himself 'Jesuiticus,' pre- 
 faces certain observations upon my first paper on the Theory 
 of Double Refraction, with the following remarks : 
 
 " The hypothesis on which Fresnel's Theory of Double 
 Refraction is founded, is the following : ( That the displacement 
 of a molecule of the vibrating medium in a crystallized body 
 is resisted by different elastic forces, according to the different 
 directions in which the displacement takes place.'" 
 
 He then proceeds to make some remarks upon the reasonable- 
 ness of this hypothesis, which it is not my present purpose to 
 dispute : but I must beg to observe, en passant, that as a matter 
 of fact it is not true that the above is the hypothesis upon 
 which FresnePs Theory of Double Refraction is based. It rests 
 on a lower level still. The true basis of the theory is the 
 assumption, that the ethereal medium consists of particles sepa- 
 rated by finite intervals (to use a well known but somewhat 
 improper mode of expression), acting upon each other by their 
 mutual attractions. From this principle the so-called funda- 
 mental hypothesis of Jesuiticus is a sufficiently easy inference. 
 I have thought it necessary however to remark upon this in- 
 accuracy, as, from the extraordinary want of precision of the 
 writers upon this subject, it is somewhat difficult to say what 
 is their real starting-point; at the same time, that in order 
 to make a proper estimate of Fresnel's theory, and of the degree 
 of skill and judgment with which he has worked it out, it is 
 very desirable that that fact should be clearly ascertained. 
 
 Jesuiticus goes on to say, " It is then proved that if any 
 particle of the ether be suddenly displaced, the other particles 
 remaining quiescent, the force of restitution developed by such 
 disturbance will not, in general, be in the direction of the dis- 
 placement, but only when such displacement is in the direction 
 
30 REPLY TO JESUITICUS. 
 
 of the aforesaid axes of elasticity. The elegant demonstration 
 of Smith, quoted by Mr. Moon, is, by Mr. Moon's own shewing, 
 fully adequate to establish the theorem as I have enunciated 
 it, which is doubtless the sense in which Fresnel (the illustrious 
 Fresnel e whose name is enrolled amongst those which pass not 
 away') doubtlessly conceived it." 
 
 Now, as to the fact which Jesuiticus appears to consider so 
 emphatically free from doubt, as to make it necessary for him 
 to reiterate his conviction to that effect almost in the same line, 
 I must confess that I do not altogether acquiesce in his opinion. 
 I confess myself willing to believe that Fresnel, in the course of 
 his long and awkward investigation (which is by no means pos- 
 sessed of that limpid clearness which characterizes Mr. Smith's) 
 had fallen into an error, rather than that he in cool blood pro- 
 posed the preposterous argument which Jesuiticus would here 
 attribute to him. Be this as it may, however, I admit that 
 Mr. Smith's demonstration, is fully adequate to establish the 
 theorem as Jesuiticus has enunciated it; but I must assure 
 Jesuiticus that, unless the demonstration establishes a great 
 deal more than the theorem so enunciated, it is not, for the 
 purpose for which it is adduced, worth the paper it is written 
 upon. What is the use of considering the impossible case 
 of a single particle suddenly disturbed while all the others 
 remain quiescent, and then reasoning upon what takes place in 
 the beginning of the motion in that case, as if the same held good 
 throughout the motion in the actual case when all the particles 
 are vibrating together, when it is perfectly certain that it does 
 not? 
 
 Jesuiticus says, " Any one who understands the subject must 
 at once acknowledge that any theory of light must be to a con- 
 siderable extent imaginative ; and that theory which can explain 
 the greatest number of facts ought to claim the attention of the 
 philosopher more than any other." 
 
 Of the justice of the remark contained in the first part of the 
 above sentence, FresnePs Theory of Double Refraction . is no 
 doubt a remarkable proof. In the sentiment of the second 
 clause I am disposed to concur, with the reservation however 
 that some portion of the credit due to a theory depends upon 
 
REPLY TO JESUITICUS. 31 
 
 its antecedent probability ; for it may happen that a theory 
 may be contrived so fantastical as to require just as much 
 explanation as the phenomena it was intended to elucidate , 
 as is the case with Fresnel's Theory of Secondary Waves 
 emanating from the general front. But mark how our friend 
 Jesuiticus proceeds : " It is to this that the undulatory theory 
 owes its great celebrity, and of all parts of the undulatory 
 theory, that of double refraction is the most extraordinary. 
 It ought to be regarded as a stupendous monument of human 
 ingenuity. It must not be forgotten how admirably the princi- 
 ples of uniaxal crystals follow from the general investigation 
 of the biaxal class ; but above all, how from this same investi- 
 gation, conical and cylindrical refraction were discovered by 
 Sir William Hamilton." 
 
 I would now ask Jesuiticus, what is the hypothesis upon 
 which Fresnel professes to explain the separation of the ray? 
 whether it is not substantially what I have stated it to be in 
 the former part of this paper ? And if so, I appeal to the 
 world whether I have not shewn incontrovertibly in my two 
 papers* on this subject, contained in the two last numbers 
 (184, 185) of the Philosophical Magazine, that Fresnel entirely 
 fails to explain the separation of the ray on that hypothesis. 
 It may be true that some of Fresnel's expressions for the 
 disturbance in polarized waves may contain in them certain 
 elements of truth (though, for my part, I should be sorry to 
 answer for any of them); but they do not on that account 
 afford any evidence of the truth of his principles, for this plain 
 reason, that they do not follow from them. It may happen 
 that from the ruins to which this great theory must soon, if it 
 be not already, reduced, some fragments may be gathered 
 which may form part of a new and more durable edifice ;f but 
 
 * The second of these is now incorporated in one, which appears in a 
 subsequent part of this work. 
 
 f Let those who think that Fresnel's wave surface may be maintained upon 
 experimental grounds, endeavour so to maintain it ; and if they succeed in 
 their endeavours, let Fresnel take the credit of the invention. But let it be 
 remembered that this has not been done yet ; and it may be observed, that what- 
 ever fame may be rescued to Fresnel from the wreck of his present great 
 reputation, must always be dimmed by his numerous and enormous errors, 
 which exceed those of any mathematical writer whatever. 
 
32 REPLY TO JESUITICUS. 
 
 Jesuiticus may take my word for it, that, notwithstanding all 
 his vapouring about Fresnel, the time is at hand when that 
 person's theory will be considered as a stupendous monument 
 of anything rather than ingenuity. As to the supposed dis- 
 coveries of conical and cylindrical refraction, had Jesuiticus 
 been aware of their real nature, he would not have ventured 
 to bring them so prominently forward. 
 
 Jesuiticus asks, " Is it likely that Mr. Airy would make such 
 a fool of himself as Mr. Moon earnestly represents ?" I never 
 had any intention of attaching to the eminent individual 
 alluded to the very unseemly epithet made use of by Jesuiticus : 
 but, notwithstanding his elegant apostrophe, and notwithstand- 
 ing the ingenious criticism contained in the latter part of his 
 paper, I am still of opinion that in the investigation quoted 
 and commented on in my first paper on Double Refraction, 
 Mr. Airy committed the error I there stated him to have com- 
 mitted, i.e. of having neglected terms without a shadow of 
 reason ; and I further reassert, that the whole investigation, if 
 meant to apply to the motion of a particle under bond fide 
 physical circumstances, is erroneous from beginning to end. 
 If Jesuiticus has any doubt with regard to the latter point, 
 I shall at any time be happy to relieve him from it. The 
 error of approximation I shall consider presently ; but I would 
 first direct attention to another assertion of Jesuiticus, respect- 
 ing Mr. Airy. He says, " that there are, and perhaps will long 
 continue to be, difficulties in the undulatory theory, none of 
 its supporters will deny. None of these difficulties are shirked 
 or glossed over in the Tract of Airy ; he plainly acknowledges 
 each as it arrives." Now I apprehend that this statement is cal- 
 culated to convey a very incorrect idea of the true state of the 
 case. Mr. Airy does indeed, for the most part, direct attention 
 to such of the difficulties connected with the subject as he 
 appears to have been aware of, but this he effects in a very 
 inadequate and partial manner. Thus, in the early and strong 
 part of the theory, a great deal is made of the difficulty arising 
 from the fact of there being nothing in the theory of sound 
 analogous to the different velocities of luminous waves in trans- 
 parent bodies, where it is in fact rather doubtful whether there 
 
REPLY TO JESUITICUS. 33 
 
 Is any difficulty at all ; while we have the following very mild 
 caveat against perhaps the wildest of all Fresnel's extravagances 
 the ascribing the change of velocity of a wave in passing from 
 the free ether into a transparent medium " to the particles of 
 ether, while retaining the same attractive force, inside of glass, 
 &c., becoming loaded with some matter which increases their 
 inertia in the ratio 1 : n without increasing their attraction," 
 (vide Tracts, 2nd edit. p. 354); respecting which Mr. Airy is 
 contented to observe, "Perhaps this supposition is hardly 
 reconcilable with that made in the last proposition." The 
 experience of so humble an individual as myself may possibly 
 have little weight with Jesuiticus ; but I can assure him that, 
 whatever he may think of Mr. Airy's fairness, it was the 
 striking difference of tone adopted by that gentleman, in the 
 parts of his work just referred to, which, when I first read the 
 Tract on Optics, as a student at the university, opened my 
 eyes to the suspicious nature of most of the speculations there 
 brought forward.* 
 
 Jesuiticus concludes as follows : " I have one word more 
 with Mr. Moon. He says, that in substituting for u^ 
 
 du , d z u h? 
 
 u - . h + - + &c. 
 dx dx z 1.2 
 
 , f du , d z u h z Q 
 
 and for u , u + n, + ^ + &c., 
 
 dx dx z 1.2 
 
 that h is considered small with respect to u. 
 
 " Does Mr. Moon know anything of analysis ? He was 
 eighth wrangler in 1838, and therefore he ought to know 
 something. His knowledge however has served him miserably 
 on this occasion. The substitutions stopping at h z merely 
 require that h should be small with respect to the length of 
 a wave, not with respect to u." 
 
 Now, it is true that if we make the special hypothesis that 
 the disturbance is of the form 
 
 2ir , t , 
 
 u = a cos -r- (vt - x), 
 A 
 
 * Jesuiticus, with characteristic bad taste, makes an allusion to the late 
 Mr. Green, of Caius College, in such terms as, to those unacquainted with the 
 real merits of that respected person, would tend to throw on his pretensions 
 some degree of ridicule. 
 
 D 
 
34 REPLY TO JESUITICUS. 
 
 this difficulty is avoided. But is it quite certain that Mr. Airy 
 can shelter himself under this plea ? If that gentleman really 
 considered that his investigation rested upon this particular 
 assumption, how did he happen not to advert to it ? His in- 
 vestigation is presented in a general form, the result deduced 
 by it is in a general form, and this result is afterwards made 
 use of in a subsequent part of Mr. Airy's work (Tracts, 2nd 
 edit., p. 354). Not a syllable is said of any limiting hypothesis 
 as to the nature of the vibration. If we are to assume 
 
 *&'/'* 
 u = cos ~Y~ (vt - x), 
 
 A 
 
 what is the use of taking the general values of u l and u', 
 deduced by Taylor's theorem? It is true that in a recent 
 edition (1842) Mr. Airy subjoins a note to his demonstration 
 (which he still retains unaltered in his text) in which he takes 
 up the precise ground suggested by Jesuiticus;* but the ques- 
 tion is, whether at the time when that demonstration was ori- 
 ginally written Mr. Airy was aware of the necessity of making 
 this assumption? If an explanation were required in 1842, 
 after the university had turned out some hundreds of students, 
 more or less fully acquainted with the undulatory theory, how 
 much more so in 1831, at a time when, as Jesuiticus himself 
 suggests, " very little of the undulatory theory was studied or 
 known at Cambridge." This mode of explanation, by the aid of 
 a particular hypothesis, has, no doubt, some date. Does it occur 
 in Fresnel's original investigation? if so let Mr. Airy, or his 
 champion Jesuiticus, cite evidence of the fact. If it does not, 
 we shall be slow to believe but that Mr. Airy committed the 
 error I stated him to have committed. If it does, it will remain 
 to be considered whether, although at one time aware of it, 
 Mr. Airy had not subsequently lost sight of it, or thought it 
 unnecessary. 
 
 I can assure Jesuiticus that there is nothing very incredible 
 in supposing Mr. Airy to have committed such an oversight. 
 
 * It is, perhaps, scarcely necessary to mention that when I first commented 
 upon Mr. Airy's investigation, I was not aware of his having adopted this line 
 of justification. 
 
UEPLY TO JESUIT1CUS. 35 
 
 It would not be the first he has made, nor perhaps the greatest. 
 Even his recent explanation betrays a singular confusion of 
 thought with regard to this subject. In the note above re- 
 ferred to, he says, " If h is so large with regard to the length 
 of a wave that the terms after h z cannot be safely neglected, 
 we may, by assuming a form for the function expressing u, 
 integrate the equation 
 
 m 
 
 How are we to know whether h is " so large with regard to 
 the length of a wave that the terms after A 2 cannot be safely 
 neglected," except by assuming 
 
 . 2zr f . N Q 
 u = A sin y- (vt - x) ? 
 A 
 
 The fact is, that so far from our knowledge of the relation 
 of h to \ leading us to this assumption, it is the assumption 
 itself which alone suggests the existence of any such relation. 
 
 Mr. Airy says, " If h is so large, &c.," as if there could 
 be any doubt of the fact! The letter h here represents the 
 ordinate of any particle measured from that whose motion is 
 being considered, although, from the turn Mr. Airy gives to 
 the investigation, it might appear to mean nothing more than 
 the distance between the rows of particles. Mr. Airy assumes 
 that the motion of the particle is principally affected by the 
 forces exerted upon it by some half-dozen particles immediately 
 surrounding it. Such a supposition is admissible only provided 
 it does not involve any considerations different from what occur 
 when the whole medium is taken into account. But if we take 
 account of the whole medium, h may have any magnitude 
 whatever ; hence it is impossible to argue from Mr. Airy's 
 investigation, as originally proposed, to the actual case. 
 
 As a further specimen of Mr. Airy's extraordinary views 
 I may observe that, in this same note he endeavours to de- 
 rive from his formula a proof that different rays may have 
 different refrangibility : in which respect (admitting his data) 
 he to a certain extent succeeds; but it follows from his 
 
 D2 
 
36 REPLY TO JESUITICUS. 
 
 result (as no one can know better than himself) that the same 
 circumstance must also obtain in the free ether, which is 
 contrary to the fact; so that at all events this is not the true 
 explanation. I have elsewhere remarked that I am unable 
 to appreciate the support which a theory can derive from 
 an explanation which no one believes ; but Mr. Airy seems 
 to entertain a different opinion in this respect. 
 
 Lastly, I would observe that Mr. Airy's investigation, let 
 it be bolstered up as it may, is incomplete, as not taking 
 account of the motion, save in one direction. When account 
 is taken of the entire motion, and the effect of the whole 
 medium is also considered, it reduces itself to a particular case 
 of Mr. Kelland's theory, which I have elsewhere shewn to be 
 entirely erroneous. 
 
 So much for the justice of my criticism on Mr. Airy now 
 for the fairness of Jesuiticus's remarks upon me. The case 
 stands thus : Mr. Airy discusses a problem (in one respect 
 at least) in a general manner. The result is in a general 
 form, and this general form is subsequently made use of by 
 Mr. Airy in another part of his work. In examining the inves- 
 tigation I perceive it to be altogether erroneous, but, to save 
 trouble, I point out one particular error which is fatal to the 
 argument as it is actually propounded. But Jesuiticus insists 
 that, although the argument is false in its actual form, it may 
 be supported by means of a subsidiary hypothesis ; and as I have 
 not adverted to this circumstance, he thinks it necessary to 
 ask whether I " know anything of analysis ? " Such a question 
 from one who has swallowed all the absurdities of Fresnel's 
 theories, occurs to me as somewhat ridiculous ; but as Jesuiticus 
 appears to have a rather vague notion of what constitutes 
 proper evidence of ignorance, I will here indulge him with 
 an illustration. 
 
 Jesuiticus considers that Fresnel's Theory of Double Refrac- 
 tion " ought to be regarded as a stupendous monument of 
 human ingenuity." He therefore, no doubt, considers that 
 theory to be true. He must, moreover, admit that the prin- 
 ciples upon which it is founded are so. He knows very well 
 
REPLY TO JESUITICUS. 37 
 
 that one of those principles is, that the equations of motion 
 of the particle xyz, are 
 
 and he cannot but be aware that, strange as it may appear, the 
 evidence by which these equations are established implies the 
 fact that the motion of the particle is entirely independent of 
 that of the surrounding medium ; or that 
 
 x = A cos (at -fa), y = B cos (bt + )3), z = C cos (ct + y), 
 
 where ABC, a/3y, are entirely independent of every circum- 
 stance connected with the rest of the medium. When there- 
 fore Jesuiticus assumes that the equation 
 
 o_ 
 
 u = A v cos - (ct - x^) 
 \ 
 
 represents the motion of the same particle, it is evident that 
 he is ignorant of the fact that he is making an assumption which 
 is consistent with the principles he has undertaken to defend. 
 
 Now what is the ignorance Jesuiticus thinks to have detected 
 in me ? Had I adverted to the particular hypothesis when 
 I wrote my criticism on Mr. Airy's investigation, I might still 
 have condemned the latter on the ground I did, as failing to 
 prove what it purported to prove. I knew the investigation 
 to be erroneous ; but in choosing my objection to it, I selected 
 that which seemed to present the most novelty, and at the same 
 time exhibited most conspicuously the gross carelessness with 
 which it had been conducted. The only ignorance Jesuiticus 
 can convict me of therefore is, that I did not know it was 
 Mr. Airy's intention to defend his investigation on this ground ; 
 a circumstance which, even admitting that Mr. Airy originally 
 had such an intention, would be perfectly compatible with a very 
 considerable knowledge of analysis. Jesuiticus's query there- 
 fore was uncalled for, as well as impertinent. If I chose I 
 might retaliate, but I can be merciful even to one who has 
 not been ashamed to indulge in a personal sneer in a publication 
 to which he has not had the courage to put his name. I have 
 shewn Jesuiticus to be guilty of a gross piece of ignorance as to 
 
38 REPLY TO JESUITICUS. 
 
 the consequences of the principles which he himself professes, and 
 I could probably expose him still further in his way ; but I do 
 not on that account think it necessary to suggest a doubt as to 
 whether he " knows anything of analysis." While entertaining 
 my own opinion as to his sagacity, I can still allow him to pos- 
 sess about as much knowledge of analysis as the generality 
 of those who study the undulatory theory at Cambridge. 
 
 As to my degree, which Jesuiticus has thought proper to 
 distinguish with so particular a notice, I can only say that I 
 have satisfaction in the enjoyment of my humble honors which 
 it is possible that he does not possess which is, that they 
 have not been attained by adopting every error of every author. 
 It is possible, moreover, that I have some claim on the respect 
 of mathematicians beyond what my degree may confer. If 
 Jesuiticus will condescend to acquaint us with his identity, I 
 shall not be afraid to measure my pretensions with his own. 
 
 In conclusion, I would recommend Jesuiticus to leave Mr. 
 Airy to fight his own battles. That gentleman is in a position 
 in which a frank confession of error will avail him more than 
 all the (masters of?) arts of the ' Society of Jesus.'* 
 
 Liverpool, Dec. 9, 1846. 
 
 * I do not know why my antagonist styles himself a * Jesuit.' From 
 his practice of begging the question, I should have guessed him a 'Mendicant.' 
 
A REFERENCE TO FORMER CONTRIBUTIONS 
 
 TO THE PHILOSOPHICAL MAGAZINE ON PHYSICAL OPTICS. 
 
 By Prof. POTTER, A.M., F.C.P.S., late Fellow of Queens' College, 
 Cambridge. 
 
 (From the Philosophical Magazine, No. 186.) 
 
 WITHOUT the slightest wish to interfere in the controversies 
 of others, I now beg to refer the readers of the Philosophical 
 Magazine to my papers in the Magazines for January 1840, 
 and May 1841. In the former, at page 20, I have shewn 
 Mr. Green's formula for the intensity of reflected light to fail 
 entirely as a representation of nature ; and in the latter I have 
 shewn the peculiar refraction near the optic axes of biaxal 
 crystals not to be represented by Sir William Hamilton's 
 analytical deductions from Fresnel's equation to the wave 
 surface in biaxal crystals. 
 
 " The anonymous correspondent ' Jesuiticus', in the last No., 
 refers to those analytical researches triumphantly in favour 
 of the undulatory theory of light. I do not write to disturb 
 the philosophical opinions of 'Jesuiticus,' but to remind the 
 readers of the Magazine where they will find the discussion 
 of the points referred to." 
 
ON DOUBLE REFRACTION. 
 
 (Continued.*) 
 (Altered from the Philosophical Magazine, No. 185, with additions.) 
 
 IN my last paper I brought under consideration the two pre- 
 liminary investigations entered into by Fresnel, with a view 
 to facilitating his theory of the separation of the ray in biaxal 
 crystals, viz. the proof of the existence of the axes of elasticity, 
 and that of the law according to which the velocity of a wave 
 consisting of rectilinear transversal vibrations may be inferred 
 from the motion of a single particle: and I shewed in the 
 first place that nothing like axes of elasticity can exist, and 
 in the second, that the proposed deduction of the kw f pro- 
 pagation is entirely erroneous. With regard to this latter 
 investigation I might have adopted a different course. The 
 scheme of FresneFs argument appears to have been, by the 
 aid of the axes of elasticity to ascertain the motion of a single 
 particle, and then, by an independent method, to investigate 
 the effect of that particle on the surrounding medium. Now 
 it so happens that, not only is the particular conclusion at which 
 Fresnel has arrived in this kst respect entirely fallacious, as 
 I have shewn, but the existence of the axes of elasticity, upon 
 which the whole of his subsequent superstructure is built, 
 affords the most convincing evidence that no motion whatever 
 can be propagated from the original particle to the particles 
 around it. 
 
 Thus, if the axes of elasticity exist, we have for the equa- 
 tion of motion of the particle x, y, z, 
 
 d?x 3 d z y clz 
 
 _ = -*, -$ = -by, jp = -c>z; 
 
ON DOUBLE REFRACTION. 41 
 
 whence by integration we get 
 
 x = A cos (at + a), 
 
 y = B cos (bt + |3), 
 
 z = C cos (c + -y), 
 
 where A, B, C, a, |3, y, are constants, which can be deter- 
 mined from a knowledge of the circumstances of the par- 
 ticle's motion at a given time : and as it assumed in the 
 investigation of the axes of elasticity that the motion of the 
 surrounding particles has no effect upon the original particle, 
 it follows that these quantities are entirely independent of 
 every circumstance connected with the surrounding medium. 
 Thus, without some special interposition of Providence di- 
 rected towards this particular particle, it will never move at 
 all ; $nd, once set in motion, it will vibrate for ever, entirely 
 irrespective of the state of rest or motion of the other particles : 
 as might have been anticipated, as the other particles exercise 
 no influence upon it, conversely it exerts none upon them : or, 
 in a word, so far from FresneVs law of propagation being true, 
 no motion whatever will be propagated. 
 
 Now, I would observe, that it would have been sufficient 
 to overthrow Fresnel's theory, had I contented myself with 
 disproving either the existence of the axes of elasticity, or 
 the investigation of the law of propagation ; or if I had 
 shewn that the existence of the axes of elasticity negatives 
 the possibility of wave motion: but I have thought proper 
 to expose the fallacy of each step of the argument, in order 
 to make known the character of the man who could propose 
 such a theory, and to expose the more effectually the enormous 
 mass of error which has so long been suffered to pass current. 
 For the same reason I shall enter into a detailed consider- 
 ation of the remaining part of Fresnel's theory, which I shall 
 here give in the words of Mr. Airy.* (Vide Airy's Tracts, 
 p. 345.) 
 
 * Sir John Herschel appears to have been so completely mystified in this part 
 of his subject by the extraordinary ratiocination of Fresnel, that he adds 
 an important blunder of his own to the already abounding provision furnished 
 by the latter. According to Sir John Herschel, (vide Treatise on Light, 
 Art. 1007,) if a plane wave be incident on a doubly-refracting crystal, having 
 parallel faces, the two emergent waves will be inclined to each other. This 
 may be true, but if so, it must revolutionize the theory of Optics. 
 
42 ON DOUBLE REFRACTION. 
 
 " Suppose that in fig. (5) MN is the front of a wave, or 
 by the definitions of (20) and the assumptions of (99) and (100), 
 all the particles in the plane, of which MN is the projection, 
 are moving with equal motions in that plane. In general the 
 force which acts on these particles in consequence of their 
 displacement, is not in the direction of the displacement, and 
 is not even in the plane MN. Resolve it into two ; one per- 
 pendicular to the plane, and the other parallel to it. The 
 former of these may be neglected, because it can only produce 
 a motion which by (101) is not sensible to the eye. The 
 latter, though in the plane, will not generally be in the direc- 
 tion of the displacement. It is impossible then to find what 
 motions will be caused by this displacement without resolving 
 it. If we can resolve it into two, such that the force produced 
 by each displacement is in the direction of that displacement, 
 then we can find for each of these separately the motions 
 that will result from it. It is clear now that we have fallen 
 on a case exactly similar to that of (104)" [the redoubtable 
 investigation professing to give the law of transmission of 
 transversal vibrations], "and the conclusion is the same, namely, 
 that there will be two series of waves travelling with different 
 velocities." 
 
 The above is meant to apply to the case of uniaxal crystals, 
 but the principle is the same when the crystals are biaxal; 
 though, on account of the more complicated analysis, the appli- 
 cation is somewhat different. In this case, if X, Y, Z be 
 the inclinations of the radius vector to the axes of elasticity, 
 a 2 cos X, b 2 cos Y, c 2 cos Z the forces upon the particle, v 2 
 the whole force parallel to the radius vector, u 2 that per- 
 pendicular to it, X lS Y 19 Z l the inclinations of u 2 to the axes, 
 we have 
 
 u 2 cos Xj = a 2 cos X - v* cos X, 
 
 = (a 2 - v 2 ) cos X, 
 u 2 cos Y l = (b 2 - v 2 ) cos Y 9 
 u* cos Z l = (c 2 - v z ) cos Z ; 
 and if the equation to the front be 
 
 Ix + my + nz = 0, 
 
ON DOUBLE REFRACTION. 43 
 
 we must have, when the whole force perpendicular to the radius 
 vector is perpendicular to the front, 
 
 7 v a o 
 
 I = cos X t = cos X, 
 
 m = cos Yi = 2 cos Y", 
 
 (i); 
 
 w 
 
 c 2 - # 2 
 
 n = cos Z, = - cos Z, 
 w 2 
 
 and we have 
 
 / cos X + w cos Y + n cos Z = : 
 
 hence, eliminating cos X, cos Y, cos Z from this last by means 
 
 of (1), we have 
 
 72 W 2 2 
 
 ra + CT + j^** ( 2 )' 
 
 an equation which gives two values of v z , and therefore two 
 sets of values of X, Y, Z corresponding to the positions of 
 the radius vector, in which the whole force perpendicular to 
 the front acts along that line. It is easily shewn that these 
 directions are at right angles to each other, and correspond 
 with the maximum and minimum values of v z , or that they 
 are identical with the greatest and least diameters of the section 
 of the surface of elasticity made by the plane of the front. 
 
 It will here be necessary, however, to advert to one or two 
 circumstances which Fresnel and his commentator appear to 
 have overlooked. 
 
 It is assumed that the resolved part of the force perpen- 
 dicular to the front " may be neglected, because it can only 
 produce a motion which is not sensible to the eye." Is this 
 quite clear? Is it quite certain that the force perpendicular 
 to the front has no effect on the motion parallel to the front? 
 Though for certain purposes it may be allowable to neglect 
 the force perpendicular to the front, it must not be forgotten 
 that it still exists; the consequence of which is that, even 
 supposing the motion parallel to the front to be rectilinear, 
 the actual motion is not so ; so that the true place of the 
 particle at a given time is not to be sought in a line in the 
 front of the wave, but in a plane perpendicular to the front. 
 
44 ON DOUBLE REFRACTION. 
 
 Let X,, Y l9 Z l be the inclinations of the true radius vector 
 to the axes of elasticity; x, y, z those of its projection on 
 the front of the wave, which we will suppose to satisfy the 
 equation (2). 
 
 Let Lx + My + Nz = Q ............ (3), 
 
 be the equation to a plane passing through the radius vector 
 perpendicular to the front; then 
 
 L cos X 1 4 M cos Y l + N cos Z l = ...... (4), 
 
 LI 4- Mm + Nn = ...... (5), 
 
 also L cos X + M cos Y 4 N cos Z = ...... (6). 
 
 The true forces upon the particle, according to the theory 
 of the axes of elasticity, are 
 
 a 2 cos X l9 b z cos Y l9 c 2 cos Z l ; 
 therefore the equations to the direction of their resultant are 
 
 a 2 cos X 1 b* cos Y l ' a 2 cos X l c 2 cos Z l ' 
 
 and, in order that this may coincide with the plane whose 
 equation is (3), we must have 
 
 La z cos X l + MW cos Y l + Nc z cos Z l = (7). 
 
 Eliminating L and M successively between (5) and (6), 
 we get 
 
 (I cos Y- m cos X) M+ (I cos Z - n cos X) N= 0, 
 (I cos Y -m cos X) L + (n cos Y- m cos Z) N= 0. 
 
 From these last, and from (4) and (7), we obtain 
 
 (n cos Y- m cos X) cos X, + (7 cos Z" - w cos X) cos Y 1 \ _ 
 
 4 (m cos X - I cos Y) cos Z l J ~ 
 
 a 2 (w cos F- m cos X) cos X x 4 J 2 (/ cos Z- n cos X) cos Y^ _ 
 
 + c 2 (m cosX-l cos Y) cos Zj = ' 
 .-. (a 2 - b 2 ) (n cos F - m cos Z) cos X x 
 
 + (c 2 - 2 ) (m cos X - I cos F) cos Z l = 0, 
 (* 2 - a 2 ) (7 cos Z - cos X) cos Y l 
 
 4 (c v - a 2 ) (m cos X - I cos F) cos Z l = 0, 
 
ON DOUBLE REFRACTION. 45 
 
 and we have 
 
 cos 2 X 1 -i- cos 2 Y l + cos 2 Z, = 1 ; 
 therefore 
 
 tan 2 Z = m cosX ~ l cos 
 
 1 " 
 
 COS - 
 
 I 
 
 J cos Z - n cos Xy J 
 
 from which it follows, that there are only two positions of 
 the radius vector in the plane Lx + My + Nz = 0, in which 
 the force upon the particle lies in that plane ; so that if at 
 any moment the particle were actually moving in a plane 
 perpendicular to the front, the next it would cease to do so : 
 from which consideration it is plain that the polarization of 
 the rays spoken of by Fresnel is entirely visionary, that his 
 law of propagation is altogether inapplicable ; and that from 
 this consideration alone, independently of every other, his 
 whole theory falls to the ground. 
 
 Nor is this all. Since it is assumed by Fresnel that we 
 can neglect the force perpendicular to the front in considering 
 the motion in the front, and as the whole force in the front 
 when the particle is disturbed along the greatest diameter is 
 parallel to that line, we may take for the equation of motion 
 parallel to the greatest diameter 
 
 where u is the ordinate measured parallel to the diameter, 
 and A is a constant. 
 
 Let r be the true radius vector of the particle (supposed 
 in the plane of the front), x, y, z its coordinates measured 
 parallel to the axes of elasticity; then the forces parallel to 
 those axes are a z x, T?y> c z z, and the whole force along the 
 greatest diameter (if a, j3, y be its inclinations to the axes of 
 elasticity) will be 
 
 cfx cos a + b z y cos j3 + c z z cos y ; 
 and this must coincide with Au where u is the projection 
 of r on the greatest diameter, 
 
 = x cos a + y cos (3 + z cos y, 
 
46 ON DOUBLE REFRACTION. 
 
 which is impossible in a doubly refracted crystal: whence it 
 follows that it is not true, as Fresnel supposes, that the effect of 
 the disturbance on the particle is the same as the combined effects 
 of two disturbances communicated in the direction of the greatest 
 and least diameters respectively, and calculated separately. Thus, 
 admitting every previous step of the investigation, admitting 
 the existence of the axes of elasticity, and waiving their in- 
 compatibility with wave motion, admitting the general pos- 
 sibility of transversal vibrations, and the truth of the law 
 according to which they are supposed to be propagated, ad- 
 mitting the assumption (of which we have not a shadow of 
 evidence, and which, in fact, may be shewn to be untrue) 
 that the force perpendicular to the front may be neglected 
 in considering the motion in the front, and passing over the 
 fact that, whatever may be the case under other hypotheses, 
 under Fresnel's no transversal vibration could take place, it 
 is plain that, if it did, it would not be such as Fresnel has 
 supposed. 
 
 I have now come to the conclusion of my examination of 
 Fresnel's Theory of Double Refraction as proposed by himself; 
 and for convenience of reference, though at the expense of 
 repetition, I shall recapitulate my objections to it, which are 
 briefly as follows: That there are no axes of elasticity 
 that if there were there would be no wave motion; that the 
 possibility of undulations consisting of transversal vibrations 
 has never been shewn much less has the law of propagation, 
 to which Fresnel assumes them to be subject, been proved; 
 that, granting all Fresnel's assumptions, there would be no 
 polarization or resolution of the ray; and lastly, that if the 
 ray were resolved into two, the components would not be 
 what Fresnel makes them out. 
 
 With those who are so ignorant of the laws which regulate 
 the exercise of our faculties, as to suppose that processes so 
 conducted could lead to a successful result, or who are so supe- 
 rior to ordinary considerations of evidence as to imagine that 
 an investigation, every step of which is erroneous, may be 
 right in the end, what I have advanced may have little 
 weight: but I am not afraid to appeal to any man who is 
 
ON DOUBLE REFRACTION. 47 
 
 accustomed to follow the dictates of his own judgment, whether 
 I have not redeemed my pledge of proving that, " to whom- 
 soever the idea of transversal vibrations may be due, the 
 researches of Fresnel have not advanced us one step beyond 
 it." 
 
 I shall hereafter consider some of the improvements upon 
 this theory which have been proposed by Fresnel's followers; 
 but I shall previously direct attention to his Theory of the 
 Reflection and Refraction of Polarized Light. 
 
ON FRESNEL'S THEORY OF POLARIZATION 
 
 WHERE THERE IS NO DOUBLE REFRACTION. 
 
 HAVING fully exposed the fallacy of Fresnel's Theory of Double 
 Refraction, I now come to the supplementary part of his theory, 
 where, by the aid of divers contrivances, he seeks to obtain 
 formulae for the intensity of polarized waves after they have 
 undergone reflection and refraction by singly refracting media. 
 I am aware that that individual's credit is by no means prin- 
 cipally founded upon his treatment of this part of his subject, 
 and that if his theory of double refraction be overthrown, 
 there are few who will undertake to break a lance in favour 
 of the strange vagaries in which he has here thought proper 
 to indulge. I have deemed it right, however, to review in 
 detail this part of the subject also, and to point out the 
 errors with which it is replete ; not from any malicious motive 
 towards those whose credit may thereby be injured, but in 
 order to make known to future ages the extravagant fallacies 
 which it is possible for very sagacious minds, when once the 
 judgment has been surrendered, to regard with complacency; 
 and at the same time to guard the philosophical student 
 against the demoralizing influence which principles and pro- 
 cesses like those I am about to consider are calculated to 
 exercise, in perverting the judgment and destroying all sound 
 habits of thought. I shall now therefore, without further 
 preface, give Mr. Airy's account of the way in which, when 
 " light, polarized in the plane of incidence, falls , on a refract- 
 ing surface of glass, &c.," Fresnel finds " the intensity of the 
 reflected and refracted ray." (Vide Airy's Tracts, p. 354.) 
 
 " Suppose that the particles of ether, retaining the same 
 attractive force, are inside of glass, &c. loaded with some 
 
49 
 
 matter which increases their inertia in the ratio of 1 : n, without 
 increasing their attraction. 
 
 " The equation of (103) would be changed to this : 
 
 d*u _ 1 / 1 \ m d *u 
 ~df ~ n \ ~J h dtf' 
 
 " If the solution before were u = <J> (vt - #), the solution would 
 now be 
 
 u = (vt - x Vra). 
 
 " The velocity of transmission is diminished therefore in the 
 ratio of Vw : 1. But we have supposed that the velocity is 
 diminished in the ratio of n : I. Consequently n = $. 
 
 " Now suppose that we have a series of equal quantities of 
 the ether in a line, and that a transverse motion is given to the 
 first, which from the constitution described in (103) it has the 
 power of transmitting to the second, &c. 
 
 " When we arrive at the surface of the glass, we must take 
 volumes of the denser ether, whose dimensions are determined 
 in the direction of the transmission of the wave by lengths pro- 
 portional to the velocity of transmission, and in the other 
 directions by their correspondence with the quantities of ether 
 
 T> J~\ 
 
 which puts them in motion. Thus in fig. (6), if DF = - , the 
 
 ether in ABDC may be considered as putting CDFE in 
 motion. Put * for the angle of incidence, f for that of refraction. 
 The proportion of the lengths in the direction of the ray is /&t : 1, 
 or sin i : sin i'. The proportion of their breadths is cos i : cos i'. 
 The proportion of the densities is 1 : yu, 2 , or sin 2 i' : sin 2 i. Com- 
 bining these proportions, the proportion of the masses is 
 
 sin i' . cos i : sin i . cos i', 
 
 " Now, if an elastic body impinges on an equal elastic body, 
 it loses its own velocity and communicates to the other a velocity 
 equal to its own : this is similar to the action of one mass of the 
 ether in vacuum on the next. Supposing the similarity of action 
 to apply to the different states of ether at the confines of the 
 medium, we must compare this with the motion of two unequal 
 elastic bodies A and J5, after the impact of A with the velocity 
 
50 ON FRESNEL'S THEORY OF POLARIZATION. 
 
 V on B originally at rest. It is known that A retains the 
 
 A 7? 2^4. 
 
 velocity - V, and that B receives the velocity . V. 
 
 Substituting for A> sin i' . cos i, and for B sin i . cos e', we find 
 
 for the motion retained by the external ether, SU . x its 
 
 sin (* -f i) 
 
 previous displacement; and for that communicated to the internal 
 
 ether - - x previous motion of external ether. Now, 
 
 sin (* 4 *) 
 
 by a succession of numerous impulses of this kind following 
 a given law, a series of waves with any law of displacement may 
 be produced; and every impulse produces parts in the two 
 media having the proportions given above. If then the original 
 
 displacement be represented by a . sin (vt - x), that retained 
 
 A 
 
 by the external ether, and which produces the reflected ray, 
 
 must be 
 
 sin (i 1 - i) . 2ir , , 
 a . . ;., ( . sin (vt - x\ 
 sin (l 4 *) A 
 
 and that transmitted to the internal ether, and which produces 
 the refracted ray, must be 
 
 2 sin i' . cos i . 2ir , . 
 
 a . --. 777 T sin (vt - x). 
 
 sin (t +t) A 
 
 With regard to the preliminary part of the above investi- 
 gation, which has for its object the explanation of the change 
 of velocity in passing from one medium to another, I would 
 observe that, being founded on the investigation of the law o 
 propagation commented on in my first paper on double refrac- 
 tion, it must with that investigation fall to the ground. But this 
 is not all I have to say regarding it; and in the first place I 
 would direct attention to the extravagance of supposing " that 
 the particles of ether, retaining the same attractive force, are 
 in the inside of glass, &c., loaded with some matter which 
 increases their inertia in the ratio of 1 : n, without increasing 
 their attraction." Mr. Airy says, indeed, in a note, " Perhaps 
 this supposition is hardly reconcileable with that made in the 
 last propositions;" but surely any objection to the hypothesis 
 
ON FRESNEL'S THEORY OF POLARIZATION. 51 
 
 on this ground is trifling, when compared with that which arises 
 from its own intrinsic absurdity. If we had to invent light, 
 such fantastical contrivances might be tolerated ; but as this 
 is not our object, and we have simply to discover the machinery 
 already devised and put in action for the bringing about that 
 great phenomenon, it would surely be better to confine our 
 efforts to following out the real indications of the phenomena ; 
 and if we cannot succeed in connecting them with agents whose 
 mode of operation is understood or can be conceived, at any 
 rate to abstain from wild conjectures as to causes which are 
 neither rendered probable nor even intelligible by analogy, nor 
 necessarily implied by the facts of the case. It is still more 
 advisable to refrain from such violent hypotheses when they 
 are, as in this case, entirely gratuitous ; for every object which 
 Fresnel had in view, in making the above assumption, is 
 equally met by supposing that the distance between the rows 
 of particles, or h, is different within the medium from what it is 
 without it. 
 
 But besides being extravagant and uncalled for, this hypo- 
 thesis is unsuccessful it does not after all explain the phenomena 
 it is intended to account for ; for, according to this theory, the 
 velocity of transmission depends solely on the nature of the 
 medium, whereas we know very well that in the same medium 
 we have very different velocities. 
 
 So much for the preliminary investigation, now for the 
 principal argument. And here one cannot but admire the 
 boldness of the genius which could assimilate the motions of 
 a number of particles situated at considerable intervals from 
 one another (being kept asunder by some inscrutable means) 
 to that of two rough parallel surfaces in contact, acting the one 
 upon the other in a horizontal direction; and which could 
 moreover suppose that the motion so produced would be pre- 
 cisely the same as that of two perfectly elastic bodies. But 
 laying aside this consideration, the investigation fails entirely. 
 Mr. Airy compares the motion of the two masses of the ether 
 within and without the medium respectively " with the motion 
 of two unequal elastic bodies A and B, after the impact of 
 -4, with the volocity V on B originally at rest:" but although, 
 
 E2 
 
52 ON FRESNEL'S THEORY OF POLARIZATION. 
 
 to start with, the ether within the glass might be considered at 
 rest, yet it can only be so momentarily. It can only be so while 
 the vibration of the external particles immediately contiguous 
 
 to it is represented by a sin -^ (vt - x), where sin -r- (vt - x) is 
 
 A A 
 
 extremely small; whereas Fresnel assumes it to hold for all 
 values of that function. The error here committed is curious, 
 as affording an additional example of the extraordinary prin- 
 ciples of investigation which that remarkable person was in the 
 habit of adopting. He proves the existence of the axes of 
 elasticity on the supposition of all the particles but one being at 
 rest, and then he takes it for granted, in the face of all evidence 
 to the contrary, that the same holds when all the particles are 
 in motion : he likewise professes to prove a law of propagation, 
 on the supposition that the disturbance is communicated in the 
 direction of the axis of the system, and then assumes that it 
 holds for eyery direction; and in this last example he proves 
 a formula which may hold for a single instant, but certainly 
 not more, and then assumes it to hold for ever. 
 
 The next investigation (vide Airy's Tracts, p. 357) has for its- 
 object, when "light polarized perpendicular to the plane of 
 incidence falls on a refracting surface, to find the intensity of 
 the reflected and refracted ray. 
 
 " We cannot here use the same kind of reasoning as in (128)," 
 (meaning the investigation which we have just considered) 
 "because the motion of displacement (being in the plane of 
 incidence and perpendicular to the path of the ray) is not in 
 the same direction for any two of the three rays. To overcome 
 this difficulty, M. Fresnel has adopted the following hypotheses 
 First he supposes that the law of vis viva holds, that is, that the 
 sum of the products of each mass by the square of its velocity 
 is constant. (This is certainly true if, as in (128), the masses are 
 supposed to act nearly as elastic bodies. Arid in all cases o: 
 mechanical action it is equal to the sum of all the integrals 
 of force x space through which it has acted, which is constant 
 in all the cases of undulation that we can strictly examine, and 
 is probably constant in this.) Next he supposes that the re- 
 solved parts of the motion, parallel to the refracting surface 
 
53 
 
 will preserve, after leaving the surface, the same relation which 
 they have there, and which, if they follow the same laws as 
 those of the impact of elastic bodies, would be thus connected : the 
 relative motions" (qu. displacements) " before and after impact 
 will be equal in magnitude but opposite in sign. (This is con- 
 fessed by M. Fresnel to be purely empirical.) Adopting these 
 hypotheses, and considering the masses to be as 
 
 sin *' cos * : sin i cos i', 
 
 and representing the displacements in the incident, refracted 
 and reflected ray, estimated in the direction nearest to that 
 of a body falling perpendicularly from vacuum on the refracting 
 surface, by a, b, c, we have the following equations, 
 
 sin *' . cos i . tf = sin i . cos i' . b* + sin i' . cos * . c 3 , 
 
 a cos i = b cos i' + c cos i. 
 Eliminating b, 
 
 (sin 2i' + sin 2t) c 2 - 2 sin 2e . ac - (sin 2i' - sin 2e) a 3 = 0, 
 
 or (c - a) {(sin 2t' + sin 2i) c + (sin 2i' - sin 2t) a} = 0. 
 
 " This equation is satisfied by c = a : but that would give 5=0, 
 and therefore expresses only total reflection, which would re- 
 quire exactly the same mathematical conditions as those that 
 we have used, but would not correspond to the physical circum- 
 stances of the problem now before us. The other is the only 
 solution which we want : it gives 
 
 tan (*' - t) 
 
 c- a- ^ /, 
 
 tan ($ -i- i) 
 
 . , cos * C tan (' - i\\ 
 
 and b=a 1 1 + - -^ A. 
 
 cos ^ [ tan (e -i- t)J 
 
 " Hence, if the displacement produced by the incident wave is 
 
 2?r / * 
 a sin (vt - x), 
 
 A 
 
 that produced by the reflected wave is 
 tan (*' - i) . 2?r 
 
 - . TV; K . Sin (^ - ), 
 
 tan(e + t) \ v 
 
54 ON FRESNEL'S THEORY OF POLARIZATION. 
 
 and that by the refracted wave is 
 
 cos i r tan (*' - i}\ . ITT f 
 
 a . 77 {l + - -r, 7} . sm (vt - fix). 
 
 cos ^ t tan (i -t- i)} X 
 
 Thus far Mr. Airy. 
 
 From the first of these formulae it follows that the tangent 
 of the polarizing angle is equal to the refractive index of the 
 medium. This result at first sight seems a remarkable corrobo- 
 ration of the truth of the expression, but there can be little 
 doubt that the formula has been expressly contrived to evolve 
 such a conclusion. Mr. Airy says, " we cannot here use the 
 same kind of reasoning as in the preceding case," but, in truth, 
 there is no difficulty in making the application by resolving the 
 motions parallel and perpendicular to the refracting surface; 
 and the only reason why this has not been done is, that the 
 result so obtained is not satisfactory. Being foiled in this first 
 attempt which, I have not a doubt, Fresnel made, he naturally, 
 and very properly had recourse to the general mechanical 
 formulae, in order to obtain an equation by which to facilitate 
 his future operations ; and for this purpose he appears to have 
 selected the equation of vis viva, or what, at least, he has 
 thought proper to call such. 
 
 The equation he obtains is 
 
 (1) sin i' . cos i . a z = sin i cos i' . b z + sin i' cos i . c~ ; 
 
 and he has the further clue that when i + i 1 = 90, c = 0. 
 Now when this last relation holds, (1) becomes 
 
 cos 8 i . a? = sin 2 i . b z , 
 so that we have cos i . a = sin i . b 
 
 = cos i' . b ; 
 
 but there seems every probability that any formula which 
 involves a and b, will involve c symmetrically with a. Hence 
 there is reason to suppose that this last equation has been 
 deduced from the equation 
 
 cos i . a = cos i' .b - cos i . c, 
 
 by putting c = 0, as this last will coincide with (1) when i+t'= 90, 
 only on the supposition that c = 0. That Fresnel reasoned 
 
ON FRESNEL'S THEORY OF POLARIZATION. 55 
 
 every step of his process exactly in the manner here indicated, 
 were perhaps asserting too much, but that he arrived at his 
 conclusion something in this way, I think there cannot be 
 a doubt; for most assuredly there is nothing in the nature 
 of things to lead us to suppose that this second equation ex- 
 presses a physical truth, so that it can be considered in no 
 other light than as a mere analytical guess. That this second 
 equation should have the good luck to express a proposition 
 which is intelligible, although we have every reason to believe it 
 untrue, need not be wondered at when we reflect that, besides 
 that which he actually selected, he had three others to choose 
 out of, namely, 
 
 cos i . a = cos *' . b - cos i . c, 
 
 sin i 1 . a = sin i . b + sin i 1 . c, 
 sin i' . a = sin i . b - sin i' . c, 
 
 any one of which would equally have served his purpose. So 
 much then for this apparent coincidence. 
 
 That such reasoning as Fresnel has here employed resting, 
 as Mr. Airy confesses, on an entirely gratuitous assumption 
 should turn out to be erroneous, need surprise nobody. That 
 it is altogether inconclusive is obvious; that it is erroneous 
 appears from the following consideration. According to the 
 above formulae the displacements of the incident and reflected 
 rays immediately contiguous to the refracting surface have, 
 
 except when sin (vt - x) is very small, the same sign ; or 
 A 
 
 the incident and reflected vibrations immediately contiguous to 
 the same point in the surface are both in the same direction, i.e. 
 both moving towards, or both moving from the refracting body, 
 which, as a general fact, may be fairly asserted to be impossible. 
 As to Mr. Airy's somewhat tardy display of discrimination 
 in speaking of one of the above hypotheses as being purely 
 empirical, it is perhaps even more injurious than his usual want 
 of that quality, as leading the student to believe that the same 
 character of gratuitous assumption does not appertain to other 
 parts of this and the preceding investigation ; for example, the 
 manner of calculating the vis viva, and the assumption as to the 
 
56 ON FRESNEL'S THEORY OF POLARIZATION, 
 
 mode of action in the preceding problem, which are as purely 
 the offspring of the fancy of their author as any other portion 
 of his ingenious theories. 
 
 The next investigation undertaken by Mr. Airy has for it* 
 object, when " light is evident on the internal surface of glass 
 at an angle equal to or greater than that of total reflection, 
 to find the intensity of the reflected ray." 
 
 " The expressions in (128) and (129)" [the preceding inves- 
 tigations} " become impossible. Yet there is a reflected ray, 
 whatever be the nature of the vibrations of the incident light. 
 And on the principle of vis viva the intensity of the reflected 
 ray ought to be equal to that of the incident ray, since there 
 is no refracted ray to consume a part of the vis viva. And 
 indeed in the last state of the expressions of (128) and (129) 
 before becoming impossible, i. e. when i' = 90, each of them 
 becomes = 1. After this the expression for the coefficient of 
 vibrations perpendicular to the plane of incidence, putting 
 fju sin i for sin *', and V(- 1) V(/* 2 sin 2 * - 1) for cos ', becomes 
 
 p sin i cos i - sin i V(- 1) V(//. 2 sin 2 1 - 1) , , 
 
 p sin i cos i + sin i V(- 1) V(X sin 2 1 - 1) 
 or cos 20 - V(- 1) sin 20, 
 
 n V(/^ 3 sin i 1) 
 where tan 9 = - 
 
 fj, cos ^ 
 
 and that for the coefficient of vibrations parallel to the plane 
 of incidence, becomes 
 
 sin * cos i - fj, sin i V(- 1) V(/-t 2 sin 2 t - 1) 
 sin i cos * + fi sin V(- 1) VC/* 2 sin 2 i - 1) * 
 
 or cos 20 - V(- 1) sin 20, 
 
 where tan = 
 
 cos i 
 
 " It is improbable that these formulee are entirely without 
 meaning : what can their meaning be ? 
 
 "M. Fresnel seems to have considered that, as the direction 
 of the reflected ray and the nature and intensity of the vibra- 
 tion were already established, there remained but one element 
 which could be affected, namely the phase of vibration. And 
 
ON FRESNEL'S THEORY OF POLARIZATION. 57 
 
 it seems not improbable that this may be affected, inasmuch as 
 the incident vibration, though it cannot cause a refracted ray 
 must necessarily cause an agitation among the particles of the 
 ether outside the glass. It would seem to us most likely that 
 the ray would be retarded (though the phenomena to be here- 
 after described compel us to admit that it is accelerated), and 
 in all probability differently according to the direction in which 
 the vibrations take place. Nothing then seems more likely than 
 that 20 and 20 should express these accelerations : and as they 
 are angles, they must be combined with angles in the ex- 
 pression for the vibration. Thus, for instance, if 
 
 a . sin (vt - x) 
 A 
 
 were the expressions for the vibrations perpendicular to the plane 
 of incidence on the supposition that they were not accelerated, 
 
 a sn 
 
 would be the expression on the supposition that they were 
 accelerated." 
 
 I forbear giving Mr. Airy's account of the reasoning by 
 which Fresnel attempts to establish that the occurrence of the 
 impossible quantity in the above expressions indicates a change 
 of 90, as it is not my purpose to comment upon it. Mr. Airy 
 is, no doubt, consistent in believing it to be " improbable" that 
 the above formulae " are entirely without meaning," as he 
 seems to have no particular horror of the way in which they 
 have been derived, and even appears to have imbibed the 
 notion " that they are fully born out by experiment" ; but to 
 any one else it would most likely appear that that has happened 
 which might have been anticipated, namely, that the formulae 
 have at last brought their inventor into an awkward dilemma. 
 A meaning they indeed bear, and that a very obvious one, 
 which is, that whatever else they may apply to, they do not 
 apply to the internal reflection of light. It is all very well for 
 Fresnel and Mr. Airy to talk of the phase being affected, but 
 what will the latter say to the refracted portion of the ray? 
 
58 ON FRESNEL'S THEORY OF POLARIZATION. 
 
 The general formula for the refracted ray, when the light is 
 polarized in the plane of incidence, is 
 
 2 sin e* '.cost' 
 
 Q> . f .j .T~ y 
 
 sin (i + ^) 
 
 and substituting p sin i for sin i', and V(- 1) V(X sin 2 i - 1) for 
 cos i', this becomes 
 
 2fju sin i . cos i 
 IJL sin * cos i + sin i V(- 1) v(/* 2 sin 2 i - 1) ' 
 
 or 2# cos * (cos - V(- 1) sin 0}, 
 
 a 
 
 where tan = 
 
 cos 
 
 If the first formula shews that a ray is reflected, surely this 
 must shew that one is refracted. 
 
 It is possible that Mr. Airy is by this time prepared to admit 
 that the investigations of the intensities do not afford a shadow 
 of evidence in favour of the formulae deduced, and that these 
 last rest solely upon the basis of experiment : but if he contends 
 that we are authorized upon experimental grounds to consider 
 the formulae for the intensity of the reflected light as so indis- 
 putably true that they must bear a meaning under all circum- 
 stances, will he argue that the formulae for the refracted light 
 are of less binding character ? and if he is not prepared to do so, 
 will he condescend to explain upon what principle we are to 
 consider the occurrence of the same fact, in the one case an 
 indication that there will be no vibration at all, and in the 
 other that there will be two vibrations succeeding one another 
 at an interval of 9 ? 
 
 Mr. Airy tells us (Tract on Light, Art. 126) that the three 
 investigations which we have just been considering " cannot 
 be considered as wholly satisfactory;" but he consoles himself 
 with the idea "that they are fully supported by experiment." 
 Who will vouch for this / Will Sir David Brewster, or Pro- 
 fessor Potter, or any other anti-undulatist ? As to Mr. Airy 
 himself, while he possesses so little candour, or so little discri- 
 mination, as to pass over without notice such a remarkable dis- 
 crepancy between facts and formulae as that I have just pointed 
 
59 
 
 out in the theory of internal reflexion, we must be permitted* to 
 entertain a doubt respecting his competency as an authority. 
 
 To an author like Fresnel no subject can present any difficulty. 
 His arguments are spiritual essence s, against which the weapons 
 of the vulgar world are unavailing : amidst the tempest of 
 objections they bear a charmed life. Listen to Fresnel, and we 
 have a theory of light perfect in all its parts, so far as inves- 
 tigated, and equal to any emergency. But there is a trifling 
 drawback on the delights of this glittering spectacle they are 
 visionary. The delightful region we are traversing is a land of 
 dreams, in which the principles which regulate human reason 
 are strangely jostled. We may if we choose for a time amuse 
 ourselves with the illusion, give the reins to the imagination, 
 shut out the necessities of real life but there must sooner or 
 later be an end of all this ; and when a returning consciousness 
 of the actual world, with its stern realities, breaks in upon us, 
 we shall find ourselves, with minds enervated and judgment 
 obscured, left to cope with difficulties which under happier 
 auspices we shrank from as insurmountable. 
 
 Liverpool, Oct. 23, 1846. 
 
ON MR. KELLAND'S "THEORY" OF HEAT. 
 
 I NOW propose to examine some of the more prominent sugges- 
 tions which have been thrown out by the followers of Fresnel, 
 with a view to obviate the glaring defects of his theory. I am 
 induced to this step, not from any particular cogency in the 
 arguments of these gentlemen which I have been able to dis- 
 cover, but from the knowledge that others have been led to 
 place in them a degree of faith greater than mine, to the extent 
 in fact of believing that many points left by Fresnel in an 
 unsatisfactory state have through their efforts been established 
 on a solid basis. As I believe that whatever objections may be 
 urged against Fresnel's processes are equally applicable to those 
 of his followers, that if the first are erroneous the latter are 
 equally so, it appears to me desirable to destroy this last refuge 
 also of a falling theory; a measure by so much the more ad- 
 visable, as in the execution of it I shall have an opportunity 
 of displaying the reality of the evil ascribed to the reception 
 of Fresnel's theory " in perverting the judgment and destroying 
 all sound habits of thought." 
 
 In the following paper I intend to consider Mr. Kelland's 
 method of treating the theory of finite intervals, as developed in 
 his Treatise on the Theory of Heat, and shall endeavour to give 
 an account of his views as far as possible in his own words. 
 
 "The particles of matter in all bodies" (Theory of Heat, 
 p. 145) " are supposed to be surrounded by particles of a subtle 
 fluid, pervading space and termed caloric, the atoms of which 
 are so much smaller than those of matter, that each material 
 particle nay be conceived to be surrounded by a large number 
 of them. 
 
 " It is supposed that the particles of caloric repel each other^ 
 but attract those of matter, whilst the particles of matter also 
 
61 
 
 repel each other, the function of the distance being always 
 the same, which we shall give reasons for assuming to be 
 that of the inverse square. 
 
 " Let xyz be the coordinates of any particle P of caloric 
 when at rest, x 4 $x, y + fy, z -f &? those of another particle Q, 
 x + Ax, y -f Ay, z + Az those of a material particle M, PQ = r, 
 PM = R, r0 (r) the law of attraction : suppose the attraction at 
 distance unity of a particle of caloric to be P, and that of a par- 
 ticle of matter M. 
 
 " Let x + a, y + j3, z + 7, be the coordinates of P at the end 
 of the time y, 
 
 x + a -f &E + Sa, y + )3 + Sy + 8)3, z + y + 82 + Sy those of Q, 
 
 ar + a + Aa: + Aa, y + j3 + A y + Aj3, z 4 y + Az + Ay those of M, 
 
 r 4 p the distance PQ, 
 
 JB + JI the distance PM." 
 Then 
 
 (1) 4- - P<7 (&c + 80) ^ (r + p) + MS (Az + Aa) ^ (12 + JT). 
 
 " To simplify this equation," Mr. Kelland supposes " that the 
 particles of the medium are arranged in symmetry about the 
 axes," and he also assumes the particles to have a vibratory 
 motion about the axis of x, represented by the equations^ 
 
 a = a . cos (nt - fac), 
 ]3 = b . cos (nt - kx), 
 y = c . cos (nt - kx) ; 
 
 and further, that the particles of matter have a similar motion 
 
 represented by 
 
 ' = A cos (Nt -Kx + B\ 
 
 &c. 
 
 where a', /3', y' denote the displacements of the material par- 
 ticle M. 
 
 These last assumptions he subsequently modifies. To shew 
 their consistency with the equation (1) is the principal scope 
 of Mr. Kelland's analysis. 
 
 We have (r + p) =* (r) -f 0' (r) p, nearly, provided p and 
 
therefore Sa, 8/3, 87, are small, and on the same hypothesis it is 
 easily seen that 
 
 + 8y 8/3 + 8*87), 
 
 {since (r + />) 2 = (&B + Sa) 2 + (fy + S/3) 2 + (& + S 7 ) 2 }. 
 Hence, if the medium be symmetrical, 
 
 a- (&c + 8a) (r + />) = a- (r) + - So? B a . 
 
 But it will be found that 
 
 da sin 
 
 oa = - a sm - + - 
 
 - - --- - - , 
 
 2 dx k 
 
 and, observing that the second term of this expression vanishes 
 under the symbol <r, we have 
 
 a &B + Sa r + = - 2ao- Lr + 
 
 Similarly for the second term in the value of -^ : hence the 
 
 w^ 
 
 equation of motion becomes 
 d 2 u ( <f>'r . 
 
 = " r + V ^ S1 
 
 <t>R + A + Jtfa'2 <j>R + &x* cos KAx. 
 
 tt J \ R ] 
 
 Similarly, 
 
 
 
 + Jf^'S +R + Ay 8 cos K&c. 
 \ R J 
 
 By substituting the values above assumed for af3y, a'P'y, 
 respectively in these last equations, we obtain 
 
 / 7 S^ \ 
 
 (4) - f w + 2or/sin 2 - SF J a = 2jP cos 
 
 (5) N* 4 2SF Sin 2 - cr/ ' = of COS ^ . a ; 
 
where P <t>r + W = f, 
 
 V 
 
 Mr. Kelland here makes a " considerable digression in order 
 to determine the form of the function 0." It is a good maxim 
 ' never to meet one's troubles half way.' This Mr. Kelland 
 seems not to have heard, or to have forgotten, otherwise in the 
 midst of his other difficulties he would not have gone out of 
 his way to prove what every one would be ready enough to 
 concede, viz. that the law of force will be that of the inverse 
 square. But let us see what additional probability Mr. Kel- 
 land throws on this assumption. 
 
 " Suppose the motion to take place in a vacuum, for which 
 we obtain the following equations, 
 
 hence therefore, j3 = b cos (nt - fas), 
 
 " Now it must be observed that the mode in which this result 
 is produced is one which takes account of the differences of 
 the disturbances on the two sides of the particle under con- 
 sideration. The magnitude of the expression will therefore 
 clearly depend greatly on the portion which arises from the 
 particles next it. In other words, (r) must be some inverse 
 function of r, such that it diminishes as r increases." 
 
 I would here observe that it does not follow that, because 
 " the mode in which" the above " result is produced is one 
 which takes account of the differences of the disturbances 
 on the two sides of the particle," " the magnitude of the expres- 
 sion will depend greatly on that of the portion which arises 
 from the particles next it ;" for, notwithstanding that such were 
 the case, if the force happened to vary as some direct power 
 
64 
 
 of the distance and we are as yet supposed to know no reason 
 why it should not it would follow that " the magnitude of the 
 expression" would not <e depend greatly on that of the portion 
 which arises from the particles next it." So far therefore from 
 this pretended argument affording us any ground for believing 
 the force to vary as an inverse power of the distance, it actually 
 assumes the point at issue. But to proceed. 
 
 Assuming $(V*) to be an inverse power of r, Mr. Kelland 
 
 considers it to follow that, " as far as this factor" (meaning I 
 / ,/ \ 
 
 presume the factor a- ( 0r + $x*\ "is concerned, the par- 
 ticles near P will have the principal influence; but within 
 a certain limit this will not be the case with the factor sin 2 - - , 
 
 which will increase at first until it becomes unity, and then 
 again diminish as &E is increased. The result is obviously this, 
 that the principal part of the term will be that which lies 
 
 within the value of &E = - ." 
 
 Upon this "result" Mr. Kelland builds his argument for 
 believing the law of force to be that of the inverse square. Its 
 " obvious" character I must be permitted to doubt. The result 
 may be true, but most certainly it is not obviously so. As x 
 
 increases from j to x, successive increments are given to the 
 
 K 
 
 value of n z , which, " so far as this factor is concerned," have all 
 the same sign. Who will assure us that the aggregate of these 
 last does not exceed the value of that portion which lies within 
 
 the value 'Bx = y- ? Is there any strong reason to believe that 
 
 K 
 
 such is not the case ? Is there anything " obvious" about the 
 matter either one way or other ? And yet the writer who is 
 too fastidious to be content with the evidence afforded by 
 analogy and the law of equal diffusion, is satisfied with a mere 
 guess at the solution of this dark mystery. 
 
 Returning however to his formulae, it occurs to Mr. Kel- 
 land that his law of force, although it " appears to satisfy the 
 requisite conditions," fails to account for one circumstance, the 
 
ON MR. KELLAND'S "THEORY" OF HEAT. 65 
 
 explanation of which appears indispensable to the theory, viz. 
 that in a vacuum there shall be an absolute freedom from vari- 
 ation of velocity due to a variation in the length of the wave." 
 
 His method of getting over "this difficulty" is sufficiently 
 characteristic. He conceives that it " arises from the imper- 
 fection of our acquaintance with the mode in which the particles 
 may be arranged." 
 
 In this respect I am compelled to differ from Mr. Kelland. 
 As a matter of fact, we have no knowledge of the existence 
 of a medium consisting of particles situate at finite* intervals 
 from one another, much less of the nature of its arrangement, 
 yet this circumstance has been no obstacle to Mr. Kelland's 
 suggesting such a medium, and even devising a law of sym- 
 metry to which he supposes it to be subject. In the same 
 way, though we have no knowledge of the mode in which the 
 particles are arranged (nor in fact are likely to have), yet 
 it was perfectly open to Mr. Kelland to suggest any arrange- 
 ment which would suit his purpose. It is the impossibility 
 of conceiving any such which constitutes the real " difficulty." 
 No ignorance of a matter of fact need disturb a disciple of 
 Fresnel. According to that school of logic, if one cannot swear 
 that a system of explanation is untrue it is sufficient. But 
 Mr. Kelland here felt it impossible to make a single step in 
 advance. The condition to be satisfied is that 
 
 nv fr ' ** ** * * x ' A'/IJ* v oTTi" _r Z* 
 
 / 
 
 is independent of k. As far as the first half wave we have 
 sin k&x k$x 1 i k^>x\ 
 
 _ I \ i r&f* 
 
 2 2 1.2.3 \ 2 ) 
 
 as far as the second, 
 
 sin k$x kx 1 
 
 ~ 2~ ~2~ ~ L2".3 \ 2 
 
 whence it is evident that 
 
 r 0' 0) i kx 
 
 " S W + ~ ^V 2 ^ sin 2 = -4 + J5X: + C& 2 + Z>^ 3 j- Ek* + &c. 
 
 \ y r y j 2 
 
 * I adopt this expression for its convenience, though sensible of its impro- 
 priety. 
 
 7T 4 &C.; 
 
66 ON MR. KELLAND'S "THEORY" OF HEAT. 
 
 Who is skilful enough to devise a medium which will be 
 so obliging as to make C finite while A, B - D, E, &c. all 
 vanish ? And until direct proof to that effect has been given, 
 who will believe that any such can exist ? 
 
 Nothing daunted by improbabilities, however, Mr. Kelland 
 without hesitation assumes that such an arrangement actually 
 obtains; but surmising that his inability to guess at its nature 
 might attach a stigma to his system, as being unable to sur- 
 mount this " difficulty," he seeks to overcome it by having 
 recourse to a subsidiary principle, which he considers to be 
 necessarily implied by his original hypothesis. "One thing," 
 says he, "is certain, that when they" (meaning the particles) 
 "are very close together, they act as a mass of particles in 
 contact." What a satisfaction to meet with ' one thing certain,' 
 one firm rock amidst the ocean of doubt on which we are 
 tossed. While examining Mr. Kelland's theory for the first 
 time, and endeavouring to ascertain the bearing of its several 
 parts amidst the confusion of principles and the errors of 
 analysis, I more than once reverted to this ( one thing certain,' 
 all in fact that I met with which laid claim to that property, 
 overlooking the circumstance, in my joy at the discovery, that 
 this claim rested solely on the credit of our author. But when in 
 a calmer moment I began to look more narrowly to the nature 
 of the proposition on whose certainty I had congratulated 
 myself, I soon found that I had been somewhat too hasty in 
 my felicitations. "When very close together they act as a 
 mass of particles in contact." What was meant by a mass of 
 particles in contact ? By some people a rigid body would be 
 considered as such, but this evidently was not intended. Neither 
 could a mass of particles like sand, kept asunder solely by 
 the mutual resistances of their surfaces, though this also seemed 
 to possess some of the qualities of the article prescribed. Kead- 
 ing on however, I became more enlightened. 
 
 " Taking it for granted that such is the case" (i. e. the above 
 certainty) " we may remove the difficulty thus : 
 
 " Let D be the density of the particles in vacuo, then the 
 equation of motion, parallel to y, becomes 
 
THEORY" OF HEAT. 67 
 
 Now though I did not pretend to understand (as I do not 
 now) why, when " they act as a mass of particles in contact," 
 this last equation should hold, the introduction of the con- 
 sideration of density seemed to imply that something like the 
 medium which is the instrument of sound was intended. If 
 this be the case, however, I must say Mr. Kelland has adopted 
 a singular mode of expressing himself. Does he consider that the 
 medium which produces sound consists of a mass of particles in 
 contact? Such a mode of regarding the construction of the 
 atmosphere is certainly somewhat novel, so much so indeed that 
 there is some difficulty in believing that this is exactly what Mr. 
 Kelland meant to express. It is possible that, as in the pro- 
 blem of sound we consider the sonorous medium as made up of 
 immediately contiguous laminae, he considered this amounted 
 to the same thing as the ultimate particles themselves being 
 immediately contiguous ; thus confounding two things very 
 distinct, the ultimate particles which constitute matter, and 
 the imaginary elements into which we may suppose it divided. 
 
 But, waiving the objection to the phraseology adopted by 
 Mr. Kelland, and assuming that he meant bond fide to compare 
 his own to the atmospheric medium, I would observe that it 
 is by no means certain that a medium such as he has sup- 
 posed, consisting of particles placed at finite intervals from 
 each other and mutually repellent, would be endowed with 
 that elasticity which is the characteristic of the medium of 
 sound. It may be true, but it is anything but certain. We 
 know, or at any rate Mr. Kelland is not the person to dispute 
 the fact, that the atmosphere is not a system of homogeneous 
 particles, but that it consists of material particles and a fluid 
 ( termed caloric;' and recent experiments have made it evident 
 that it is to the presence of this last that the elasticity is due. 
 This fact may well beget a doubt as to whether any single 
 fluid consisting of homogeneous particles can be subject to 
 the same laws as those which govern the motions of the atmo- 
 sphere. It may be said that Mr. Kelland here takes for 
 granted no more than what has been conceded to every writer 
 on the undulatory theory; but I would observe that previous 
 writers have for the most part taken for their postulate nothing 
 
 F2 
 
68 ON MR. KELLAND'S 
 
 further than this, that the medium by which light is produced 
 is subject to the same laws as those which govern the sonorous 
 medium, while in the present instance Mr. Kelland wishes us to 
 consider as certain not merely that the real medium is like the 
 medium of sound, but that his medium is so ; thus implying, 
 beyond the ordinary hypothesis which is the basis of the undu- 
 latory theory, this additional fact, that his is the true medium, 
 which, it must be permitted to add, is so far from being certain 
 that all the evidence yet adduced militates against it. 
 
 I have said that Mr. Kelland considers the supplemental 
 principle to which he has here had recourse to be necessarily 
 implied by, or at any rate to be consistent with, his original 
 hypothesis; but so far is this from being the case, the latter 
 is contradicted by the former in a material point. This will 
 appear from the consideration, that he has not only supposed his 
 medium to consist of mutually repelling particles, but likewise 
 that the motions of the particles are small compared with their 
 actual distances ; whereas, in a medium consisting " of particles 
 in contact" the motions are exceedingly great compared with 
 their distances inter se. It is true that this is only of importance 
 as regards the particles in the immediate neighbourhood of that 
 whose motion is being considered; but, as the force varies 
 inversely as the square of the distance, these last are precisely 
 those whose effect is most important. Hence on this ground 
 also it is not ( certain' that Mr. Kelland's medium will ac 
 as a mass of particles in contact. 
 
 Lastly, with respect to the only undulations of the sonorous 
 medium with which we are acquainted, the vibration is in 
 the direction of transmission, while it is all along Mr. Kelland's 
 object to shew that the vibrations are transversal. Hence, i 
 the fact were true which Mr. Kelland states as certain, it would 
 be a fatal blow to the theory. 
 
 By a novel and ingenious combination of the principle oi 
 the last equation with that of (3), Mr. Kelland deduces a value 
 for the velocity in a transparent medium, 
 
 ( 6) J = V* + 2 (Q - M) 
 
ON MR. KELLAND'S "THEOIIY" OF HEAT. 69 
 
 where Z>, D' are the densities in vacuo and within the body 
 respectively, V the velocity in vacuo, Q the repulsion at dis- 
 tance unity of a mass of particles which would occupy the 
 space of a material particle : a conclusion which is so far 
 satisfactory that it gives different values of v corresponding to 
 different values of k, or the length of the wave ; it also exhibits 
 a uniform velocity in vacuo. This last triumph however has 
 a considerable drawback attendant upon it, for, according to 
 Mr. Kelland, in a VACUUM there can be no variation in the length 
 of a wave. From the equations 
 
 /3 = b . cos (nt - kx), 
 it is evident that - n z = k z ri* . D, 
 
 or 1 = k* . D. 
 
 But * = T^> ' A 2 = 47r 2 Z), 
 
 fC 
 
 a conclusion equally novel and satisfactory. 
 
 It does not appear that Mr. Kelland has troubled himself 
 to compare the above value for the velocity with that which 
 results from his original equation. It will be found however 
 that the two are altogether irreconcileable. This will appear 
 as follows. 
 
 Since the medium is one of symmetry, we have 
 
 v As 2 + Ay 2 + A* 2 - 
 
 and similarly of=Q. Hence the equations (4) and (5) become 
 
 + 2<7/sin 2 --} a = - 2SFsin 2 a', 
 2 ) 2 
 
 Mr. Kelland assumes that K = h, N = n, which he says 
 is " evident." Now this may be a very allowable hypothesis, 
 but it is certainly not a necessary one ; as in fact the phe- 
 nomena of double refraction would lead us to suppose it pos- 
 
70 ON MR. KKLLAND'S " THEORY" OF HEAT. 
 
 sible that the vibrations of the particles of ether and of 
 the material particles are propagated with different velocities. 
 Assuming the above, however, since by adding the two last 
 equations we obtain 
 
 (7) N z a + n z a = 0, 
 
 it is evident that a = - a', and equation (3) becomes 
 
 - 2P<rf sin 3 - 
 ,.%., 2 
 
 which gives t? = = 
 
 i fC 
 
 This will agree with (6), provided that 
 
 
 But the second member of this equation represents the square 
 of the velocity in vacuo = V z , 
 
 . sn 
 
 and = a constant, 
 
 KI 
 
 which is impossible, or at any rate untrue. 
 
 From the above we see pretty clearly the fallacy of Mr. 
 Kelland's explanation of " the variation of velocities in trans- 
 parent media." The fallacy of the entire theory may be 
 surmised from the equation (7): for from that equation it is 
 evident that a and a' have always different signs. But a and 
 a' represent the motions of any particles taken at random, the 
 one from the caloric, the other from the material substance 
 of the body : whence it appears that all the particles of the 
 caloric at any given time must be moving in the contrary 
 direction to some one material particle, in other words all the 
 particles of caloric at any given time must be moving in the 
 same direction ; a circumstance which precludes the possibility 
 of wave motion. 
 
71 
 
 But no longer to keep my readers in suspense as to the fate 
 of this theory, I shall hasten to the catastrophe. 
 " To find the value of n* and N\ 
 " The equations (2) and (3) are now reduced to 
 
 d*a . 2 
 
 - = 2Pacr - -- sin - , 
 r 2 
 
 Ta + U x a suppose, 
 
 " But by addition, 
 
 T + T + T = 0. U + U + U = 0. 
 
 * y z * x y e 
 
 " Also it is clear that T y = T z , U y =U s ; 
 
 .. r=-2T, *7=-27; 
 
 . /, 
 
 or if 
 
 " The solutions of these equations are 
 
 J?5 j3 = b cos (w - ^), 
 
 &c. 
 
 tf But we assumed the solution of the first equation to be 
 a = a cos (nt kx). 
 
 " It is necessary therefore to alter that assumption in such 
 a manner as still to satisfy the values of Sa, Aa, found in (157). 
 
 (8) " Let a = a ( NV3 -' - -"*') cos (kx + c) = a cos (kx + c), 
 
7 Oi\ MR. KELLAXDS " THEORY" OF HEAT. 
 
 then a will, by the same operation as in (157), give the same 
 results; and 
 
 Aa = A' cos (jkx + c + JTA.r) - a 
 
 da k&x 
 = a COS KLX + Sin a, 
 
 dx a 
 
 which is also the same as before. 
 
 " The above therefore is a solution of the equations subject 
 to all its assumptions." 
 
 Not quite. One of those assumptions was that Sa was small ; 
 but it is evident from (8) that a and Sa may attain any mag- 
 nitude as the time increases (except at certain points at regular 
 intervals in the medium, at which cos (kx + c) = 0) ; whence it 
 follows that the whole of the preceding investigation is entirely 
 fallacious. 
 
 Liverpool, Dec. 1, 1846. 
 

 ON PROFESSOR POWELL'S ACCOUNT OF 
 
 THE THEORY OF FINITE INTERVALS. 
 
 N the following paper I propose to examine some of the 
 results collected by Professor Powell in his " General and 
 Elementary View of the Undulatory Theory as applied to the 
 Dispersion of Light."* It is with some regret that I have 
 felt obliged to select, as the subject of a not very compli- 
 mentary criticism, one of the few attempts which Oxford has 
 of late years put forward to vindicate her claim to successful 
 rivalship with her sister university, even in the peculiar line 
 of the latter : but the circumstance of this being the only pub- 
 lication, so far as I am aware, which contains any tolerable 
 account of the different views which have been promulgated 
 on these topics, and the impossibility of referring the ordinary 
 class of readers to the original sources from which Professor 
 Powell has derived the materials of his summary, have com- 
 pelled me to this course. 
 
 The general nature of Professor Powell's views may be sur- 
 mised from the following paragraph which occurs in his Intro- 
 duction : 
 
 " The primary object in a mechanical theory is to connect 
 the simple mathematical representation of waves with acknow- 
 ledged principles derived from dynamics. Such principles 
 were in some degree established by Huyghens, and the more 
 precise dynamical theory of vibrations of an elastic medium, 
 as in the case of sound, investigated by Euler," &c. " was soon 
 seen to apply (at least generally) to the analogous case of undu- 
 lations of ether constituting homogeneous light. But with re- 
 ference to many peculiar properties, as of heterogeneous light, 
 and even in the instance of the propagation of transverse vibra- 
 
 * London, 1841. 
 
74 ON PROFESSOR POWELL'S 
 
 tions (which Young had shewn to be a necessary supposition), 
 further modifications and extensions of the theory were required. 
 Such extension in any degree of generality was first made by 
 Fresnel, principally in his Memoir on Double Refraction, {Mem. 
 Acad. Vol. vii.), who clearly pointing out the too limited nature 
 of the ideas of ether hitherto adopted, laid down that of mole- 
 cules separated by finite intervals, by which the whole system 
 of transverse vibrations, of polarization, and the wave surface, 
 were established, by an analysis which must ever remain a 
 monument of the profound and comprehensive genius of its 
 author." 
 
 My readers must by this time be tolerably well aware that 
 I entertain a very different opinion of the success of Fresnel's 
 attempt to " extend the theory" from Professor Powell. Into 
 this question, however, I shall not at present enter further 
 than by saying that, whatever may be the merits of Fresnel's 
 " analysis/' it is not reasoning. It may be divination, but it 
 most assuredly does not (to use Professor Powell's somewhat 
 singular phraseology) " connect the simple mathematical repre- 
 sentation of a wave with acknowledged principles derived from 
 dynamics ;" at least not by any rules of evidence which are 
 ordinarily recognized as such by the human understanding. It 
 is satisfactory, however, to find that Professor Powell is not 
 ashamed of his principles. There are those who, while they 
 give up every step of Fresnel's process, imagine that somehow 
 or other his theory is right as a whole. Professor Powell is 
 more candid if not more discriminating. He contributes his 
 mite of admiration to the process adopted as well as to the 
 inferences deduced. 
 
 But although entertaining this high opinion of Fresnel's merits, 
 Professor Powell is not altogether blind to his defects. " The 
 mechanical principles are expounded chiefly in general terms, 
 and the results are not deduced from a mathematical develop- 
 ment of equations of motion. Thus these researches, though 
 of the highest order, were soon found not to possess in all 
 points that comprehensiveness of principle which would satisfy 
 the increasing demands of discovery; and in some instances 
 the theory was confessedly left defective." To give an account 
 
THEORY OF FINITE INTERVALS. 
 
 75 
 
 
 of the supplementary hypotheses by which these defects have 
 been sought to be obviated, is a principal object of Professor 
 Powell's work. 
 
 " Let the coordinates in space (vide Powell on the Undu- 
 latory Theory, p. 1 6) of any molecules m, m, m", Sec. be respec- 
 tively 
 
 m ...... x, y, z, 
 
 m ...... x + Ax, y + Ay, z + Az, 
 
 m" ...... x + A#', y + A?/, z + Az', 
 
 &c. 
 
 " When the system is disturbed after a time t, let the dis- 
 placements be respectively, of 
 
 Af, 
 
 A*?, f + A?, 
 
 r the distance between m, m at rest,/?- the law of force. Then 
 
 whence we obtain 
 
 -~ 
 
 = Sm 
 
 r) 
 
 -f (r) A?/} 
 
 (r) 
 
 (a), 
 
 [where 
 
 (r) = r 4 ^ (r) . rAr]. 
 With a view to " the integration of these equations" (p. 21) 
 
76 ON PROFESSOR POWELI/S 
 
 Professor Powell adopts " a method which, though apparently 
 prolix and capable of simplification, it is yet desirable to follow 
 as it includes other important consequences." 
 
 He assumes, in the first place, that the ray is in the direc- 
 tion of the axis, which he conceives may be done " without loss 
 of generality." If the vibration is transversal this implies that 
 
 so that, observing that 
 
 rAr = A#Af + Ay AT; + 
 the equations (a) reduce to 
 
 = 2m {0 (r) Ay + + (r) Ay ( Ay AT? + A*A?)} 
 
 y =2m{0(r)A?+iKr) 
 
 It is next assumed that 
 
 T? = 2 {a sin (nt - kx)} 1 , ' 
 
 which gives us 
 
 f . k&X / , 7 ^ 7 A / a 7 
 
 AT? = 2a { - 2 sin - - sin (w - fee) - sin k&x cos (w - kx) } , 
 
 I 2 'J 
 
 /3 cos 6 < - 2 sin 2 - sin (nt - kx) - sin k&x cos (nt - kx) , 
 
 + )3 sin b < + sin k&x sin (nt-kx)- 2 sin 2 - cos(w-&) I. 
 I 2 0' 
 
 Al ^ 
 
 Also - = - m 2a sin (w^ - fee), 
 ar 
 
 -3-2 
 
 2 {|3 cos & sin (w - kx) - j3 sin 5 cos (nt- kx)}. 
 
 Substituting these values for AT?, A?, ~ , \ in equations 
 
 d? dfr 
 
 ()3) respectively, and equating to zero the coefficients of sin (nt-kx) 
 
 and cos (nt - kx) in the results, we obtain the four following 
 
 equations 
 
 (1) - W 2 2a = sin S (j3? sin 20) 
 
 - cos S (j3y 2 sin 2 0) - 2 (op 2 sin 2 0), 
 

 THEORY OF FINITE INTERVALS. 77 
 
 (2) = - 2 (op sin 20) 
 
 - cos 52 (J3q sin 20) - sin 52 (j3? 2 sin 2 0), 
 
 (3) - w 2 cos 52f3 = sin 52 (f3p' sin 20) 
 
 - cos 52 (fy>' 2 sin 2 0) - 2 (aq 2 sin 2 0), 
 
 (4) - n z sin 52/3 = - 2 (ay sin 20) 
 
 - cos 52 (/Sjp' sin 20) - sin 52 (ftp' 2 sin 2 0, 
 [where p = m {0 (r) 4 i/; (r) A?/ 2 }, 
 
 />' = m (0 (r) + i (r) As; 2 }, 
 
 ? 
 
 20 = ^ 
 
 The two last may be replaced by the following, 
 
 (5) - w 2 2/3 = - 2 (j3/>' 2 sin 2 0) 
 
 - cos 52 (aq 2 sin 2 0) - sin 52 (aq sin 20), 
 
 (6) = 2Q3p'sin 20)-sin52(a^2sin 2 0) + cos 52(asin 20). 
 
 Professor Powell next proceeds to draw certain deductions 
 from these equations. He says, 
 
 " In the case of elliptic polarization a and ft are equal for all 
 the ellipses, so that we have 
 
 2a = ha, 2/8 = hft, 
 where A is a constant, and we can obtain from (1) and (5), 
 
 P' sin 2 0) + 2 2 (p sin 20) 
 
 + 2aft cos 5 .2^ 2 sin 20} ; 
 and from (4) we get 
 
 . , _ a2 (q sin 20) 4 ft cos 52 (>' sin 20) 
 n*hft - ftZ (p 2 sin 2 0)~ 
 
 From this Professor Powell draws the conclusion that, " upon 
 the whole the formulae (7) for elliptically polarized light, in- 
 volving the above value of n, is the solution of the differential 
 equations (ft) for the motion of a system of molecules constituted 
 as at -first supposed." 
 
 It is somewhat remarkable that it should not have occurred 
 to Professor Powell that the above values for n z and sin 5 form 
 but a meagre result horn four equations. To make my remarks 
 
78 ON PROFESSOR POWELL'S 
 
 upon this subject more clear, I must direct attention to an 
 oversight which occurs in the early part of the above process, 
 which throws the whole into almost inextricable confusion. By 
 assuming 
 
 77 = So sin (nt - kx), 
 
 it is to be presumed that Professor Powell meant the symbol 2 
 to stand for the sum of such a series as the following : 
 
 (8) a sin (nt - kx) + a sin (n't - k'x) + a" (sin n't - k''x) + &c., 
 
 (and similarly with regard to ) ; but he elsewhere uses the 
 same symbol to indicate a totally different thing, namely, the 
 sum of all the different values assumed by such functions as 
 the following, 
 
 m {0 (r) + ^ (r) Ay 2 } 2 sin 2 U -~ &c. 
 
 when different values are assigned to A#, Ay, A#. 
 If we rectify this error by putting 
 
 7? = cr {a sin (nt - kx)}, 
 
 f = o- { sin (nt - kx + b)} , 
 
 it is obvious that a material change will take place in the 
 nature of the equations (l), (2), (5), (6); for we shall now have 
 from (/3) 
 
 ^? = {sin b . 2 (q sin 20) - cos &S (0 2 sin 2 0)} x cr { sin (nt - kx)}, 
 
 - {cos b . 2 (7 sin 29) - sin S (y 2 sin 2 0)} x o- [ft cos ^ - ^)}, 
 
 - 2 Qt> 2 sin 2 0) x o- {a sin (?^ - &&)} - 2 (/? sin 20) 
 
 x o- { a . cos (w^ - ^)1 ; 
 
 d z 
 
 and similarly for | , 77 and f But, as Professor Powell 
 ut 
 
 assumes that 
 
 crfi sin (nt - kx) = sin (nt - kx) <rj3, 
 
 era sin (nt - kx) = sin (nt - kx) era, 
 
 &c. 
 
 it is obvious that we may dispense with the symbol a altogether, 
 and may take 
 
 7? = a sin (nt - kx), f = y8 sin (nt - kx + &). 
 

 THEORY OF FINITE INTERVALS. 79 
 
 Hence, if we put 
 
 S (p 2 sin 2 9) = P 13 2 O' 2 sin 2 0) = P',, 2 (? 2 sin 2 0) = Q I} 
 2 (> sin 20) = P 2 , 2 (/, sin 20) = P' 2 2 (? sin 20) = Q 2 , 
 we may replace equations 1.2.5.6 by the following : 
 
 (7) & (Q 2 sin b -Q l cos b) - a (P l - n z ) = 0, 
 
 (8) j3 (Qj sin b + Q 2 cos V) + P 2 =0, 
 
 (9) a (Q 2 sin 5 + Q 1 cos 6) -f /3 (P' t - n z ) = 0, 
 (1 0) a(Q l sin 6 - Q 2 cos ) - /3 P' 2 = 0. 
 
 Eliminating -^ . and w 2 , from these we shall have a relation 
 
 between P^, P'-fv Q^, i>e> between k and the coordinates 
 of the point of rest of the particles. But k depends on the 
 length of a wave. Hence, upon this principle the length of 
 a wave may be expressed in terms of the coordinates of the 
 medium. Now from the equation thus obtained in PQ, an 
 infinite number of real values of k may be found, but there may 
 be not one. The onus of proof rests with Professor Powell to 
 shew whether any such values can exist. Till this is done 
 it admits of doubt whether " the formula for elliptic polarization 
 involving the above," or any other " value of n, is the solution 
 of the differential equations (/3) for the motion of a system of 
 particles constituted as at first supposed." 
 
 I next observe that, eliminating ~ from (8) and (10), we get 
 
 (Q, 2 sin 2 b - Q 2 3 cos 2 b} + P 2 P 2 = o, 
 whence sin 2 b = ~ .............. ( 1 1), 
 
 Now sin b and cos b are necessarily less than 1. It follows 
 therefore that a relation must exist between Q t Q z) P 2 P 2 > ^ n 
 order to satisfy this condition. It rests with Professor Powell 
 to point out from what hypothesis as to the constitution of the 
 medium such a relation would result. 
 
80 ON PROFESSOR POWELL*S 
 
 Lastly, eliminating from (7) and (8), we get 
 
 (Q/ 2 sin 2 I - Q* cos 2 b) + (P, - n 2 } (P\ - n z ) = 0, 
 and eliminating b from this by (11) and (12), we have 
 
 Now admitting Pf^ Pfv Q^ Q. to be all possible, which is 
 not certain, as I have shewn, it remains to be proved that n~ 
 is so. Again, admitting n~ to be possible, it will in general have 
 two values, or we shall have two waves propagated with dif- 
 ferent velocities. It is for Professor Powell to shew how this 
 difficulty can be obviated. 
 
 I am aware that these objections may partly be met by sup- 
 posing a symmetrical disposition of the medium ; but Professor 
 Powell repudiates the necessity of any such supposition in the 
 case of elliptic polarization. He says (Art. 105) "the formulas 
 for unpolarized light, as well as the formula for plane polarized 
 light, are only solutions, provided" certain " conditions are ful- 
 filled in the original equations ....... While on the other hand, 
 
 in the case of elliptically polarized light, it is important to bear 
 in mind that the solution has been obtained solely from the 
 conditions of elliptic polarization." 
 
 I shall presently adduce other objections to the case of ellip- 
 tically polarized in common with that of ordinary light ; but in 
 the mean time enough has been said to shew that it is not true, 
 that in the case of elliptically polarized light a solution has 
 been obtained solely from the conditions of elliptic polarization. 
 
 It is obvious from what has gone before that Professor 
 Powell's investigation, although he seems to have persuaded 
 himself to the contrary, relates simply to the case of elliptically 
 polarized light. As his reasoning, however, may be so modi- 
 fied as to apply to the general case, I shall here proceed 
 to give some account of it. Putting b = 0, in (1), (2), (3), (4), 
 he obtains four equations, respecting which he observes, that 
 in all of them "it is evident that, since a and ft are by the 
 original condition wholly arbitrary and independent, both of each 
 other and of the other quantities, these equations can only hold 
 
THEORY OF FINITE INTERVALS. 81 
 
 good for all values whatever of a and /3, if each of the terms 
 involving respectively a and j3 are separately = a." Now the 
 fact is, there is no reason whatever to suppose that a and |3 are 
 " wholly independent" of each other, and of the other quan- 
 tities, or. that the equations must hold " for all values whatever 
 of a and /3." If the equations did not hold for some particular 
 values of a and j3, it would be no opprobrium (rather the con- 
 trary) to the theory; for it would only shew that the medium 
 refuses to transmit rays of a certain character, which we have 
 every reason to believe to be the fact. Passing over the defect 
 of logic however, from the last-named equations Professor 
 Powell deduces the following : 
 
 = S (op sin 20), = S ($q sin 20), 
 
 = 2 (ftp' sin 28), = S (aq sin 20), 
 
 = S ($q 2 sin 2 0), = S (aq 2 sin 2 0), 
 
 (13) = rc 2 Sa - 2 (op 2 sin 2 0), = 2 S)3 - S (fy>' 2 sin 2 0) (14), 
 
 which, although they have in strictness been suggested by the 
 particular case of elliptically polarized light likewise occur in 
 the general case. But in this latter they contain only an im- 
 perfect representation of the truth. For taking the series () 
 to represent TJ, it is evident that besides the four equations 
 (1), (2), (5), (6), corresponding to the first term a sin (nt - kx) 
 of the series, we shall have four exactly similar equations cor- 
 responding to every other term as a' sin (n't - k'x) &c.; in this 
 latter instance a'p'n'k'b' recurring in the place of aflnkb ; thus 
 
 giving us the same equations for the determination of , ri, K, |3' 
 
 a 
 
 as for -r , n, k, b. If we choose to satisfy these independently 
 
 of , we must have 
 
 P 2 = P' 2 = Q t = Q 2 = o, P l - n z = 0, P\ - n z = 0. 
 
 Professor Powell afterwards enters (Art. 108, et seq.) into 
 some considerations to shew how the first four of these equa- 
 tions may be satisfied by "an hypothesis respecting the ethereal 
 molecules in space, viz. that they are distributed uniformly." 
 
 G 
 
8% ON PROFESSOR POWELL'S 
 
 Now I admit that on this hypothesis we satisfy the equations 
 P 2 = 0, F z = 0, Q 8 = 0. But the case of Q l = 0, or 
 
 7. A y, 
 
 2wi/> (r) Ay Az 2 sin 2 - 
 2 
 
 presents more difficulty. Professor Powell says, whenever the 
 three first " are evanescent it is easy to shew that 
 
 7, A / 
 
 2 
 
 r) AyAz 2 sin 2 - - = 0, 
 2 
 
 by a simple transformation of coordinates :" and then assuming 
 
 y = y cos t - z' sin t, 
 
 z = z ' cos t + y sin t, 
 
 he shews that, since (putting P = i// (r) 2 sin 2 0) 
 S(PAyAz) = {2(PAy' 2 )-S(PAz') 2 }sm 20- 2S(PAz'Ay>os20, 
 
 we may, by properly assuming tan 20, cause 2 (PAyAz) to 
 vanish. 
 
 I must here, however, recall to Professor Powell's recollection 
 a circumstance which seems to have escaped him. When we 
 
 d z 
 
 assume that , A, and | all = 0, the first of the equations (a) 
 at 
 
 requires certain conditions to be satisfied. It is evident, in fact, 
 that in this case we must have 
 
 (15) = 2m {</, (r) &x (AyAiy + AzA?)}, 
 
 which, substituting for A?7 and Af their values, implies the four 
 following equations, 
 
 = Sm ^ (r) AzAy . 2 sin 2 ^ , 
 = 2wi/ (r A#A . 2 sin 
 
 = 2m;// (r) A#Az 2 sin 2 - , 
 
 = Smi// (r) A#Az sin 
 
 The second and fourth of these may be satisfied by the hypo- 
 thesis of uniform distribution, but the first and third cannot, 
 and the only way of getting over this difficulty is by trans- 
 forming the coordinates. Hence it is evident that, instead of 
 
THEORY OF FINITE INTERVALS. 83 
 
 a simple transformation in the plane of yz, an entire trans- 
 formation of the system must be effected, in order to satisfy the 
 three conditions, 
 
 = S (PAyAz), = S (PAzAz), = S (PAyAs). 
 
 I believe it will be found that there is only one position of the 
 axes which will satisfy this condition, as M. Cauchy, who 
 appears in this instance to have been rather clearer sighted 
 than Professor Powell, admits (md. Powell, Art. 141): and 
 as it has been assumed that the ray coincides in direction with 
 the axis of x, we are thus led to the somewhat astounding 
 conclusion that there are only three directions, at right angles 
 to one another, in which a wave of given length can be pro- 
 pagated. Truly this is having ( axes of elasticity' with a ven- 
 geance. 
 
 But this is not all. It will be seen that the equations (13), 
 (14), give us two values of n*, namely P a and P\. Professor 
 Powell considers these as identical (md. Art. 97), and this 
 apparently without any reference to the nature of the medium. 
 For my own part I confess myself unable to conceive how this 
 could occur unless we suppose the particles to be arranged 
 in square order, and that the ray is propagated along one of 
 the sides of the square. But even this wretched fragment of 
 a result (if any one should think it deserving the least attention) 
 will be found to rest on no solid basis. Recurring to the equa- 
 tions (a) it is obvious that the assumptions 
 
 = 0, A = 0, ^=0, 
 
 involve some further conditions beyond those involved in the 
 equation (15). We have 
 
 d\ ffr d fr Ar d z fr Ar 2 1 , 
 
 = Sm H- + -T- - + -7-2 - - + &c. V (Az + A). 
 
 df \r dr r 1 dr* r 1.2 j^ 
 
 Now, in order that - may = 0, besides the equations 
 
 ~.. A.*Ar +- . A 
 {dr r r j 
 
 G 2 
 
84 ON PROFESSOR POWELI/S 
 
 [the second of which is identical with (15)], we must have 
 
 fd 2 fr Ar 8 d (fr Ar 
 
 = 2m {-T. j.^-. . Az + K . -- 
 \dr* r 1.2 dfe \r i 
 
 and similarly of every other term in the equivalent of -yv . It is 
 
 ctt 
 
 obvious that these conditions cannot be satisfied by the hypo- 
 thesis of uniform distribution, whence it follows that there will 
 be an accelerating force on the particle parallel to the axis 
 of x. Now, though this may be small, its effects in a given 
 time may be very important; as, by means of it, instead of 
 being 0, or even very small, may come to have any assignable 
 magnitude, It is incumbent therefore on the supporters of 
 this theory to shew that the variations of will be confined 
 between small limits. I apprehend that this object can hardly 
 be effected without breaking up the entire theory. 
 
 Having thus considered the theory which Professor Powell 
 has treated of most in detail, I shall say a few words with regard 
 to the " other solutions of the Differential Equations," of which 
 he gives an account in his 4th Section. The first of these is 
 Mr. Kelland's,* who " deduces, as above, the fundamental 
 equations. Then developing A, A^, A, each as functions of 
 the three variables x, y, z y we have 
 
 , 
 
 dx dy dz 
 
 d^ A^ d^ Ay 2 d*t A^ 
 + dx* 2 + dy* ~2~ ^dz* ~2~ 
 
 &c., 
 
 j 
 
 dxdy dxdz dydz 
 
 and similar expressions for A-7 and A." Then " adopting the 
 hypothesis of symmetrical distribution, and thus rejecting the 
 sums of products of odd dimensions in A#, Ay, &z ; rejecting 
 also products of the same terms of four dimensions as insensibly 
 small," he reduces the equations to an integrable form. 
 
 But it is obvious that this process is entirely fallacious, as 
 
 * Mr. Kelland appears to have subsequently abandoned this theory, and to 
 have adopted that which I have discussed in the preceding paper. 
 
THEORY OF FINITE INTERVALS. 85 
 
 depending upon a false approximation. Instead of A#, Ay, Az 
 being so small that products involving the fourth powers of those 
 quantities may be neglected as insensible, these quantities may 
 have any magnitude whatever. This is so obvious that I cannot 
 but feel some surprise that such a method should ever have 
 been proposed. 
 
 Mr Kelland, however, is not singular in this error. " Mr. 
 Tovey (in his first papers in the Journal of Science, Vol. VITI. 
 pp. 7, 270, 500)," as Professor Powell informs us, " after estab- 
 lishing the fundamental equations, proceeds to develop I, ??, f as 
 each a function of A#. He thus finds 
 
 dt ; d^ Az 2 
 
 A = Ax + -f &c., 
 
 dx dx* 1.2 
 
 and similar expressions for A?; and A?." Now we can make no 
 use of this series for Af , for, in the first place, as A# may have 
 any magnitude, we are not sure that even the whole series 
 would be an equivalent for A, and a fortiori we cannot take 
 a limited number of its terms to represent that quantity. 
 
 " Sir John Lubbock (Journal of Science, Vol. xi. Nov. 1837) 
 developing A/> as a function of r, 
 
 dp d z p Ar 2 
 
 Ao = -f- Ar + -= . + &c.," 
 dr dr* 1.2 
 
 obtains the following equations 
 
 d*t v d*p v d"p 
 
 = a? cos X + a 2 cos X f . 
 dt* dr* dr* 
 
 &c. &c. 
 
 But this is evidently untenable as depending on the same 
 false approximation as before, since Ar is not small. Professor 
 Powell makes an elaborate comparison between Sir John Lub- 
 bock's results and those of Fresnel in connexion with the axes 
 of elasticity, and seems to think that the former correspond with 
 the latter. The fact therefore of Sir John Lubbock's process 
 being erroneous is in complete accordance with my views as 
 to that of Fresnel. 
 
 It would be beside my purpose to enter into the consider- 
 ation of Professor Powell's formulae for the 'diffusion' ; for, as 
 
86 ON PROFESSOR POWELI/S THEORY OF FINITE INTERVALS. 
 
 those formulas have not upon theoretical grounds the smallest 
 claim on our attention, the only result of such a step would 
 be to expose, possibly, further error and inconsistency. If Pro- 
 fessor Powell thinks his formulae are to be defended on experi- 
 mental grounds, there are those who are both able and willing to 
 combat them upon those grounds. In this respect, however, 
 I would warn my readers against the too easy reception of 
 boldly asserted claims, and I would recommend them to judge 
 for themselves of the probability of formulae thus arrived at, 
 giving any real exact representation of the phenomena of nature. 
 The formulae of Fresnel and his followers may truly represent 
 the phenomena of light, and so may some of those given in 
 Dr. Peacock's Examples on the Differential and Integral Cal- 
 culus ; but as there appears to be no other ground for preferring 
 the former to the latter beyond that of ascertained accordance 
 with experiment, it appears to be necessary to sift with the 
 utmost keenness the evidence which may be adduced with 
 regard to this point. 
 
 Liverpool, Jan. 6, 1847. 
 
87 ) 
 
 ON MR. GREEN'S THEORY OF 
 
 REFLECTION AND REFRACTION. 
 
 THE late Mr. George Green, of Caius College, in a paper " On 
 the Laws of Reflection and Refraction of Light at the common 
 surface of two non-crystallized Media" (read before the Cam- 
 bridge Philosophical Society, Dec. 11, 1837), appears, as far as I 
 am able to judge from a very cursory inspection, to have arrived 
 at some conclusions respecting the intensity of reflected and 
 refracted light, which agree more or less with those of Fresnel. 
 How operations conducted apparently upon different principles 
 should, though equally erroneous, lead, to a certain extent, to 
 similar results, it is not my purpose to explain, though I 
 apprehend that in this instance it would not be difficult to do 
 so. It is, for the present, sufficient to shew, as I am prepared to 
 do, that Mr. Green's results, whatever they are, are based upon 
 a false analysis. 
 
 The explanation of Mr. Green's theory, as far as is necessary, 
 I shall give in his own words. 
 
 "Let us conceive a mass composed of molecules acting on 
 each other by any kind of molecular forces, but which are 
 sensible only at insensible distances, and let moreover the whole 
 system be quite free from all extraneous action of every kind. 
 Then, x y t z being the coordinates of any particle of the medium 
 under consideration when in equilibrium, and 
 
 z + u, y + v, z 4 w, 
 
 the coordinates of the same particle in a state of motion [where 
 , v, w are very small functions of the original coordinates 
 (x, y, z) of any particle and of the time ()], we get, by com- 
 bining D'Alembert's principle with that of virtual velocities, 
 
 fd z u ~ d*v ~ d z w ~ \ _. 3 x.v 
 
 SAw ( -j- $u + 3 St> + -gj- $w ] = VDv . fy (1), 
 
88 ON 
 
 Dm and Dv being exceedingly small corresponding elements of 
 the mass and volume of the medium, but which nevertheless 
 contain a great number of molecules, and the exact differen- 
 tial of some function and entirely due to the internal actions 
 
 of the particles of the medium on each other 
 
 " Let us now take any element of the medium, rectangular in 
 a state of repose, and of which the sides are dx, dy, dz, the 
 length of the sides composed of the same particles will in a 
 state of motion become 
 
 dx' = dx (1 + sj, dy = dy (1 + s 2 ), dz dz (1 + s 3 ) 
 where s lf s z , s 3 are exceedingly small quantities of the first 
 order. If, moreover, we make 
 
 dy 1 dx dx 
 
 a = cos < /, , p = cos < , , 7 = cos < , , 
 dz dz dy 
 
 a, ft 7 will be small quantities of the same order. But 
 whatever may be the nature of the internal actions, if we 
 represent by 
 
 Sfydxdydz 
 
 the part of the second member of the equation (1), due to the 
 molecules in the element under consideration, it is evident that 
 $ will remain the same when all the sides and all the angles 
 of the parallelepiped, whose sides are dx , dy', dz' , remain 
 unaltered, and therefore its most general value must be of the 
 form 
 
 = function (s l S 2 s 3 , a ft 7). 
 
 But s l9 S 2 , S 3 , a, ft 7 being very small quantities of the first 
 order, we may expand in a very convergent series of the form 
 
 = 00 + 01 + 02 + 03 + &C '> 
 
 , 1? 2 , &c. being homogeneous functions of six quantities 
 a, |3, j) s l9 s z , s 3 of the degrees 0, 1, 2, &c., each of which is 
 very great compared with the next following one. If now, 
 p represent the primitive density of the element dx dy dz, we 
 may write p dx dy dz in the place of Dm in the formula (1), 
 which will thus become, since $ is constant, 
 
 rrr j j j fd*U ~ d*V ~ d*W 
 
 Jffpdxdydz j $u + 2 6e + ^ 
 = /// dxdydz (80, + 80, + &c.) 
 
THEORY OF REFLECTION AND REFRACTION. 89 
 
 the triple integrals extending over the whole volume of the 
 medium under consideration. 
 
 te But by the supposition, when u - 0, v = 0, w = 0, the 
 system is in equilibrium, and hence 
 
 = /// dx dy dz 80, 
 
 seeing that 0, is a homogeneous function of * lf s t , s a , a, j3, y of 
 the first degree only. If therefore we neglect 3 , 4 &c.,~which 
 are exceedingly small compared with 2 , our equation becomes 
 
 rrr j j i ~ ~ 
 
 fffpdxdydz fc+ _+ _ 
 
 = /// dx dy dz S0 2 , 
 
 the integrals extending over the whole volume under con- 
 sideration ........ 
 
 " If now we can obtain the value of 2 , we shall only have to 
 apply the general methods given in the Mecanique Analytique. 
 But 2 being a homogeneous function of six quantities of the 
 second degree will in its most general form contain twenty-one 
 arbitrary coefficients. The proper value to be assigned to each 
 will of course depend on the internal constitution of the 
 medium. If however the medium be a non-crystallized one, 
 the form of 2 will remain the same, whatever be the directions 
 of the coordinate axes in space. Applying this last considera- 
 tion, we shall find that the most general form of 2 for non- 
 crystallized bodies contains only two arbitrary coefficients. In 
 fact, by neglecting quantities of the higher orders, it is easy to 
 perceive that 
 
 du dv dw 
 
 s i = ~r> 5 2 = :r j s * = T~ ; 
 dx dy dz 
 
 dw dv n d^v du du dv 
 
 = +-, p=-r~ + :r> v ~ ~j~ + ~r > 
 dy dz dx dz dy dz 
 
 and if the medium is symmetrical with regard to the plane (xy) 
 only, 2 will remain unchanged when - z and - w are written 
 for z and w. But this alteration evidently changes a and /3 
 to - a and - j3. Similar observations apply to the planes 
 (xz\ (yz). 
 
 " If therefore the medium is merely symmetrical with respect 
 
90 
 
 to each of the coordinate planes, we see that <p 2 must remain 
 unaltered when 
 
 -2, - W,- a,- j3,-j CZ, W, a, (3, 
 
 or - y, - v, - a, - 7, 1 are written for \ y, v, a, 7, 
 
 or - x, - w, - /3, - -yj U, w, /3, 7. 
 
 " In this way the twenty-one coefficients are reduced to nine, 
 and the resulting function is of the form 
 
 & Y + H( d + i d * 
 
 .. j . 
 
 dy fe dx dz dx dy 
 
 " Probably the function just obtained may belong to those 
 crystals which have three axes of elasticity at right angles to 
 each other." 
 
 Mr. Green then proceeds still further to simplify the equa- 
 tions, preparatory to the discussion of the case which he has 
 made the object of his analysis, viz. of reflection and refraction 
 in non-crystallized media. But unfortunately the simplification 
 which has already been made is altogether untenable. Mr. 
 Green talks of the medium being symmetrical. How is it pos- 
 sible that a medium can be symmetrical in a state of motion ? 
 We are at perfect liberty to make use of the consideration of 
 symmetry as regards the coordinates (#, y } z) of the points of 
 rest of the particles, but most assuredly the same principle 
 cannot be extended to the variable coordinates (u, v, w) of 
 the particles in a state of motion, and yet it is nothing short 
 of this that Mr. Green's theory requires. The nearest approach 
 which I can imagine to a state of symmetry when there is 
 motion, is in the case of a series of indefinitely extended plane 
 waves advancing perpendicularly to their points. But this is, 
 properly speaking, regularity not symmetry, most certainly not 
 that kind of symmetry which Mr. Green contends for, and 
 upon which his simplification of the equation is based. More- 
 over, the motion we are now considering is not an actual motion 
 which we are at liberty to assume of such simple character as 
 may suit our convenience, but an arbitrary geometrical one; 
 
THEORY OF REFLECTION AND REFRACTION. 91 
 
 a motion in fact which is restricted solely by the conditions 
 of being small and of preserving the continuity of the medium. 
 There is not the smallest reason for imagining the slightest 
 relation between the motions in opposite quadrants, beyond 
 what the ' equation of continuity' may afford. If any one shall 
 be found of sufficient force of thought to wrest from that equa- 
 tion the fact of such a symmetrical arrangement of the medium 
 being necessary, as Mr. Green's theory implies, then I shall be 
 ready to admit that gentleman's simplification of his equation ; 
 but till then I must contend that that simplification is altogether 
 inadmissible, and that every result founded upon it must be 
 entirely rejected. 
 
 February 13, 1847. 
 
ON PROFESSOR MACCULLAGH'S THEORY. 
 
 IN a paper read to the Royal Irish Academy, Dec. 9, 1839, 
 and published in the Transactions of that body, Professor Mac- 
 Cullagh enters upon a somewhat different branch of the wave 
 theory from that, the discussion of which by Mr. Green we 
 have just been considering. The latter gentleman, as may be 
 recollected, confines his attention to the Reflection and Refrac- 
 tion of Light in non-crystallized media. The object of Professor 
 MacCullagh, on the other hand, is to make an fe Essay towards 
 a Dynamical Theory of Crystalline Reflection and Refraction." 
 But the results of Professor MacCullagh, equally with those 
 of Mr. Green, are entirely erroneous, the error in the two cases 
 being so far of the same character as that each consists in a false 
 simplification of the same equation. It is not my intention to 
 occupy the time of my readers with vain regrets at the waste 
 of ingenuity displayed in this theory a waste arising, as far 
 as I am able to judge after a very cursory examination, from 
 a single inadvertence, which in its turn owes its origin to the 
 besetting sin of Fresnel and his followers the attempting to 
 establish a foregone conclusion by the pure force of analysis. 
 Unlike every other writer on the undulatory theory whom 
 I have met with, Professor MacCullagh appears to be fully 
 aware of the "altogether unsatisfactory" nature of Fresnel's 
 theory ; but he appears to have been dazzled by the brilliant 
 results which that theory professed to elicit, and seems to have 
 come to the conclusion that, whatever its defects, somehow or 
 other, Fresnel had hit upon the ,right track ; and that all that 
 was requisite was to establish a firm communication between 
 the ordinary regions of philosophy and the species of flying 
 island which the ingenious author of the theory of double 
 refraction appeared to have brought into existence. But it is 
 
ON PROFESSOR MACtTULLAGH's THEORY. 93 
 
 time that my readers should have an opportunity of deciding for 
 themselves upon the justice of these strictures, and therefore 
 without further preamble I shall proceed to give an account 
 of Professor MacCullagh's theory. 
 
 " 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 the 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 
 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 a differential 
 element of volume ; and, for the present, we shall consider the 
 mutual actions of these particles to be sensible only at distances 
 which are insensible when compared with the length of a wave. 
 " Let x y z be the rectangular coordinates of a particle 
 before it is disturbed, and x + f , y + vj, z + its coordinates at 
 the time t, the displacements f , 77, f being functions of x, y, z 
 and t. Let the ethereal density, which is the same in all media, 
 be regarded as unity, so that dx dy dz may, at any instant, 
 represent indifferently either the element of volume or of mass. 
 Then the equation of motion will be of the form 
 
 where V 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. . 
 
94 ON PROFESSOR MACCULLAGH*S THEORY. 
 
 " We come now to investigate the particular form which 
 must be assigned to the function V, in order that the formula 
 (1) may represent the motions of the ethereal medium. For 
 this purpose conceive the plane of x 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 dx' dy' dz' , having its edges parallel 
 to the axes of x, y' , z, 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 
 parallelepiped which are perpendicular to the edge dz' will be 
 shifted, each in its own plane, in a direction parallel to the axis 
 of y, but their displacements will be unequal and will differ by 
 dy, so that the edges connecting their corresponding angles will 
 no longer be parallel to the axis of z ', but will be inclined to it 
 
 at an angle K whose tangent is . 
 
 dz 
 
 " Now the function V can only depend on the directions of 
 the axes of x' y' z with respect to fixed lines in the crystal, ad 
 upon the angle K which measures the change of form produced 
 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 z', determines the other two, we may regard V as 
 depending on the angle K and on the direction of the axis of x 
 alone. But" it may be shewn " that the angle K and the angles 
 which the axis of x makes with the fixed axes of x, y, z, are all 
 known when the quantities X, Y, Z are known. Consequently 
 Vis a function of X YZ" 
 
 On this basis it is not difficult to reduce S V to the form 
 
 - SJ^= a 2 XdX + VYdY + JZdZ, 
 but unfortunately this basis is entirely erroneous. 
 
 It is a mistake to suppose" that "the function V can only 
 depend on the directions of the axes of x' y' z with respect to 
 fixed lines in the crystal and upon the angle K which measures 
 the change of form produced in the parallelepiped by vibra- 
 tion." It is not " the change of form produced in the paral- 
 lelopiped by" any actual "vibration," but that which arises 
 from the arbitrary geometrical motion which the "methods 
 
95 
 
 given in the Mecanique Analytique contemplate. It is not true 
 that the hypothesis as to V depending on the directions of the 
 axes of x' y' z with respect to fixed lines in the crystal and 
 upon the angle u" is the most general supposition which can 
 be made. The following paragraphs may serve to prove to 
 those who have any doubt on the point what were the views 
 of Lagrange with regard to it. 
 
 "Nous nommerons dans la suite* (vide Mecan. Anal. p. 196) 
 ces differentielles marquees par S, des variations pour les dis- 
 tinguer des autres marquees par d qui se trouvent dans la 
 meme formule, et qui expriment les accroissemens ou decrois- 
 semens successifs des variables, a raison du terns et du mouve- 
 ment des corps ; tandis que les variations sont relatives au 
 changement instantanee des corps, et qui est tout a fait inde- 
 pendant de leur mouvement effectif. 
 
 " Ensuite en ayant egard aux equations de condition, donnees 
 par la nature du systeme propose, entre les coordonnees des dif- 
 ferens corps, on reduira les variations de ces coordonnees au 
 plus petit nombre possible, ensorte que les variations restantes 
 soient tout-a-fait independantes entr' elles et absolument arbi- 
 traires. Alors on egalera zero la somme de tous les termes 
 affectes de chacune de ces dernieres variations ; et 1'on aura 
 toutes les equations necessaires pour la determination du mouve- 
 ment du systeme." 
 
 Now what are the equations of condition in the present case ? 
 Clearly they reduce themselves to one, the equation of incom- 
 pressibility. Professor MacCullagh has introduced the hypo- 
 thesis that the medium is incompressible, and this (however 
 improbable in itself) for the purpose of discussion is perfectly 
 allowable. But it is not allowable to introduce the further 
 assumption, which Professor MacCullagh in effect does intro- 
 duce, that the motion is constrained to take place in the direc- 
 tion of certain fixed lines. It is one thing to suppose that 
 a series of plane waves may be propagated in which the vibra- 
 tion is rectilinear and transversal, and another to assume that 
 the medium is incapable of any other kind of motion. 
 
 It is not, however, altogether without reflection that Professor 
 MacCullagh has fallen into this error. The point seems to 
 
 * Paris, 1788. 
 
96 
 
 have been brought before him, although he has arrived at a 
 wrong conclusion respecting it. He says (vide p. 4) : 
 
 " 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 described 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 indefinitely near each other, can have 
 nothing but its thickness altered by the displacements : and 
 since the assumed preservation of density requires that the 
 volume of the slice should not be altered, nor consequently its 
 thickness, it follows that the displacement must be in the plane 
 of the slice, that is to say, they must be parallel to the wave 
 plane." 
 
 But this, although a fair account of the actual motion (if any 
 motion of the kind supposed could exist under the assumptions 
 which are made) has nothing to do with the arbitrary motion 
 with which we have now to deal, which is quite independent 
 of the actual movement. The only conditions to which the 
 medium is subject are those of continuity and incompressibility. 
 If Professor MacCullagh thinks that from those conditions it 
 follows that the portion of the function V due to the element 
 dx', dy ', dz is independent of /A, v (the other two angles into 
 which the parallelepiped may be forced), and depends solely on 
 the directions of x y' z and the angle K, let him prove it : but 
 I must observe that he has not done this yet, and until he has 
 done so I am under the necessity of affirming that his system 
 is erroneous, and his results inadmissible.* 
 
 February, 13, 1847. 
 
 * I may observe that the existence of two transversal waves polarized at right 
 angles to each other, which Professor MacCullagh deduces from his theory, is 
 precisely what, tinder such premises, might have been expected to occur ; for 
 the condition that the motion is constrained to take place in a direction 
 transverse to the course of the wave, is what actually occurs in the case of a 
 stretched cord, so that we are necessarily led to the equations applying to that 
 
REPLY TO SOME REMARKS OF MR. AIRY 
 
 Contained in a Recent No. of the Philosophical Magazine. 
 
 SINCE the publication of my two papers on Double Refrac- 
 tion, the individual who is generally looked upon as at once the 
 most zealous and the most able champion of Fresnel's theory in 
 this country, has thought proper to issue a manifesto expressing 
 the nature of his faith therein, which, as indicating the grounds 
 upon which the controversy must for the future be conducted, I 
 shall now lay before my readers. It is contained in a paper 
 inserted in the Philosophical Magazine (No. 189, p. 469), en- 
 titled, " On the Equations applying to Light under the action 
 of Magnetism," in which Mr. Airy states as follows : 
 
 "In order to justify my intruding upon you with a sugges- 
 tion which is exceedingly imperfect, I think it right to state 
 to you my opinion upon the present state of the optical theory, 
 and upon several steps which, though leading to nothing con- 
 clusive, have nevertheless contributed to the real intellectual 
 progress of science. 
 
 " Of the truth of the undulatory theory, as regards the geo- 
 metrical representation of light by undulations based upon 
 transversal vibrations, the resolution of which into vibrations at 
 right angles to each other constitutes polarization, I have not 
 the shadow of a doubt. These undulations, whatever may be the 
 way in which they may have been originally created, I conceive 
 to be propagated by mechanical laws applying to the attractive 
 or repulsive forces of the particles of the medium, the assumed 
 aether, or the medium and the aether combined. But I have 
 seen no mechanical theory to which I attach much importance 
 or any unqualified belief. Nevertheless I think that the inves- 
 tigation and publication of these mechanical theories have been 
 
98 REPLY TO SOME REMARKS OF MR. AIRY. 
 
 advantageous to the science by shewing that mechanical laws 
 may be able to explain effects never before ascribed to mecha- 
 nical laws. As regards the progress of intellect it has been very 
 important to shew that variation of velocities, as depending on 
 the period of the oscillations, is mechanically possible ; it has 
 been very important to shew that transversal vibrations are 
 mechanically possible ; it has been very important to shew that 
 crystalline separation of differently polarized rays is mechanically 
 possible. It is not that I believe completely in any one of 
 the mechanical explanations which have been given, but that 
 a priori difficulties have been removed, and that it may now 
 be considered that there is a fair chance of reducing the whole 
 to mechanical explanation." 
 
 This is Mr. Airy's present confession of faith. It is satis- 
 factory to find that in the course of fifteen years he has made 
 some small progress towards a just appreciation of Fresnel's 
 theory. In 1831 we find him, in the preface to the 2nd edition 
 of his Tracts, making the following statement. " The mecha- 
 nical part of the theory, as the suppositions relative to the con- 
 stitution of the ether, the computation of the intensity of 
 reflected and refracted rays, &c., though generally probable. 
 I conceive to be far from certain." Thus, while in 1831 
 Mr. Airy, although conceiving the mechanical part of the 
 theory to be far from certain, thought it ' generally probable,' 
 in 1846 he declares he has seen no theory to which ' he attaches 
 much importance,' much less f unqualified belief.' I am afraid 
 that this confession says very little for the progress of the theory 
 in the mean time ; but be this as it may, I am ready to accept 
 the controversy exactly as Mr. Airy has proposed it. 
 
 Mr. Airy thinks that " the investigation and publication of 
 these mechanical theories have been advantageous to the 
 science, by shewing that mechanical laws may be able to explain 
 effects never before ascribed to mechanical laws." Has this been 
 done ? Have the mechanical theories of which Mr. Airy speaks 
 been even of the limited utility he ascribes to them ? Mr. Airy 
 thinks it ' very important to shew that the variation of velocities, 
 as depending on the period of the vibrations, is mechanically 
 possible .' To what single investigation will Mr. Airy point as 
 
REPLY TO SOME REMARKS OF MR. AIRY. "99 
 
 effecting this purpose ? In the last edition of Mr. Airy's own 
 Tract on the Theory of Light, we find an investigation 
 professing to have accomplished this object : but independently 
 of the preposterous absurdity of the process adopted^ the result 
 obtained is altogether untenable ; for if that result were true, it 
 would follow that we should have dispersion in the free ether 
 just as much as in a mixed medium an idea which cannot be 
 entertained for a moment, and which Mr. Airy himself would 
 be the last to advocate. 
 
 Possibly Mr. Airy, like Sir John Herschel, thinks it enough 
 that a phenomenon be explained on a certain hypothesis, even 
 though we are perfectly certain that that hypothesis has no 
 foundation in fact. 
 
 In the 'Treatise on Light' (Encyc. Met. Art. 989) we find 
 the following extraordinary specimen of ratiocination. 
 
 "Let us take with M. Fresnel (Annales de Chimie, xvu. p. 1 79, 
 et seq.) the case of a crystal with one axis. We may regard 
 this, or rather the ether within it, modified in its action by 
 the molecular forces of the crystal, as an elastic medium in 
 which the elasticity in a direction perpendicular to the axis is 
 different from that in a direction parallel to it, that is, in which 
 the molecules are more easily compressible in the one than in 
 the other direction ; but equally so in all directions perpen- 
 dicular to the axis, oh whatever side the pressure be applied. 
 To aid our conceptions in imagining such a property, we may 
 assimilate an uniformly elastic medium to an assemblage of 
 thin, elastic, hollow, spherical shells in -contact; and such a 
 medium as we are considering to a similar assemblage of 
 oblate or prolate hollow ellipsoids, arranged with all their axes 
 parallel to one common direction, which is that of the axis 
 of the crystal. It is evident that the resistance of the spherical 
 assemblage to pressure must be the same in all directions, but 
 that of the spheroidal must differ according as the pressure is 
 applied perpendicularly or parallel to the axis. Thus, it is 
 easy to crush an egg by a force applied in the direction of 
 its shorter diameter, which will yet sustain a violent pressure 
 applied at the extremities of its longer. It is moreover evident, 
 if any molecule of such an assemblage were disturbed so as 
 
100 REPLY TO SOME REMARKS OF MR. AIRY. 
 
 to throw it into vibration, that, provided always the amplitude 
 of its excursions were extremely small compared to the dia- 
 meter of each ellipsoid, the immediate tendency of the vibra- 
 tion will be to communicate motion to two strata only of mole- 
 cules, viz. that in which the axis and equator of the disturbed 
 molecule lie respectively, since it is only at the poles and 
 equator that they touch, and therefore only through these 
 points that motion can be communicated. Consequently, any 
 motion communicated to a molecule of such a mass could 
 only be propagated by vibrations performed in planes parallel 
 and perpendicular to the axis. Hence, if a vibratory motion 
 in any plane be propagated into such an assemblage of particles 
 from without, it will immediately, on its reaching it, be resolved 
 into two in the planes aboved named, and these, by reason 
 of the different elasticities, will be propagated with different 
 velocities. 
 
 " The reader must not suppose that this is intended for an 
 account of the real mechanism of crystallized bodies. It is 
 merely intended to shew that it is not absurd or contrary to sound 
 mechanical principles) to assume that such MAY be their constitu- 
 tion, that vibrations can be propagated through them by molecular 
 excursions executed in planes parallel and perpendicular to their 
 axes." 
 
 Such arguments may be current in the skies with which 
 Mr. Airy and Sir John Herschel are familiar, but they certainly 
 are not such as commonly circulate in this lower sphere. I 
 have elsewhere expressed myself at a loss to understand the 
 support which a theory can derive from an explanation which 
 no one believes. Mr. Airy and Sir John Herschel however 
 appear to be of a different opinion. 
 
 In the same note to the second edition which I have before 
 adverted to, Mr. Airy refers his readers to Professor Powell's 
 work for further explanation as to the doctrine of different 
 refrangibility. I have in an earlier part of this work examined 
 each of the methods treated of by Professor Powell, as also 
 some others which have been proposed from different quarters, 
 and I have shewn that they are all alike erroneous. I assert 
 therefore, and I challenge Mr. Airy to gainsay ^ the assertion, 
 
REPLY TO SOME REMARKS OF MR. AIRY. 101 
 
 that, it is not true " that the variation of velocities, as depending 
 on the period of the vibration, has been shewn to be mechani- 
 cally possible." 
 
 Let us now, however^turn to the second declaration of 
 Mr. Airy. He says, " It has been very important to shew that 
 transversal vibrations are mechanically possible." Again I ask 
 has this been done ? In Mr. Airy's own work we find an 
 investigation having for its object to shew the possibility of 
 transversal vibrations, which investigation I have shewn to be 
 entirely erroneous. If Mr. Airy relies on Professor Powell, or 
 Mr. Kelland, or Mr. Green, or Professor MacCullagh for aid 
 in this particular, I must beg to assure him that the processes 
 adopted by those gentlemen are as fallacious as his own, and 
 that they fail as completely as he does himself to demonstrate 
 that transversal vibrations are possible. 
 
 Lastly, Mr. Airy considers, " it has been very important to 
 shew that the separation of differently polarized rays is me- 
 chanically possible." Does Mr. Airy allude to Fresnel's 
 theory ? I think I may without vanity presume that my 
 attack on that theory was the immediate cause of Mr. Airy's 
 present declaration : but I cannot flatter myself that he has done 
 me the honour of reading my papers perhaps he has been too 
 much en-" gaged." But whatever the cause, as the fact appears 
 certain, I must trouble Mr. Airy with a brief recapitulation of 
 the objections which may be urged against Fresnel's theory of 
 the separation of the ray. 1. In the first place, then, there are 
 no axes of elasticity such as Fresnel has supposed. 2. If these 
 last existed, there would be no undulation. 3. Granting all 
 Fresnel's postulates there would be no polarization. The 
 motion of the particles in a plane perpendicular to the point of 
 the wave is, even upon his own principles, clearly impossible. 
 These positions suffice to establish that it is not true that 
 Fresnel's hypothesis shews " that the separation of differently 
 polarized rays is mechanically possible." 
 
 One cannot but admire the rhetorical skill with which 
 Mr. Airy endeavours to acquire the support of every inves- 
 tigation connected with the subject of finite intervals, at the 
 same time that he seeks to avoid being responsible for any 
 
ItEPLY TO SOME REMARKS OF MR. AIRY. 
 
 one. In speaking of the class of operations for which Mr. 
 Airy has claimed so small a degree of credit, I have, from 
 the generality of his remarks and the impossibility of con- 
 sidering every case, confined myself to those which are best 
 known or are most accessible. It is possible that Mr. Airy 
 may have had in view other investigations than those which I 
 have particularized. If he will take the trouble to specify any 
 such, I will undertake to give them my best consideration and to 
 publish my opinion concerning them, whether favourable or 
 unfavourable ; in the latter case supporting my views by 
 argument. But in the mean time I would suggest that, of the 
 cases instanced above, some have been distinctly sanctioned by 
 Mr. Airy himself, and that so lately as the year 1842 ; and that 
 as his eulogy extends to the whole class of the " mechanical 
 theories," the above must be considered as participating in it. 
 It is a grave charge to make, but until Mr. Airy assures us 
 to the contrary, we must conclude that, at the time of writing the 
 above paragraph, after fair warning, he deliberately entertained 
 the opinion that those unparalleled investigations were of impor- 
 tance " as regards the progress of intellect." 
 
 Liverpool, November 25, 1846, 
 
PROFESSOR CHALLIS'S THEORY OF 
 
 LUMINOUS RAYS. 
 
 PROFESSOR Challis appears to have adopted a somewhat novel 
 view of the constitution of light derived from the consideration 
 of the solar spectrum, and which though ' simple' appears to 
 have escaped the notice of all his predecessors. 
 
 " If," says he,* " a beam of sunlight pass through a narrow 
 aperture about one-thirtieth of an inch in breadth and be 
 received on a glass prism, the edges of which are parallel to 
 the borders of the aperture, a spectrum is formed by the trans- 
 mitted light, which, when magnified and properly looked at, 
 exhibits, as is well known, a large number of dark lines parallel 
 to the refracting edge of the prism. If instead of passing 
 through an aperture with parallel borders, the light passed 
 through a circular aperture, one-thirtieth of an inch in diameter, 
 *a spectrum of diminished width would be seen, but of the same 
 length as before, and crossed by the same dark lines. The 
 transmission through the prism has produced no change in the 
 light: it has only brought into view the parts of which the 
 incident beam is composed. Taking, for instance, a portion of 
 light immediately contiguous to any one of the dark lines, the 
 prism informs us that the incident beam contains light of that 
 particular refrangibility, abruptly terminated in a plane passing 
 through the axis of the beam perpendicular to the edge of the 
 prism. The existence of this abrupt termination is owing to the 
 cause, whatever it may be, which produces the dark line, and 
 has nothing whatever to do with the transmission of light 
 through a small aperture. Let now the prism be turned about 
 the axis of the beam to any other position. The spectrum will 
 present exactly the same appearance as before, and light of the 
 same refrangibility (not necessarily the same light) as in the 
 former case, will still be bounded by a dark line ; and so for 
 
 
 * Camb. Phil. Trans., Vol. viu. part iii. p. 362. 
 
104 
 
 every position the prism be made to take, by being turned 
 about the axis of the incident beam. This experiment proves 
 that every beam of white light contains portions of light of 
 a definite refrangibility, the sides of which are turned in all 
 directions from the axis of the beam. This fact is at once ex- 
 plained by supposing light to consist of rays : and it does not 
 appear possible to give any other explanation of it. Although 
 the experimental evidence applies immediately only to the por- 
 tions of light contiguous to dark lines, yet a very strong pre- 
 sumption is afforded by it that all light is in the form of rays. 
 The existence of the dark lines themselves is most simply ac- 
 counted for by supposing that certain rays of sun-light are in 
 some manner extinguished." 
 
 This, as an explanation, is not very happy. Professor Challis 
 may know very well what he means when he speaks of " the 
 sides of" a portion of light being '* turned in all directions 
 from the axis of the beam," but I apprehend that to most of 
 his readers it will be anything but obvious. Again, one cannot 
 but be struck by the statement that " the transmission through 
 the prism has produced no change in the light," and with % 
 another equally round assertion, " that the existence of this 
 abrupt termination has nothing whatever to do with the trans- 
 mission of light through a small aperture," which every one 
 knows to be untrue, and the Professor as well as the rest, or 
 why did he stipulate for an aperture only one-thirtieth of an 
 inch wide ? The rashness of this latter remark, however, may 
 be considered as compensated for by the perfectly unobjection- 
 able character of another which occurs in connexion with it, 
 to wit, that the " abrupt termination is owing to the cause, what- 
 ever it may be, which produces the dark line,'* of which the 
 triteness somewhat verges upon the ridiculous. 
 
 To speak seriously however. The broad fact to be accounted 
 for is, that the light in the spectrum terminates abruptly, and 
 this Professor Challis seems to think may be explained " by 
 supposing light to consist of rays," and "that certain rays of 
 sun-light are in some manner extinguished." 
 
 No doubt this explanation is perfectly competent as far as 
 it goes, but it is somewhat to be marvelled at how such extinc- 
 
LUMINOUS RAYS. 105 
 
 tion as here spoken of takes place. Let us take the case 
 originally proposed by Professor Challis of the narrow aperture, 
 and suppose a lens placed between the prism and the aperture. 
 
 A spectral image of the latter may then be received on a 
 screen which, according to the Newtonian theory as to the com- 
 position of white light, ought to consist of a series of images of 
 equal breadth, corresponding to rays of different refrangibility, 
 and overlapping each other. But Professor Challis seems to be 
 of opinion that this is not the case. He seems to think that 
 instead of a series of images of nearly equal breadth overlapping 
 each other, the majority of the images capriciously take it upon 
 them to contract themselves to dimensions bearing no proportion 
 to those of the rest : that, for instance, while red rays give 
 an image ^th of an inch in breadth, blue only give ^th ; the 
 remaining T goths which would be due to them upon the New- 
 tonian principle and the laws of plane optics being absolutely 
 " extinguished in some manner." 
 
 Has Professor Challis reflected in what manner this would 
 be possible ? If only a fifth part of the geometrical image due 
 to the blue rays is illuminated, what has become of the rays 
 which should illuminate the remaining portion? If the rays 
 corresponding to one-fifth of the image reach the screen, there 
 can be nothing in the constitution of the prism to prevent those 
 corresponding to the remaining four-fifths from arriving there 
 also. The missing rays cannot have committed suicide by the 
 way. It is very easy to speak of certain rays of sun-light being 
 in some manner extinguished, but unless Professor Challis can 
 throw discredit upon Newton's theory of the composition of 
 white light, or impugn the laws of the formation of images 
 derived from geometrical optics, or shew that the transmissi- 
 bility of a ray through a glass prism depends upon the length 
 of its course through the prism, and that it is in fact a dis- 
 continuous function of that length, it must be admitted that 
 the mode of explanation which he proposes, so far from being 
 ( simple,' presents some of the gravest difficulties that have 
 ever been suggested to the philosophical mind. 
 
 It is somewhat remarkable that Professor Challis should 
 have selected the dark lines of the solar spectrum, perhaps the 
 
106 puoFEsson en ALMS'S THEORY OF 
 
 most obscure branch of the entire subject, as the source from 
 which to illuminate us as to the theory of light. Nor does 
 the particular purpose to which he applies them tend to 
 diminish our surprise at the fact of their being thus introduced 
 to our notice. A plain man, I should imagine, would be rather 
 astonished to be told that philosophers had at length arrived 
 at the conclusion that light consisted of rays, and he might 
 probably be disposed to smile if we were to assure him that 
 this profound result had only been established by means of 
 an experiment which no one ever heard of till within the last 
 fifty years. 
 
 Those mystical beings, the dark lines of the solar spectrum, 
 which appear to have taken such hold of the Professor's imagina- 
 tion, are nothing more than the shades of the departed aperture ; 
 and a sharp shadow is an apparition which, however difficult 
 to account for, has too long been familiar to us to admit of 
 its being an object of terror. Does Professor Challis think that 
 this difficulty has slept in forgetfulness till the time of Frauen- 
 hofer ? Suppose the prism removed, and let the naked aperture 
 be then "properly looked at;" shall we not in that case also 
 have a sharply defined shadow ? If not, I must beg to suggest 
 that the prism, so far from having " produced no change in the 
 light," has produced a most material change in it. 
 
 It is to be regretted that in reintroducing the terms ( beam ' 
 of light and ' ray ' of light, which have long been dismissed 
 from the service in both the two physical theories, Professor 
 Challis does not define the meaning which he attaches to them. 
 In a subsequent part of his paper (p. 3 6 8) he appears to consider 
 a vibration propagated along " a slender cylindrical portion of 
 ether" as equivalent to a ray ; but if this be the case, it must 
 be admitted that the fact of the beam being bounded by sharp 
 edges is but meagre evidence of its being composed of a 
 number of distinct threads of vibration. Supposing light to 
 consist of undulations, I apprehend that by a ' beam of light ? 
 must be meant an undulation or series of undulations confined 
 between certain planes. Now the agitation of a slender cylin- 
 drical portion of ether, of which Professor Challis speaks as 
 if it were what he means by a ray, is simply an undulation 
 
LUMINOUS RAYS. 107 
 
 confined between certain planes. Thus, according to Professor 
 Challis, the most simple way of accounting for the fact of an 
 undulation being confined between plane limits, is by supposing 
 it to consist of other undulations likewise confined between 
 plane limits, "and it does not appear to him possible to give 
 any other explanation of it." This may be an explanation, 
 bat it occurs to me as being of that kind in which nothing is 
 explained. 
 
 It is natural that Professor Challis should seek to enhance 
 the difficulty from which he believes his theory to free us. 
 " The facts," he tells us, " are perfectly consistent with the 
 Theory of Emission, and the advocates of that theory might 
 justly appeal to them as evidence in its favour." Yet I appre- 
 hend that the most eminent of those advocates, that the illus- 
 trious author himself to whom the Theory of Emission owes 
 its every claim to our regard, would have been very little apt 
 to make the appeal which Professor Challis is so ready to allow 
 to his followers. If Professor Challis thinks the abrupt ter- 
 mination in perfect accordance with the Theory of Emission, 
 what does he think of the bearing upon that theory of the 
 fading fringes which occur in diffraction? If the one class 
 of phenomena accord with the Theory of Emission, it will at 
 any rate be by no means easy, it will imply considerations 
 anything but " simple" to reconcile with it the other ; and yet 
 upon a conclusion thus hastily adopted Professor Challis would 
 found a new theory of Optics ! 
 
 The bearing of the phenomena of the pure spectrum upon 
 the constitution of light has been completely misapprehended 
 by Professor Challis. It seems to have escaped his observation 
 that the fusion of colours which occurs when the lens and tele- 
 scope are removed, is as actual a fact as the accurately defined 
 boundaries which are brought to light by means of those 
 adjuncts. The simple conclusion to be drawn from these two 
 facts taken together is this: that after emerging from the 
 prism, we shall have, corresponding to every point of the 
 aperture and corresponding to every degree of refrangibility 
 of the transmitted light, a diverging pencil of variable intensity, 
 that intensity being greatest in the middle and diminishing 
 
108 PROFESSOR CHALLIS'S THEORY OF 
 
 gradually towards the sides. It is not true that, "taking a 
 portion of light immediately contiguous to any one of the dark 
 lines, the prism informs us that the incident beam contains light 
 of that particular refrangibility, abruptly terminated in a plane 
 passing through the axis of the beam perpendicular to the edge 
 of the prism." "We might as well conclude from seeing a 
 window-frame cutting sharp against the sky, that the beam 
 which grazes the frame and enters the eye contains " light 
 of that particular refrangibility abruptly terminated in a plane 
 passing through the" edge of the frame and the position of the 
 eye. If vision were confined to the use of the transit instru- 
 ment, we might indeed arrive at so preposterous a conclusion ; 
 but the faculty of locomotion spares us this absurdity. 
 
 Satisfied, however, as to the justice of his ray theory upon 
 experimental grounds, and nothing daunted by the unpromising 
 aspect which it presents to the application of the undulatory 
 doctrine, Professor Challis boldly sets to work to shew how 
 the existence of rays may be accounted for on the latter hypo- 
 thesis. To a practical mechanician such an attempt would 
 seem about as hopeful as the endeavour to make water stand 
 up above the sides of a glass ; but to one who has at his com- 
 mand the differential equations of fluid motion which controul 
 the very elements themselves, what shall be impossible ? 
 
 The Professor sets out from a transformation of the equation 
 .of continuity, 
 
 dp dpV 
 
 (in which p denotes the density, V the velocity at any point, 
 and R, K the greatest and least radii of curvature at any point 
 of the surface, whose differential equation is d^ = 0,) and the 
 ordinary integral of the general equation of fluid motion, 
 
 a 1 log' p + J^- ds + ~ = F(<), 
 
 " the variation with respect to space being," in these two 
 equations, " from point to point of the line of the motion. By 
 differentiating this equation with respect to s and t successively, 
 
LUMINOUS RAYS. 109 
 
 we get cfdp 
 
 .(2). 
 and 
 
 " Hence substituting for -~ and (- from the preceding 
 
 pat pas 
 
 equations, and differentiating with respect to p, the result is 
 
 ,, Tr2 , - dF dV dVf\ 1 
 
 _ (a - y z } _ 4 2 V _ + 2 - -- h 2 _ I __ i __ 
 ^ } ds z dsdt ds dt ds \R R 1 
 
 for J72 = JJ2' = (/5. Now putting - for F", and integrating 
 
 Gf5 
 
 with respect to s after the substitution, it will be found that 
 
 " Lastly, for q put + % (), the function % () being such 
 that JP(Q -x (t}= 0. 
 
 Then = =r, and 
 
 __ _ 
 
 ~ ds 2 ) ds* * fo' dsdt Ts\R + ~R')~ 
 
 " If the surface normal to the directions of motion be a plane, 
 R and R are each infinitely great, and the equation strictly 
 applying to this case of motion is 
 
 __ 
 df ^d? 2 d'didi~ 
 
 " It is well known that this equation is exactly satisfied by 
 a particular integral applying to motion propagated in a single 
 
 direction, namely, ~ = F {{ a + -^-}t - s\', and that at the 
 ds \\ dsj J 
 
 same time -^ = a log p. From these two equations it follows 
 
 uS 
 
 that a given state of density and velocity is carried through 
 space by the propagation and by the motion of the particles 
 
 
110 PROFESSOR CHALLIS'S THEORY OF 
 
 with the velocity a + - . The rate of propagation is therefore 
 
 US 
 
 strictly a, whatever be the velocity and density of the particles. 
 Unless this were the case, the velocity and arrangement of den- 
 sity in a given wave would change by propagation, however 
 small the motion of the particles might be." 
 
 The conclusion here arrived at is somewhat extraordinary. 
 " A given state of density and velocity is carried through space 
 
 .... with the velocity a + - . The rate of propagation is 
 
 as 
 
 therefore strictly a." I confess myself unable to apprehend the 
 distinction between " velocity" of propagation and " rate of pro- 
 pagation" ; and supposing that no such distinction is intended, 
 I am unable to appreciate the force of the argument that be- 
 cause the rate of propagation is a + - it is therefore a. If the 
 
 as 
 
 velocity a 4 ~ seems absurd or incomprehensible, it may be 
 us 
 
 a reason for rejecting the integral or the theory upon which it 
 is founded, but it cannot justify us in an arbitrary adaptation 
 of results to suit collateral purposes.* But to proceed, 
 
 " In order that equation (4), in which R and R' are not 
 supposed to be indefinitely great, may apply to motion in which 
 the type of the wave remains altogether unchanged by propaga- 
 tion, it must be of the same form as equation (5). This will be 
 the case if 
 
 the resulting equation being the same as (5), with the difference 
 of having a' in the place of a. 
 
 a 
 
 Professor Challis next enters into some disquisition to shew 
 that the function i//, of which he is in search, must contain three 
 arbitrary functions which will enable him to satisfy three assump- 
 
 * The velocity of propagation in this case is a and not a + , as may be 
 readily shewn. 
 
LUMINOUS RAYS. Ill 
 
 tions, which he proceeds to make in order to facilitate his further 
 progress. The three assumptions are, first, " that the propa- 
 gation of the motion is in a single direction ; next, that the 
 motion of the particles, situated in a fixed straight line which 
 I shall call the axis of 2, shall be entirely in that line ; thirdly, 
 that for the motion along the axis of z the equation (7) is 
 satisfied." 
 
 To these is afterwards added a fourth, namely, that the conden- 
 sation at any point whose coordinates are xyz is ^ (z, t) x f(xy\ 
 \vhere (j) l (zt} denotes that at any point in the axis of z. Upon 
 this latter basis the Professor then deduces values for u, v, w, 
 namely, 
 
 and finally arrives at the equation 
 
 = cos n (2 - a'f) ............ (8), 
 
 which gives the velocity 
 
 -~ = m sin n (at - z) ............ (9). 
 
 " It results from the foregoing reasoning, that if the small 
 vibrations of the aether in the direction of propagation follow 
 the law expressed by the equation last obtained, the conden- 
 sation in any plane perpendicular to the axis of rectilinear 
 propagation may vary at a given time from point to point, and 
 at the same time the propagation be uniform. A consequence 
 of this result is, that a very slender cylindrical portion of the 
 aether may continue in agitation while the contiguous portions 
 are at rest: and since the law above obtained is that which 
 has been found by experience to apply to the phenomena of 
 light, the existence of rays of light, which was proved experi- 
 mentally at the commencement of this paper, is accounted for 
 theoretically." 
 
 The facility with which some writers persuade themselves 
 of the perfect conformity with nature of results deduced from 
 premises which have so very slight a connexion with the facts 
 they are intended to account for, is somewhat remarkable. The 
 above result is entirely illusory. 
 
112 PROFESSOR CHALLIS'S THEORY OF 
 
 Since " the first powers only" of the velocities are " taken 
 account of," and since in this case, as Professor Challis assures 
 us (Camb. Phil. Trans. Vol. vn. part 3, p. 375), 
 
 I L L(^. ^ &$ dv \ 
 
 R + R ~ V \dx* + dy* + dz* " ds ) 
 
 dx* 
 
 (since f = r), 
 
 substituting this value in (4) we have, neglecting the terms 
 
 , ,. fd6\* , dd> d*<b 
 
 depending upon -/- and -f- . -^ -, 
 \ds J ds dsdt 
 
 a., 
 
 Had Professor Challis commenced his operations with this 
 equation he would have served every purpose of his inves- 
 tigation, and have saved himself and his readers an infinity 
 of trouble by withdrawing from their attention the ponderous 
 
 processes depending upon the term + , which are of no 
 
 use whatever. He would also have avoided the impropriety of 
 expressing by the same symbol < two totally different things, 
 namely at one time the motion at any point, and afterwards 
 merely the motion in the axis of z. To obviate this ambiguity 
 I shall write equations (7), (8), (9), as follows : 
 
 ,L *tf -t, $f /> d<f> 
 
 dx dy dz 
 
 m , 
 
 9 n = cos n (z a t), 
 n 
 
 -~ - m sin n (at - z\ 
 az 
 
 Now, observing that 
 
 d<f> d(j> d(f> 
 
 dx dy dz 
 
LUMINOUS RAYS. 113 
 
 and therefore that 
 
 _. _ 
 
 dx 2 ~ dx* dy z 9o dy* dz* dz z ' 
 
 substituting these values in (10) and eliminating * by means 
 of (8), we obtain 
 
 P-M+ *(%+%).*, 
 
 , a /a - a 2 7 
 
 where - - = k. 
 
 a 3 
 
 But Professor Challis gives us for the determination of f 
 the following equation, 
 
 d* 1 <f 1 , a 1 
 
 Hence, comparing this with (1 1), we get 
 
 j&JH*. 1 "' 
 
 therefore -j- = 0, -j- = 0, 
 
 and y = a constant, or the propagation will be the same as in 
 a cylindrical tube where a uniform wave perpendicular to the 
 axis in the direction of that line. So much for the theory of 
 luminous rays. 
 
 The theory of rays failing, I conclude that that of polarization 
 will fall with it. 
 
ME, STOKES'S THEORY. 
 
 THE Theory of Mr. Stokes, which I now propose to consider, 
 is contained in a paper, " On the Theories of the Internal 
 Friction of Fluids in Motion, and of the Equilibrium and 
 Motion of Elastic Solids," read to the Cambridge Philosophical 
 Society, April 14, 1845, and published in the Transactions 
 of that body, (Vol. vin., part in. p. 287). 
 
 Mr. Stokes's Temple of Truth is evidently no Hall of Apollo, 
 in which each object shines resplendent beneath the God of 
 Rays, but rather a dismal cavern in which the trembling votary, 
 scared by unfamiliar forms, is initiated into mysteries which are 
 calculated to bewilder rather than to enlighten. Whatever de- 
 mand the theories we have hitherto considered may make on 
 our credulity, it must be admitted that they do not tax very 
 heavily the understanding. With Mr. Stokes's theory the case 
 is altogether different. It is only by a painful effort that his 
 reasoning can be understood and his arguments appreciated. 
 Like as in some kinds of magic, the visions of truth are only 
 to be apprehended through the medium of an artificial darkness. 
 It is precisely from the consideration of a state of circumstances 
 which the boldest mind might recoil from as impracticable, that 
 Mr. Stokes professes to elicit the principles of his theory. 
 
 The first section of Mr. Stokes's paper commences with an 
 attempt " to define or fix the precise meaning of a few terms" 
 which he has occasion to employ. And first with regard to the 
 expression " the velocity of a fluid at any particular point." 
 
 " If we suppose a fluid to be made up of ultimate molecules, 
 it is easy to see that these molecules must, in general, move 
 among one another in an irregular manner, through spaces 
 comparable with the distances between them, when the fluid 
 is in motion. But since there is no doubt that the distance 
 
115 
 
 between the adjacent molecules is quite insensible, we may 
 neglect the irregular part of the velocity, compared with the 
 common velocity with which all the molecules in the neigh- 
 bourhood of the one considered are moving, or we may con- 
 sider the mean velocity of the molecules in the neighbourhood 
 of the one considered apart from the velocity due to the 
 irregular motion. It is this regular velocity which I shall 
 understand by the velocity of a fluid at any point, and I shall 
 accordingly consider it as varying continuously with the co- 
 ordinates of the point." 
 
 If Mr. Stokes finds his idea of the " velocity of a fluid at 
 any point" rendered more precise by the above definition, it 
 is to be feared that his previous notions on the subject must 
 have been very indefinite. We have here in fact two defi- 
 nitions : first, we have a regular and an irregular motion, 
 and we are told that we are to neglect the latter; but we 
 are not informed how we are to distinguish the one from 
 the other : and secondly, we are to take " the mean velocity 
 of the molecules in the neighbourhood of the one considered 
 apart from their irregular motion." Now with regard to this 
 latter definition, which alone appears in any way intelligible,, 
 (though in fact just as objectionable as the other), I would 
 observe that, if the velocities of the different molecules in 
 a small geometrical element differ materially, it cannot be of 
 the smallest use to know the value of their mean, since if we 
 had such knowledge it would give us no clue to the actual 
 motion either of the molecule " considered," or of any of its 
 neighbours. Moreover, according to the above definition, the 
 term " the velocity at any point" under such circumstances 
 would have no definite meaning, since the value of the mean 
 | velocity would depend upon the size of the element and the 
 mode in which the division into elements takes place, both of 
 which are perfectly arbitrary.* On the other hand, if the varia- 
 tion of velocity is so slight that it may be neglected, why speak 
 of taking the mean of the velocities of the different molecules 
 when that of any one molecule will serve the same purpose ? 
 
 * A slight change in the origin of coordinates might revolutionize all the 
 averages. 
 
 1 2 
 
116 MR. STOKES'S THEORY. 
 
 The true principle here to be assumed, and which Mr. Stoke 
 has involved in so much mystery, is this, that the velocity o 
 the different molecules in any small geometrical element of 
 fluid may be considered uniform. This principle I look upon a 
 a characteristic of fluids, as it may be also of rigid bodies whe 
 subjected to internal disturbance. It is this principle alon 
 which enables us to pass from the consideration of a fluid 
 consisting of a series of separate molecules divided by interval 
 to that of a continuous medium in which motion may be referre 
 to pressure. For if this principle do not hold, it is impossib' 
 to attach any definite signification to the expression " the pre 
 sure at any point," since this expression amounts to nothin 
 more than a mode of speaking of the aggregate effect of a nun 
 ber of contiguous particles ; and unless we suppose the effe< 
 of each particle to be sensibly the same, the value of th 
 aggregate will depend upon the number and position of th 
 particles we select, and therefore may be made to assum 
 (within certain limits) any assignable value. 
 
 No doubt this view of the subject is opposed to Mr. Stokes 
 idea of the irregular motion of the molecules " through space 
 comparable with the distances between them," upon which h 
 whole theory is based : but I would ask, what reason have w 
 to believe in this irregular motion? Has any one seen it? Is 
 it anything else than a gratuitous assumption ? It has so Ion 
 been the fashion to regard a fluid as a collection of high! 
 volatile particles acting upon each other by their mutual attrac 
 tions and moving in the most independent manner, that thf 
 possibility of a quiet-going medium in which the contiguoi 
 particles act in some sort of concert, seems to have been entire!) 
 overlooked. And yet we have no reason to believe the thing 
 impossible. Rigid bodies are generally considered to consi 
 of particles acting upon each other by their mutual attractions 
 yet I am not aware that any one considers the particles in them 
 be moving te among one another in an irregular manner througl 
 spaces comparable with the distances between them." A flui 
 after all is something different from a heap of sand. If it be aske 
 how the one differs from the other, I would say that in the on 
 case we have a connexion between the motions of the particle 
 
MR. STOKES'S THEORY. 117 
 
 which does not exist in the other. Nor is this connexion 
 of motion between the particles of a fluid a merely speculative 
 property, but a physical fact of which we have familiar ex- 
 perience in the phenomenon of viscosity, a property which 
 belongs in a greater or less degree to all fluids which can be 
 made the subject of vision, and which may be inferred of those 
 which cannot be so subjected. But to return. 
 
 After giving a definition of relative equilibrium, of which the 
 accuracy may be doubted, but the obscurity is unquestionable, 
 and after illustrating the difference between it and actual equi- 
 librium in a way little calculated to enlighten the student, 
 Mr. Stokes proceeds to state the nature of the forces acting 
 on a small element. These he makes to consist of the external 
 forces (such as gravity) and the pressures arising from the 
 molecular action of the surrounding medium. It is somewhat 
 remarkable that he here omits to mention the forces which 
 the different molecules of the element itself exert upon each 
 other, more particularly as he considers it an " undoubted 
 result of observation" that such forces " are forces of enormous 
 intensity," and as in point of fact there is some reason to think 
 that he makes these identical forces play an important part in 
 the ultimate development of his theory. Passing this however, 
 Mr. Stokes next observes that "in consequence of the intensity 
 of the molecular forces," which he has already told us is 
 " enormous," " the pressures arising from the molecular action 
 on the element will be very great compared with the external 
 moving forces acting upon it." He consequently neglects the 
 external forces in all his subsequent operations. 
 
 Now it may be observed that it does not necessarily follow 
 that because the molecular forces are enormous the pressures 
 arising from them are so, nor, if the latter were of the mag- 
 nitude Mr. Stokes supposes, would it follow that the external 
 forces may be neglected in comparison with them, since the 
 motion is affected only by the difference of pressure and not by 
 the absolute pressure. 
 
 Waiving this objection however, we next find Mr. Stokes 
 taking for granted that in estimating the pressure at any point 
 we may neglect any motion of rotation about an axis passing 
 
118 
 
 MR. STOKES S THEORY. 
 
 through the particle considered and common to all the con- 
 tiguous particles, and that we may also neglect any motion 
 of translation common to the whole element. From the latter 
 of these assumptions it follows that if u, v, w, be the velocities 
 parallel to the axes at the point P, whose coordinates are xyz, 
 and x + x, y + /', z + z are the coordinates of another point P' 
 near to the former, the effect of the motion at P f on the pressure 
 at P will depend on the velocities 
 
 du 
 
 -j:.y 
 
 du 
 
 T x 
 
 dx 
 
 du , 
 dz 
 
 dx* 
 
 dv 
 
 dv 
 
 9 + *.* iiy. 
 
 dw , dw , dw , 
 dx dy dz 
 
 These velocities Mr. Stokes shews may be replaced by three 
 linear velocities parallel to the respective axes and an angular 
 velocity common to all the points in the neighbourhood of P. 
 The latter he neglects, and then shews that during a single 
 instant the axes may be so taken that the residual motion about 
 P may be represented by 
 
 Tj == s> /> T/ _._ p itj XT' ~ ^'"^ 
 
 Without wishing to detract from Mr. Stokes's merit in sug- 
 gesting this simplification, I may observe that the same result 
 may be obtained somewhat more readily upon a different 
 principle, namely, by assuming that the pressure at any point 
 depends only upon that part of the motion around it which 
 takes place towards or from it, or that that part only of the 
 motion at P' is effectual in producing pressure at P, which 
 takes place along the line joining the two points. For, the 
 whole velocity at P' parallel to this line 
 
 du 
 
 dv 
 
 dw 
 
 du 
 dy 
 
 dv_ 2 
 dy y 
 dw 
 
 -j- y< 
 
 dy 
 
 du 
 
 xz, 
 dz 
 
 dv 
 dz **> 
 
 dw . 
 
MR. STOKES'S THEORY. 119 
 
 which can always be reduced to the form 
 - (Ax 12 + By' z + Cz 12 ) 
 
 by transformation of coordinates; and this last expression is 
 evidently the same as would have been produced if we had had 
 three velocities parallel to the new axes 
 
 Ax, By, Cz. 
 
 Mr. Stokes next proceeds to replace the motion represented 
 by the above expressions for U, V, W, by " a motion of dila- 
 tation, positive or negative, which is alike in all directions, 
 and two motions which he calls motions of shifting, each of the 
 latter being in two dimensions and not affecting the density," 
 
 Without entering into any discussion as to the propriety of 
 the latter term, which is not very obvious, I may observe that 
 the motion ( of shifting' parallel to xy consists of a simultaneous 
 contraction parallel to x, and dilatation parallel to y, or vice versa, 
 and is represented by the velocities crx, - cry, parallel to x 
 and y respectively. The other motion of shifting parallel to xz 
 produces velocities crx, - cr'z, parallel to x and z. " If 8 be the 
 rate of linear extension corresponding to a uniform dilatation," 
 we have 
 
 8 = 1 (e 1 + e" + e'"\ cr = \ (e 1 + e'" - 2e"), cr' = !(>' + e" - '2e'"). 
 
 " Denoting then by p + p , p + p", p + p", the pressures on 
 planes perpendicular to the axes of xyz, we must have 
 
 p = (e', e", e"'\ p" = (e", e 1 " , e' ,) p" = (e", e 1 , e"), 
 
 $ (e , e" , e"} denoting a function of e ' , e", e" , which is sym- 
 metrical with respect to the two latter," " p being the pressure 
 which would exist about the point P if the neighbouring mole- 
 cules were in a state of relative equilibrium." " The question 
 is now to determine, on whatever may seem the most probable 
 hypothesis, the form of the function 0." To the discussion 
 of this question I must request my reader's particular attention. 
 " Let us first take the simpler case in which there is no 
 dilatation, and only one motion of shifting, or in which e" = - e, 
 e'" = 0, and let us consider what would take place if the fluid 
 
120 MR. STOKES'S THEORY. 
 
 consisted of smooth molecules acting on each other by actual 
 contact. On this supposition it is clear, considering the mag- 
 nitude of the pressure acting on the molecules compared with 
 their masses, that they would be sensibly in a position of rela- 
 tive equilibrium, except when the equilibrium of any one of 
 them became impossible from the displacement of the adjoining 
 ones, in which case the molecule in question would start into 
 a new position of equilibrium. This start would cause a cor- 
 responding displacement in the molecules immediately about the 
 one which started, and this disturbance would be propagated 
 immediately in all directions, and would soon become insensible. 
 During the continuance of this disturbance, the pressure on 
 a small plane drawn through the element considered would not 
 be the same in all directions, nor normal to the plane : or, which 
 comes to the same, we may suppose a uniform normal pressure^ 
 to act, together with a normal pressure p ai and a tangential force 
 t //9 p u , and t a being forces of great intensity but short duration, 
 that is, being of the nature of impulsive forces. As the number 
 of molecules comprised in the element considered has been 
 supposed extremely great, we may take a time r so short that 
 all summations with respect to such intervals of time may be 
 replaced by integrations, and yet so long that a very great 
 number of starts will take place in it. Consequently we have 
 only to consider the average effect of such starts, and moreover 
 we may without sensible error replace the impulsive forces 
 such as p, and t fit which succeed one another with great rapidity, 
 by continuous forces. For planes perpendicular to the axes of 
 extension, these continuous forces will be the normal pressures 
 
 P'>P">P'"- 
 
 " Let us now consider a motion of shifting differing from the 
 former only in having e increased in the ratio of m to 1 . Then 
 if we suppose each start completed before the starts which 
 would be sensibly affected by it are begun, it is clear that the 
 same series of starts will take place in the second case as in 
 the first, but at intervals of time which are less in the ratio 
 of 1 to m. Consequently the continuous pressures by which 
 the impulsive actions due to these starts may be replaced must 
 be in the ratio of m to 1. Hence the pressures />',/", /?'", must 
 
MR. STOKES'S THEORY. 11 
 
 be proportional to e', or we must have 
 
 p' = Ce 1 , p" = C'e', p" = We 1 ." 
 
 Such is the principle upon which Mr. Stokes professes to 
 elucidate the theory of fluid motion, the " method of starts," 
 and certainly a most startling method it is. It is somewhat 
 extraordinary that any person of ordinary judgment should 
 have been deluded into the belief that reasoning of such a 
 nature, which may be correctly characterized as the argument 
 from the incomprehensible, can carry with it a particle of con- 
 viction to the human mind ; and yet a similar kind of argument 
 has been adopted by one far more eminent than Mr. Stokes, 
 and who for upwards of thirty years has been looked up to 
 as the great authority on subjects of this nature. 
 
 Mr. Stokes says, "It is clear, considering the magnitude of 
 the pressures acting on the molecules compared with their masses, 
 that they would be sensibly in a position of relative equilibrium 
 except when the equilibrium of any one of them became impos- 
 sible from the displacement of the adjoining ones." How does 
 the consideration of the " magnitude of the pressures" enter 
 here ? Whether the forces be great or small, it would be very 
 extraordinary if one particle, in the midst of a number of others 
 which are all at rest, should of its own motion start from its 
 position of equilibrium, even assuming it to have all the vola- 
 tility which Mr. Stokes appears to ascribe to it. Mr. Stokes 
 says, " the molecule in question would start into a new position 
 of equilibrium." No doubt it would do its best, but it might 
 happen not to find a new position of equilibrium. Does Mr. 
 Stokes really believe that a succession of starts takes place at 
 intervals in the way he describes ; or does this method amount 
 to nothing more than dividing the continuous motion of a par- 
 ticle into an indefinite number of short intervals, and considering 
 the separate effect of the motion in each interval ? If anything 
 depend on the individuality of the starts, I must beg to state 
 that we have not, nor are ever likely to have, the smallest 
 evidence of their existence : and if Mr. Stokes relies on the 
 division into intervals of the continuous motion, I must be per- 
 mitted to assert that his argument is nothing more nor less than 
 
MR. STOKESS THEORY. 
 
 a flat assumption that the pressures will be proportional to the 
 velocity. Mr. Stokes says, " if we suppose each start completed 
 before the starts which would be sensibly affected by it are 
 begun, it is clear that the same series of starts will take place 
 in the second case as in the first, but at intervals of time which 
 are less in the ratio of 1 : m." Is this so clear ? Undoubtedly 
 the fact may be so, but have we any particular reason to believe 
 that it is ? Let us take an analogous case. If I tap with my 
 knuckle against a window pane, all the circumstances occur 
 which Mr. Stokes supposes in his account of the effect of one 
 motion of shifting ; the external molecules of the glass " start 
 into new positions of equilibrium," " these starts cause a cor- 
 responding displacement in the molecules immediately about 
 those which started, and this disturbance is propagated imme- 
 diately in all directions, the nature of the displacement however 
 being different in different directions, and soon becoming insen- 
 sible," &c. &c. Now if I increase the velocity of my hand in 
 the ratio of m to 1 , according to Mr. Stokes, the " same series 
 of starts will take place in the second place as in the first, but at 
 intervals of time which are less in the ratio of 1 to m"; and this 
 would be all that would occur. But we know as a matter of 
 fact that the effect in the one case may be nothing more than 
 to cause to " start" some fair tenant of the chamber if the blow 
 be communicated from without, or to " displace" some unfor- 
 tunate mendicant if administered from within, while in the 
 other the glass may be broken. We may see, in like manner, 
 that in Mr. Stokes's original case it is not necessarily true that 
 the operation of an acceleration of the velocity of any displaced 
 particle will be simply to shorten the time in which the effects 
 of the displacement will be developed, but, on the contrary, 
 effects of a totally different character may result in other words 
 it is not "clear," nor have we the smallest reason to believe 
 it to be the fact, that the same series of starts will take place in 
 Mr. Stokes's second case as in his first, either at intervals of 
 time which are less in the ratio of 1 to m, or otherwise. 
 
 From the above formulae for p' 9 p", p"', Mr. Stokes deduces, 
 reasonably enough, the following, 
 
 p' = - 2fjie, p" = - 2/ue", p" = 0. 
 
123 
 
 He then proceeds to consider the effect of a second motion 
 of shifting, which he shews in a manner sufficiently allowable 
 (i. e. assuming the validity of his first equations) would give us 
 
 (A) / = -2^', p" = -tye", p'" = -2rf". 
 
 After arriving at these desirable results Mr. Stokes enters 
 into some further disquisitions on the theory of starts, for which 
 I must refer such of my readers as may have any curiosity 
 respecting them to his original paper. He next observes, 
 " there remains yet to be considered the effect of the dilatation. 
 Let us first suppose the dilatation to exist without any shifting : 
 then it is easily seen that the relative motion of the fluid at the 
 point considered is the same in all directions. Consequently 
 the only effect which such a dilatation could have would be to 
 introduce a normal pressure p,, alike in all directions, in ad- 
 dition to that due to the action of the molecules supposed to be 
 in a state of relative equilibrium : but since these displacements 
 take place, on an average, indifferently in all directions, it 
 follows that the actions of which p, is composed neutralize each 
 other, so that p t = 0." This extraordinary conclusion, namely, 
 that " a uniform motion of dilatation, positive or negative, which 
 is alike in all directions," has no effect upon the pressure at the 
 point from which the motion radiates, is more than I was pre- 
 pared for, even after experience of the strange reasoning in 
 which Mr. Stokes had previously indulged. Mr. Stokes con- 
 siders the relative pressure at a point to depend upon the 
 relative motion about that point, and yet he proposes to neglect 
 the effect of a general motion of all the surrounding particles 
 towards the point in question ! If all the surrounding particles 
 are rushing towards a point with the same velocity, does Mr. 
 Stokes really conceive that they will produce no pressure at the 
 point? Or if all the particles are moving away from a point, 
 does he conceive that such motion will not tend to diminish 
 the pressure at the point ? If Mr. Stokes neglects the motion 
 towards or from the point, what motion will he take notice of? 
 Will he rely upon the transverse motion? and if he does, 
 how will he get rid of the angular velocity, the dismissal of 
 which is the leading point in his theory? The same principle 
 
MR. STOKESS THEORY. 
 
 of the motion of dilatation producing no effect upon the pressure 
 is made use of by Mr. Stokes in another attempt to arrive at 
 expressions for the pressures p', p" , p"'. He there says, 
 
 " If we had started" (i. e. in the general case) " with assuming 
 (e, e", e"') to be a linear function of e' 3 e" , and e", avoiding all 
 speculation as to the molecular constitution of a fluid, we should 
 have at once p' = Ce + C' (e + e"), since p is symmetrical with 
 respect to e"and e'"', or changing the constants, p'= [j. (e" + e" - 1e) 
 + K (e + e" + e'"). The expressions for p" and p" would be ob- 
 tained by interchanging the requisite quantities. Of course ice 
 may at once put K = 0, if we assume that in the case of a uni- 
 form motion of dilatation the pressure at any instant depends 
 only on the actual density and temperature at that instant, and 
 not on the rate at which the former changes with the time." 
 
 Of course we may put K = in the case supposed, but it is 
 neither a matter of course nor a matter of fact that we are 
 entitled to make any such assumption as Mr. Stokes here 
 supposes. If we can neglect a uniform motion to or from the 
 point whose state of pressure we are considering, pray what 
 motion may we not neglect ? If we consider that odd creature 
 of Mr. Stokes's, a motion of shifting parallel to xy, we have the 
 two velocities crx, cry, parallel to x and y respectively ; and 
 confining ourselves to the former, we have on the opposite side 
 of y equal and opposite velocities, whence, on Mr. Stokes's 
 principle, they ought to destroy one another and at the same 
 time produce no pressure ; so that we might get rid of the 
 motions of shifting just as readily as of the motion of dilatation. 
 
 But strange as is the conclusion arrived at by Mr. Stokes, the 
 use he makes of it is still more extraordinary. Having shewn 
 that the motion of dilatation has no effect, an unsophisticated 
 person might have thought that we had for ever done with it, 
 and that the motions of shifting and the equations (A) would 
 have been left in quiet possession of the field. Far from it. 
 
 " If the motion of uniform dilatation coexists with two motions 
 of shifting, I shall suppose, for the same reason as before, that 
 the effects of these different motions are superimposed" (i.e. as 
 the effect of the motion of dilatation is nothing, either nothing is 
 superimposed on the motions of shifting, or the motions of 
 
MK. STOKKS'S THEORY. 
 
 shifting are superimposed on nothing). " Hence subtracting -8 
 from each of the three quantities e, e", and e'" 9 and putting the 
 remainders in the place of e ', e" ', and e" in equations (A), we have 
 
 (B) p' = n (e" + e 1 " - 2e'), p" = /i (e" + e' - 2e"), 
 
 Of this singular step Mr. Stokes does not give one syllable 
 of explanation. Does it follow, because the motion of dila- 
 tation has no effect, that in estimating the effect of the motions 
 of shifting we may substitute e - 8, e" - 8, e" - 8, respectively, 
 for e', e" , e'" in the formulae (A) ? Mr. Stokes has such strange 
 notions of logic, that when he writes in this elliptical style it 
 is somewhat difficult to divine how he means to fill up the 
 gap in his argument: but if there be any argument here at all, 
 I imagine it must be something like the following " As the mo- 
 tion of dilatation produces no effect, it is all one as if there were 
 none; hence we may assume 8 = 0."!!! To which it may be 
 replied that, assuming the motion of dilatation to have no direct 
 effect on the pressure, it would nevertheless not be generally 
 true that B = 0, and it would therefore not be generally true 
 that the quantities e - 8. e'- S, e" - 8, are identical with e, e", e"'; 
 and it follows therefore, even on Mr. Stokes's own shewing, that 
 the equations (B) do not generally hold. 
 
 I have thought it necessary to discuss in detail the process 
 by which Mr. Stokes arrives at his fundamental formulae, as 
 1 consider that process to be replete with principles of the most 
 dangerous character, and which in fact, if once admitted, must 
 tend to destroy all confidence in the deductions of the human 
 mind. The fallacy of these formulae however may be shewn in 
 a very direct and simple manner, which, on account of the 
 bewildering tendency of arguments such as I have just been 
 considering, it may be desirable that I should point out. 
 
 If e , e, e" are all positive, and we confine ourselves to the 
 relative motion, the tendency of all the particles in the neigh- 
 bourhood of that which is being considered must be to fly from 
 it. Now that the retreating of the particles along the axis of x 
 from the proposed particle should have the effect of diminishing 
 the pressure along that axis, as the first of the formulae (B) 
 
126 
 
 would make it do, is highly reasonable ; but that the recession of 
 particles along the axis of y should actually have the contrary 
 effect of increasing the pressure parallel to x 3 as the same 
 formula? would imply, is a circumstance too incredible to be 
 believed. Upon this ground alone, therefore, I think we are 
 warranted in concluding that this formula is erroneous, and 
 similarly of its two companions. But upon these formulae Mr. 
 Stokes rests his whole theory of Fluid Motion, and also that of 
 the Motion of Elastic Fluids. It follows therefore that both 
 those theories must fall to the ground. 
 
 London, Nov. 5th, 1847. 
 
( 127 
 
 MR. O'BRIEN'S THEORY.* 
 
 THE theory I now propose to consider contains a singular 
 mixture of discrimination and error. Whenever Mr. O'Brien 
 finds the views of preceding writers inimical to his own, he 
 seems to have found little difficulty in demonstrating the ab- 
 surdity of the former. Nevertheless his own theory is open 
 to objections just as fatal as those of his predecessors. It is 
 not my intention to consider this theory in detail, but I shall 
 content myself with pointing out in it such an amount of fallacy 
 as may serve to shew that the many important results which 
 Mr. O'Brien claims to have discovered do not possess a 
 shadow of value. 
 
 Mr. O'Brien forms the general equations of undulatory 
 motion pretty much in the same way as Mr. Kelland. In 
 the simplification of these equations, however, he adopts an 
 entirely different process at the same time that he makes use of 
 the same leading principle, viz. that the displacement of any 
 two particles is very small compared with the distance between 
 them. " This assumption," Mr. O'Brien tells us, so far as he is 
 " aware, has been made by every one who has written on the 
 subject of undulations, whether in the case of light or sound." 
 
 Now in that theory of Sound which is attributed to Bernoulli, 
 and which is generally looked upon as one of the greatest 
 triumphs of modern analysis as applied to physical science, 
 and as such is not to be lost sight of in the consideration 
 of this matter, I am not aware that any such assumption as 
 Mr. O'Brien here speaks of has been made : nor do I see very 
 clearly how any such could be made having reference to the 
 principle of investigation there adopted since in that theory 
 
 * On the Propagation of Luminous Waves in the Interior of Transparent 
 Bodies. By the Rev. M. O'Brien, M.A., late Fellow of Caius College, 
 Cambridge. Camb. Phil. Trans. Vol. vn. part iii. p. 397. 
 
MR. OBRIENS THEORY. 
 
 we do not profess to deal with particles separated by intervals, 
 but with geometrical elements in immediate contiguity with 
 each other. To those indeed who, like Mr. O'Brien, look 
 upon it as a certain fact that all matter consists of spherical 
 particles separated by finite intervals, it is possible that even 
 the Bernouillian method implies the truth of some such prin- 
 ciple as he here assumes : but I must protest against citing 
 those great writers, to whom we owe everything of value in 
 connexion with this subject, as authorities for assumptions 
 which they never recognised, and in the embarrassment of 
 which they had too much discrimination to involve themselves. 
 
 Mr. O'Brien indeed seems to think that " this is no assump- 
 tion in the present investigation," and gives several reasons 
 in support of this view ; amongst others, the distance of the 
 sun from the earth, and the law of the intensity varying as the 
 inverse square of the distance an argument which might 
 possess some weight if there were no such thing as candlelight. 
 It is suggested to us also that " if this be not true, the principle 
 of the superposition of small motions cannot be applied to the 
 ethereal undulations, and the whole undulatory theory must fall 
 to the ground;" a circumstance which, it may be as well to 
 hint to Mr. O'Brien, would rather operate against his assumption 
 than in its favour, in the consideration of some of his readers. 
 
 Allowing to pass, however, this and other minor matters, 
 I shall now advert to what I regard as the cardinal error of 
 Mr. O'Brien's theory. Having arrived at the equation 
 
 m dr z 2 \da? dy* dz* 
 
 2 \dy 2 dz* dxdy dxdz 
 + differential coefficients of the fourth and higher order in which 
 mM= 2w {/(r) &r}, 
 
 mN = 2m [-/'O 
 
 mP = Sm (i/ f (: 
 
 lr y 
 
MR. O'BRIEN'S THEORY. 
 
 and assuming that " af3y will be of some such form as 
 c sin - (yt - px - qy - sz)" we are indulged with the follow- 
 ing remarks. 
 
 " We know that the molecular forces of all ordinary bodies 
 are quite insensible at the smallest distances that can be mea- 
 sured ; and therefore they must be so at the distance A, which, 
 though small, is measurable : hence we may suppose that the 
 particles of ether exercise no sensible force at the distance A; 
 
 r* 
 
 and this being the case, S/(r) ^ c must be extremely small 
 
 A. 
 
 r z 
 
 compared with ^f(r) ^ c, since r is the distance between two 
 A 
 
 particles which exercise a sensible force upon each other." 
 
 The contrivances by which different writers on the hypothesis 
 of finite intervals delude themselves into the belief that they 
 lave simplified the equations of motion are remarkable, and 
 Mr. O'Brien's contrivance is no exception to the rule ' Because 
 we may neglect the force due to particles whose distance is 
 considerable, therefore we may neglect any function of that 
 distance.' Thus, supposing the law of force to be (what, by 
 the way, Mr. O'Brien does not suppose it) that of the inverse 
 
 YYL 
 
 square, if we neglect S -3 , where r is equal to or greater 
 
 than A, according to Mr. O'Brien we may, under the same 
 
 , m r 2 _ m r* _ m r 
 
 circumstances, neglect S z ^- 3 , 2* ^ ^ 4 , . . . . S : a con- 
 clusion, I must be permitted to suggest, which is so far from 
 obvious, that any theory resting upon it must be entirefy 
 rejected. 
 
 I think it proper to observe, that in a subsequent part of 
 his paper Mr. O'Brien professes to prove " two remarkable 
 and very general theorems respecting transverse and normal 
 vibrations," which he believes " capable of very important ap- 
 plications"; and which really would be so if they were true, 
 mt they in fact are grounded upon an entire misconception 
 f the nature of transversal undulation. 
 
130 MR. O'BRIEN'S THEORY. 
 
 " To shew that the condition of the vibrations being trans- 
 versal is da d $ ^ 
 
 __ -| -- - -f c= 0. 
 
 dx dy dz 
 
 " Let the equation to any surface in which all the particles 
 are in the same phase of vibration (i.e. any wave surface) be 
 
 where u is a parameter which does not vary as long as xyz 
 belong to the same wave surface, but is different for different 
 wave surfaces : for example, if the wave surface be spherical 
 we may take u to represent the radius, or if it be plane 
 we may take u to represent the perpendicular upon it from 
 the origin. In the former case the above equation would be 
 
 u z , 
 
 and in the latter px + qy + sz = u, 
 
 where p, q, s represent the cosines of the angles which the 
 surface makes with the coordinate planes. It is evident that 
 in both these cases u varies only when we pass from one wave 
 surface to another. 
 
 " Now, supposing that t is constant, the phase of vibration 
 or, which is the same thing, a, |3, 7, can vary only when 
 u varies ; hence a, |3, 7 must be functions of u and t only, &c. 
 
 It is evident that Mr. O'Brien has not studied the geometri 
 conditions which govern undulations consisting of transve 
 vibrations. His definition of a wave surface is evidently this 
 that it is a surface for every point in which a, |3, 7 are invariable 
 
 Now let a, b, c be the coordinates of any particle on th 
 surface of a sphere whose equation is 
 
 a* + b z + c* = r z ; 
 
 and let a + x, b + y, c -f z be the coordinates of any other particl 
 on the same surface. Then if each particle receive the sam 
 small transversal displacement, whose resolved parts are repr 
 sented by 0)87, since each particle will still lie on the surfa 
 of the sphere, we shall have 
 
 (a 4 of + (4 4 ft* + (c 4 y) 2 = r\ 
 (a 4 a + xj + (b + + yf 4 (c 4 7 4 z) ! = r\ 
 
MR. O'BRIEN'S THEORY. 131 
 
 and we therefore have the equation of condition, 
 
 2 + y z + z* + 2 (a + a) x + 2 (b + /3) y + 2 (c + y) 2 = ... (1); 
 
 from which it is evident that all the particles on the surface 
 of the sphere are not capable of receiving the same displace- 
 ment^ but only such as lie in the intersection of the curve 
 represented by the equation (1) with the sphere. We may 
 ,hus see that it is only in the case of a plane wave that all 
 the particles in the wave surface can receive the same dis- 
 )lacement, and Mr. O'Brien's " simplification" therefore cannot 
 >e entertained. 
 
 In the report of the proceedings of the Cambridge Philo- 
 
 ophical Society, contained in the number of the Philosophical 
 
 Magazine for the present month, is inserted a notice of a paper 
 
 f Mr. O'Brien's recently read to the Society, one of the objects 
 
 )f which is " to shew that the equations of vibratory motion of 
 
 a crystallised or uncrystallised medium may be obtained in their 
 
 most general form, and very simply, without making any as- 
 
 jumption as to the nature of the molecular forces." 
 
 It does not appear whether Mr. O'Brien considers his new 
 
 heory as an improvement upon or a variation from his former 
 
 >ne ; and of its merits I must confess myself quite incapable 
 
 >f judging, from the imperfect notice which is contained in the 
 
 eport in question : nevertheless, my conviction of the impos- 
 
 ibility of arriving at any practical conclusion by means of a 
 
 direct attack on the general equations of motion is such, that 
 
 ! am ready to pledge myself to do one of three things namely, 
 
 o shew the fallacy of the theory ; to shew that it rests upon 
 
 some unauthorised assumption ; or to confess myself in error.* 
 
 London, Nov. 21, 1847. 
 
 * It appears to have escaped ray notice during the cursory examination 
 D Mr. O'Brien's theory, upon which the above criticism was founded, that 
 Mfr. O'Brien makes the same false approximation which occurs in Mr. Kelland's 
 fast theories, vide ante, p. 84. I should otherwise not have entered so fully into 
 ;he consideration of the former gentleman's paper. 
 
 March 24, 1848. 
 
 K2 
 
( 132 ) 
 
 A NEW THEORY OF TRANSVERSAL VIBRATION 
 
 WHATEVER difference of opinion may be entertained wit 
 regard to the validity of the processes adopted in the differen 
 theories of transversal vibration which have been hitherto pro 
 posed, I think there can be none as to their entire failur 
 to familiarize the mind with the peculiar mechanism by whic 
 that remarkable kind of motion is produced. With regard t 
 other great mechanical theories the case is different. Th 
 general theory of the planetary motions the mode in whic 
 a central force, combined with a transverse velocity, leads t 
 the production of a re-entering orbit the general theory o 
 sound the manner in which an agitation of a small portioi 
 of the elastic medium may be propagated to the remotes 
 distances these are ideas which can be rendered familiar t 
 any person of ordinary understanding, without the aid of a lin 
 of analysis. But if any person were to attempt, from the exist 
 ing theories upon the subject, to frame a short explanatior 
 of the cause of transversal vibrations, I apprehend he woulc 
 experience considerable difficulty : and granting that he migh 
 be successful in establishing the possibility of that kind o 
 motion, I apprehend that he would wholly fail to convey tc 
 the mind that degree of conviction and acquiescence which 
 in the case of elementary principles, can only result from the 
 simplest and most familiar considerations. 
 
 It is precisely to this point that I am desirous of directing 
 the attention of my readers. It is my object to make th< 
 idea of transversal vibration as familiar as that of the vibratioi 
 which occurs in sound; to shew that that species of motioi 
 is not so completely sui generis as to make it necessary foi 
 us, in order to explain it, to change all our ideas respecting 
 
A NEW THEORY OF TRANSVERSAL VIBRATION. 133 
 
 the nature of the ethereal medium, or to have recourse, as 
 the sole means for its elucidation, to doubtful deductions from 
 obscure and insignificant equations^ It is my object to shew 
 that, so far from being of a character altogether novel and 
 extraordinary, it is what, upon the occurrence of a very simple 
 modification of the circumstances of direct vibration, would 
 naturally result to shew that it is not merely what may, but 
 what might be expected to take place ; and if in the accom- 
 plishment of this aim I shall suggest no considerations but 
 such as are of the simplest character, and such as would 
 require no great stretch of thought to devise, I trust that 
 the results which will be obtained will not be thought alto- 
 gether devoid of interest or importance. 
 
 I conceive that much of the error and confusion of thought 
 which prevail with regard to this subject, have arisen from 
 the non-recognition of the geometrical conditions which govern 
 undulations consisting of transversal vibrations. What is meant 
 by a spheroidal undulation in which the vibrations are trans- 
 versal? In the case of normal vibrations the definition of a 
 wave is easy : it is a portion of ether which may be divided 
 into concentric laminaB of equal thickness, such that the motion 
 of each small element of any particular lamina is exactly the 
 same. 
 
 And in the case of transversal vibrations, if the wave be 
 plane, the same definition will apply; for, however incon- 
 ceivable the mechanism by which parallel laminae of ether can 
 be made to move upon one another in a direction transverse 
 to their common normal, there seems to be no geometrical 
 difficulty in the way of our adopting such an idea. But when 
 we come to the case of spherical or spheroidal undulations, 
 the matter is altogether different. It is geometrically impossible 
 to devise a standard according to which the motion in any 
 spherical or spheroidal lamina can be said to be the same 
 throughout.* If we consider the spheroidal wave in uniaxal 
 crystals, in which, according to the received theory, the motion 
 takes place in planes passing through the axis of the spheroid, 
 
 * The fallacy of Mr. O'Brien's attempt in this respect has been already 
 pointed out. 
 
134 A NEW THEORY OF TRANSVERSAL VIBRATION. 
 
 a very little reflection will serve to convince us of the truth 
 of this statement. For even if we grant which, if we consider 
 the medium to be of the same nature as that of sound, would 
 be impossible, and in any case is incredible that the amount 
 of motion of each particle is the same; and if we assume 
 the pole of the spheroid as the standard of direction which 
 is the most advantageous supposition that can be made still, 
 for the particles immediately about the pole, even this standard 
 would fail, since, when any particle arrived at the pole, it 
 would be impossible to say what course it should take. And, 
 although it may to some appear of small importance what 
 becomes of a few obscure particles at the pole of the wave, 
 provided the law hold generally, yet, to those who are accus- 
 tomed to note the admirable exactitude of the operations of 
 nature, the absolute failure of the rule, even in this extreme 
 case, will afford a convincing proof of the necessity for its 
 modification. 
 
 Although, however, it is impossible to ascribe to spherical 
 and similar waves, in which the vibrations are transversal, that 
 absolute uniformity of motion which may exist in the case 
 of normal vibrations, we may still assign to them such a dis- 
 tinctive and recognizable character as will enable us to reason 
 with respect to them with clearness and precision. Recurring 
 to the case of the spheroidal wave in uniaxal crystals, we 
 are instructed by the existing theory to consider the vibrations 
 in this case as taking place in planes passing through the axis 
 of the spheroid (i. e, along meridians passing through the poles), 
 and no further information is given to us respecting the dis- 
 tribution of the motion along the surface of the wave. If, 
 however, we assume the motion to be symmetrical with respect 
 to the axis, i.e. that the motion throughout any section made 
 by a plane perpendicular to the axis is the same; and if we 
 further suppose that all the particles between the equator and 
 the pole are moving in the same direction, i.e. that they are 
 all moving towards or all moving from the pole, a very im- 
 portant consequence will follow. 
 
 It is evident that, if all the particles are moving from the 
 equator to the poles, there must be a suffusion of particles 
 
A NEW THEORY OF TRANSVERSAL VIBRATION. 135 
 
 upon the latter; or in that part of the wave we shall have 
 condensation, at the same time that the efflux of particles from 
 the equator must occasion a rarefaction at the equator. This 
 state of things evidently can only be temporary. The con- 
 densation at the pole will tend to relieve itself by the ether 
 in that part flowing along the surface of the wave towards the 
 equator, where the density is less than its own; so that in 
 the end we shall have condensation at the equator where 
 before we had rarefaction, and rarefaction at the pole where 
 before we had condensation. We shall thus have a recurrence 
 of the original motion, the condensation at the equator relieving 
 itself by producing a motion towards the poles, and so repeating 
 the condensation there. It is this alternate motion from the 
 equator to the pole, and vice versa, which, speaking generally, 
 I consider to constitute transversal vibration. 
 
 To descend, however, more into particulars. Suppose our 
 wave to consist of a single condensation, followed by a single 
 rarefaction. It is obvious on this hypothesis that, if the density 
 of each spheroidal element were uniform, there could be none 
 of the lateral motion along the surface of which we have been 
 speaking. For, if we consider the case of a single pulse 
 uniform throughout its whole extent, it is obvious that the 
 effect of the condensed portion of the wave must be, 1st, To 
 flow forwards, so as to produce condensation in front where the 
 density is the mean density : 2nd, To flow backwards, so as 
 to fill up the hollow occasioned by the rarefaction behind it; 
 and this would be the whole motion which would occur.* 
 But, if we make the hypothesis that the intensity of the wave 
 is not uniform throughout, i.e. if we suppose that at the pole, 
 for example, the condensed portion is more condensed and 
 the rarefied portion more rarefied than anywhere else, we 
 shall have precisely that kind of motion of which we have 
 been speaking. The condensed portion of the wave, besides 
 the motion in front and rear which it will have, as in the 
 previous case, will tend to flow laterally from the pole towards 
 the equator, since the density in this portion of the wave is 
 
 * I consider the above to be the rationale of the undulations which 
 occur in sound. 
 
136 A NEW THEORY OF TRANSVERSAL VIBRATION. 
 
 less at the equator than at the pole ; at the same time that, 
 in the rarefied portion of the wave, the effect will be exactly 
 the reverse, since in that portion the density is greater at the 
 equator than at the pole. These effects, however, will by-and- 
 by be inverted. The particles which were at first condensed 
 will, by the progress of the wave, become rarefied, and there- 
 fore instead of moving from the pole will tend to move towards 
 that point. Thus, as in the first half of the motion each 
 element has a tendency to move forwards, in the direction 
 of transmission, and laterally away from the pole, in the second 
 half it will tend to move backwards towards the centre, and 
 laterally towards the pole. Thus, corresponding to the dif- 
 ference of phase as regards the direct vibration, we have a 
 difference of phase as regards the transversal motion, or a direct 
 and a transversal motion will be communicated simultaneously, 
 will be contemporaneous in duration, and will be propagated 
 together with the same velocity.* 
 
 I have above supposed the condensation of the condensed 
 half of the wave, and the rarefaction of the rarefied portion 
 to be respectively greatest at the pole; and further, that in 
 the first half of the wave the condensation diminishes, cceteris 
 paribus, as we recede from the pole, and similarly of the 
 rarefaction in the rarefied portion. Now this is equivalent 
 to considering that the intensity is a maximum at the pde, 
 
 * As the considerations above suggested, though simple, may not be 
 familiar, and as the essence of my theory is comprised in the last few sen- 
 tences, I must trouble such of my readers as may be desirous of making 
 themselves masters of it, to refer to fig. (7) at the end of this work. 
 
 DdfF represents a section of the wave made by a plane passing through 
 the axis AO, DdeE being the condensed, EefF the rarefied portion. 
 
 If we have a single wave only, the condensation in DdeE will tend to 
 relieve itself by moving 1st, Forwards from the centre, since the ether in 
 front has the density of equilibrium ; 2nd, Backwards towards the centre, 
 since the density behind is less than the density of equilibrium, the ether 
 there being rarefied. Now if the density throughout any spheroidal element 
 of the wave be uniform, this is the whole motion which can occur. But 
 if we suppose the density in DdeE to be greatest, cceteris paribus, along 
 the line AB, and to diminish gradually as we recede from that line along 
 the surface of the wave towards D and d, besides the motion backwards and 
 forwards which it will have, as in the preceding case, there, will likewise 
 be a lateral motion from AB towards DE and de ; and if we further suppose 
 that the rarefaction in EefF is greatest along BC, and diminishes gradually 
 from that line, there will be a motion from EF and ef towards BC ; so that 
 the lateral motions in the two divisions of the wave will be in opposite 
 directions. 
 
A NEW THEORY OF TRANSVERSAL VIBRATION. 187 
 
 and that it diminishes as we recede from that point; and 
 we thus arrive at the important conclusion, that transversal 
 vibration depends upon variation of intensity in the direct wave. 
 This variation has no necessary effect upon the velocity of the 
 wave (since waves of all intensities may be propagated with 
 the same velocity), and may therefore occur in spherical waves 
 just as readily as in waves of any other form. 
 
 Upon the mode of distribution of the intensity in the se- 
 condary waves depends the nature of the polarization, and this 
 distribution will of course vary according to circumstances. 
 
 When light is reflected from glass at the polarizing angle, 
 we must suppose the secondary waves (which in this case are 
 spherical) to have their positions of maximum intensity at the 
 extremities of some particular diameter (which may be called 
 the polarizing axis), and that the intensity in each hemisphere 
 diminishes gradually in proportion to the distance from those 
 points respectively. It is easy to see how an arrangement 
 of this kind would give rise to a rectilinear vibratory motion 
 in the direction of great circles of the sphere made by planes 
 passing through the polarizing axis. This axis must evidently 
 be either parallel or perpendicular to the plane of incidence, 
 according as the vibration is supposed to take place in that 
 plane or perpendicular to it. On the latter hypothesis the 
 position of the axis would be absolutely denned, but in the 
 former it would be still indefinite, as almost any position of 
 the plane of incidence would answer the conditions of the 
 problem : nevertheless, to fix our ideas I would suggest that, 
 if the latter hypothesis as to the direction of the vibration 
 be adopted, we should consider the axis to be parallel to the 
 front of the general reflected wave. 
 
 When light suffers ordinary refraction, or reflexion otherwise 
 than at the polarizing angle, since in neither of these cases 
 is the vibration rectilinear, it is obvious that the distribution 
 of intensity which we have assumed to obtain in the case 
 last considered will not account for the phenomena; inasmuch 
 as that distribution, as we have already seen, necessarily 
 implies rectilinear vibration. If, however, we assume a double 
 distribution of intensity about two axes at right angles to each 
 
138 A NEW THEORY OF TRANSVERSAL VIBRATION. 
 
 other, each of which, if it existed separately, would be of the 
 same character as that supposed in the last case, it is obvious 
 that the imperfect polarization which obtains in the present 
 instance may be completely accounted for. It must be ob- 
 served, however, that although the refracted and reflected waves 
 on this hypothesis, each undergo, as it were, two simultaneous 
 and opposite polarizations, these polarizations will be unequal 
 in degree. Thus assuming the polarizing axes in the two 
 waves to be respectively parallel and perpendicular to the 
 plane of incidence, and assuming the vibration in a wave re- 
 flected at the polarizing angle to take place in that plane, the 
 polarization of any other reflected wave with respect to the 
 axis in the plane of incidence will be greater than its po- 
 larization with respect to the axis in the opposite plane ; while 
 with the refracted waves the case will be exactly the reverse : 
 and as the polarization of the reflected wave with respect to 
 the axis in the plane of incidence goes on diminishing as the 
 angle of incidence deviates more and more from the polarizing 
 angle, so the polarization of the refracted wave with respect 
 to the same axis, under the same circumstances, will continually 
 increase ; while with regard to the polarization of the two waves 
 respectively in the plane perpendicular to the plane of incidence, 
 the effects will be precisely opposite. Thus taking the two 
 waves together we see that, speaking generally, the amount 
 of action is the same in every direction, or on the whole an 
 equilibrium of action is maintained. I shall hereafter take 
 occasion to advert to this circumstance. 
 
 If we now turn our attention to the ordinary wave in uniaxal 
 crystals, according to the received theory, we are informed that 
 the motion of each particle in the ordinary (secondary) wave 
 takes place in a section of the sphere made by a plane per- 
 pendicular to the axis of the crystal. It is therefore evident, 
 whatever physical theory be adopted, that if this distribution 
 of motion obtains, the mechanical action in the ordinary must 
 be of a totally different character from that which occurs in the 
 extraordinary wave ; a circumstance which, so far as I am 
 aware, there is no experimental evidence to warrant us in 
 believing. There would be no difficulty, however, in applying 
 
A NEW THEORY OF TRANSVERSAL VIBRATION. 139 
 
 the principles which I have above suggested to account for the 
 kind of motion under consideration, it being in fact merely 
 requisite for this purpose to suppose that the intensity along 
 any section of the sphere made by a plane through the axis 
 of the crystal is uniform. At the same time I confess that 
 I am unwilling to acquiesce in any hypothesis which would 
 recognize a specific difference between the ordinary and extra- 
 ordinary waves, other than that of form, and I am therefore 
 strongly inclined to regard the polarization of the ordinary as 
 the same in character as that which occurs in the extraordinary 
 wave, and in the waves which arise when light is reflected from 
 glass at the polarizing angle ; or, in other words, that it must be 
 attributed to a symmetrical distribution of intensity about a 
 single axis perpendicular to the axis of the extraordinary wave. 
 A very little consideration, however, will shew that, if this 
 hypothesis be adopted, it will be impossible to consider the axes 
 of both the waves as fixed lines within the crystal ; since if they 
 were so, there would be certain directions in which the trans- 
 mitted (general) waves would be polarized in the same plane ; 
 as, for instance, when the direction of the transmitted waves 
 was in the plane passing through the axes of polarization of the 
 two waves. This consideration leads me to infer that the axis 
 of polarization of at least one of the two waves must change its 
 position with the angle of incidence. If only one of the axes 
 was variable, we should naturally conclude that the axis of the 
 spheroidal wave, which, according to the received theory, coin- 
 cides with the axis of form, would be that which was fixed. 
 I confess, however, that, consistently with the principles above 
 laid down, I am unable to discern any reason why the polarizing 
 axis of the spheroidal wave should be in any way more fixed 
 than that of its spherical companion. For although at first 
 sight it might appear most probable that the spheroidal wave 
 should be polarized with respect to its axis of form ; yet if we 
 consider that the form of the wave depends entirely on the 
 velocity, which is absolutely independent of the intensity, to 
 which alone, according to the above theory, the polarization 
 is attributable, it is obvious that, consistently with that theory, 
 there is no reason to expect that the spheroidal wave should 
 
140 A NEW THEORY OF TRANSVERSAL VIBRATION. 
 
 be polarized with respect to its axis of form rather than any 
 other diameter.* Moreover, there would be some difficulty in 
 the application of the above principles if it were supposed that 
 the extraordinary (general) wave could under any circumstances 
 meet its secondary at the extremity of the polarizing axis of the 
 latter. On these several accounts, therefore, I am led to the 
 conclusion, that the polarizing axes of the two waves are alike 
 variable in position according to the direction of the incident 
 light, but that they are always at right angles to each other. 
 In order to fix our ideas I shall make the further assumption 
 that the axes are respectively parallel to the directions of vibra- 
 tion in the general waves. 
 
 It will thus be seen that, in opposition to the received 
 theory, I do not conceive double refraction to depend in any 
 degree upon the direction of vibration. I conceive the opposite 
 polarization, which we find to exist in the two waves, to be the 
 consequence, not the cause of double refraction. If by any 
 means two waves are called into play, it appears to me pretty 
 much a matter of course that they should be oppositely polarized. 
 The principle of the equilibrium of action which we have seen 
 to hold in the case of the reflected and refracted rays in 
 ordinary media would lead us to anticipate that such would 
 be the case. How two waves come to be propagated is indeed 
 a dark enigma, and one which I pretend not to unravel. 
 Sometimes it has occurred to me that the ether within the 
 crystal and the molecules of which it is composed act as 
 separate conductors of the incident vibration, which is thus 
 separated into two vibrations propagated with different veloci- 
 ties ; while in those bodies which possess only single refrac- 
 tion, either the two waves agree in velocity, or one only is 
 transmitted. One thing however is certain, that the scheme 
 upon which the received theory is conducted of ascribing the 
 variation of velocity to the different elasticity of the medium 
 according to the direction in which the vibrations take place 
 is entirely erroneous, since the occurrence of such a property 
 
 * If the polarization be supposed to be with, respect to any diameter, the 
 lines of equal intensity will consist of elliptic sections of the spheroid made by 
 planes parallel to the tangent plane at the extremity of the polarizing axis. 
 
A NEW THEORY OF TRANSVERSAL VIBRATION. 141 
 
 within the crystal would produce effects entirely different from 
 those which we find to occur. This may readily be shewn. 
 It is clear that at any moment we shall have within the 
 crystal a series of concentric similar spheroidal laminae or waves 
 diverging from each point on the surface, the breadth of any 
 one of which at any point will vary as the diameter passing 
 through that point. Hence the breadth of the m th wave at the 
 equator will be to that of the same wave at the pole, as the 
 equatoreal radius is to the polar radius. But the breadth of 
 the m th spherical wave is the same throughout and equal to the 
 polar breadth of the m th spheroidal wave. Hence, comparing 
 the velocities in the ordinary and extraordinary waves along 
 any diameter perpendicular to the axis of the crystal, it is 
 obvious that, supposing the velocity of transmission to depend 
 solely on the length of the wave* and to be irrespective of the 
 direction of vibration, the velocities will be actually such as we 
 find them, i. e. in the ratio of the greatest and least radii of the 
 spheroid : and it follows that, if the velocity depended on the 
 direction of vibration, as well as on the length of a wave, the 
 velocities would not be in the ratio of those radii. This con- 
 clusion is so obvious that I cannot but feel some surprise at its 
 not having been adverted to. My attention was first directed 
 to it whilst following out to their consequences the principles 
 I have above endeavoured to unfold. 
 
 It results, therefore, that the distinctive character of the two 
 waves is impressed upon them immediately upon entering the 
 crystal, and that once within it, a wave of a given type will be 
 propagated with a velocity which is quite independent of the 
 direction of its vibration. How two waves are called into play, 
 as I have stated before, I do not profess to explain ; but after 
 all, this may be a less anomalous result than is generally sup- 
 posed. How does it happen that in double refraction each 
 wave is rectilinearly polarized, while in ordinary refraction we 
 have always partial or elliptical polarization? We have seen 
 that the partial polarization which occurs in ordinary refraction 
 may be resolved into two rectilinear polarizations with respect 
 to axes at right angles to each other. Does not this seem to 
 imply that in ordinary as well as in crystalline refraction two 
 
142 A NEW THEORY OF TRANSVERSAL VIBRATION. 
 
 waves in reality exist, which coincide indeed in form and mag- 
 nitude, but still give indications of their separate origin by the 
 fact of their opposite polarization ? It may be considered, to 
 diminish the force of this analogy, that in ordinary refraction 
 the polarization with respect to the two waves is unequal, 
 while in double refraction the two waves are of equal intensity. 
 It may be observed, however, that the inequality of the 
 polarizations in ordinary refraction only exists when the 
 polarization is referred to axes respectively parallel and per- 
 pendicular to the plane of incidence. The wave, however, 
 may be split into two of equal intensity though differing in 
 phase, and polarized with respect to axes lying in a plane per- 
 pendicular to the plane of incidence and inclined at angles of 
 + 45 and - 45, respectively, to that plane. If x and y be the 
 coordinates of a particle, where x is measured along the inter- 
 section of the plane of the front with the plane of incidence ; 
 and x, y, the coordinates of the same particle referred to rect- 
 angular axes in the plane of the front inclined at the above 
 angles respectively to the plane of incidence ; if 
 
 2?r / * i 27r f * 
 
 x = a cos (vt - x), y = o sin (vt - x\ 
 
 A A 
 
 we shall have 
 
 x l = V2 . I a cos (vt - x) - b sin (vt - x} 
 t A A 
 
 y l = V2 . < a cos (vt - x) + b sin ~ (vt - x} I , 
 
 and these last will correspond to two waves of equal intensity 
 consisting of rectilinear transversal vibrations, the polarization 
 being referred to the new axes respectively.* 
 
 It will be observed that in the above theory the secondary 
 waves of which I have treated are considered to have an actual 
 existence, and not to be of that ideal character merely which 
 some writers would appear to assign to them. This was the 
 
 * In this case, as in those which have gone before, I consider the axes of 
 polarization of the secondary waves to be parallel to the front of the general 
 reflected or refracted wave. The quantities a and b will vary with the angle 
 of incidence. 
 
A NEW THEORY OF TRANSVERSAL VIBRATION. 143 
 
 view of the subject taken by the illustrious founder of the 
 undulatory theory, who conceived the luminous waves to be 
 broken up at every encounter with material particles. It was 
 upon this hypothesis that he established the laws of ordinary 
 reflexion and refraction, and to his modification of the principle 
 as adopted in the two latter cases is to be attributed the greatest 
 triumph of the undulatory theory the discovery of the law 
 of refraction of the extraordinary wave in uniaxal crystals. 
 I might add that it has been entirely owing to my having 
 proceeded upon that hypothesis that I have arrived at the 
 theory of transversal vibration contained in this paper ; respect- 
 ing which I must be permitted to observe that, whatever 
 opinion may be formed as to whether light is really due to 
 transversal vibrations, or, granting such to be the case, as 
 to whether the transversal vibrations which occur are of the 
 kind I have described, it is the only theory of transversal 
 vibration which has been hitherto proposed, which is in any 
 way intelligible. And lastly, it has been by proceeding upon 
 that hypothesis that I have pointed out for the first time the 
 'certain fact that the variation of velocity in the extraordinary 
 wave is not due to its polarization. It will not, therefore, be 
 on slight grounds that I shall give up the actual existence of 
 secondary waves called into play when light is incident upon 
 the surface of a transparent body. I am, however, far from 
 thinking that in applying the principle of secondary waves to 
 account for transversal vibration, considerable modifications of 
 the above theory may not be satisfactorily made, and I am even 
 of opinion that the same mechanical principles may be applied 
 without the aid of the principle of secondary waves. 
 
 Thus it may not be necessary to consider the motion through- 
 out each spheroidal lamina of the extraordinary wave to be in 
 the same direction, i.e. altogether towards or altogether from 
 the pole. It is sufficient that the direction of motion in each 
 section perpendicular to the axis should be the same ; so that 
 we might in fact have a series of annular transversal undulations 
 radiating from the pole of the spheroid along the surface of the 
 wave. This idea may be pictured to the mind by supposing 
 a ripple to occur on the surface of a soap bubble, descending 
 
144 A NEW THEORY OF TRANSVERSAL VIBRATION. 
 
 from the highest point, and resembling the ripple which occurs 
 when a stone is thrown into water, with the difference of taking 
 place on a spherical instead of a plane surface. Of course I do 
 not mean that the surface of the wave is elevated and depressed 
 in the case of an ethereal transversal undulation in the same 
 way as the surface of the water, but merely that an alternate 
 condensation and rarefaction takes place in each indefinitely 
 thin lamina resembling or corresponding to the alternate eleva- 
 tion and depression of the fluid surface. 
 
 The fig. (8) represents rudely a section of a wave in which 
 we should have both direct and transversal vibration, without 
 the aid of the machinery of secondary waves ; the shaded parts 
 representing the condensations, and the white spaces the rare- 
 factions. It is evident that one effect of the condensations at A 
 and B must be to fill up the hollow between them at d, so as 
 ultimately to produce condensation there, to which purpose the 
 condensation at D will also contribute. On the other hand, we 
 shall at the same time have a rarefaction produced at A and B ; 
 and we shall have in like manner a condensation at e, and a 
 rarefaction at C; so that after the lapse of a certain period the 
 first half of the wave, AC, will present the same appearance as 
 the second, DE, actually presents. It is easy to see how, under 
 the above circumstances, a transversal oscillatory motion would 
 be maintained and propagated. 
 
 It would not be difficult to multiply examples of the appli- 
 cation of the above principles. Those which have been already 
 given, however, may suffice. In the foregoing paper it has not 
 been so much my object to propound a settled theory of polari- 
 zation, as to put the student in possession of a novel and 
 powerful machinery, which I believe to be capable of the most 
 extensive application in the theory of optics. 
 
 Liverpool, January 17, 1848, 
 
ON THE SUPPOSED INCOMPATIBILITY OF 
 
 TRANSVERSAL VIBRATION WITH THE SONOROUS MEDIUM. 
 
 THE substance of the preceding theory was communicated to 
 the Cambridge Philosophical Society at a meeting of that body, 
 which took place in the spring of last year. It was upon the 
 same occasion suggested that the impossibility of transversal 
 vibration in a medium resembling that of sound had been 
 established by Poisson from the consideration of the general 
 equations of fluid motion, and by the kindness of Mr. Hopkins 
 I was referred to several memoirs of that eminent author in 
 which this point is attempted to be proved. Having a pretty 
 strong conviction of the impossibility of deriving any useful 
 result from the general equations of fluid motion, I cannot say 
 that I felt much alarm at such an intimation, and the result 
 has shewn that I had good ground for my confidence. 
 
 In the Journal de VEcole Poly technique, tome vn. p. 334, 
 M. Poisson commences his attack upon the theory of fluid 
 motion by a transformation to polar coordinates of the well- 
 known equation 
 
 If z = ru, x = r V(l - u*) cos co, y = r \/(l - u*) sin a>, 
 
 this becomes 
 
 , , 2 d.r$ 
 
 ' ( ~ U} ~ 
 
 d\r<j> 
 
 dt 2 ' I dr* r*du r z (1 - u*) dv? 
 
 Of this equation M. Poisson obtains an exact integral; but 
 one which, as may readily be surmised, is of no practical 
 utility : although the author appears to have thought that it 
 
 L 
 
146 ON THE SUPPOSED INCOMPATIBILITY OF 
 
 enabled him to establish as a general fact the uniformity of the 
 velocity of sound in space. This however, which, so far as 
 I am aware, is the sole result obtained by our author indepen- 
 dently of any special assumption as to the nature of the vibra- 
 tion, is erroneous, as will be presently shewn. After attempting 
 to establish this proposition, M. Poisson immediately deserts 
 the general integral, and proceeds to integrate the equation by 
 approximation. The formula he assumes for this purpose is 
 the following : 
 
 x, u, eo) + &c., 
 
 r j \~> > "v ' i"vi\~> w ) ~j ' 2 
 
 where x = r-at; or rather, he takes for the complete value 
 of T(J) the sum of the above series and an exactly similar series 
 in which z = r + at: but for the sake of simplicity I shall confine 
 myself to the above, as the consideration of the second series 
 introduces no new feature. He then proceeds to determine 
 the values of the functions f^ / 2 . . . . according to the ordinary 
 principles for ascertaining indeterminate coefficients, and he 
 so arrives at an equation similar to the equation of Laplace's 
 coefficients. 
 
 Now it is manifest that the above form of the integral has 
 been suggested by the integral of the ordinary equation for the 
 motion of sound in a cylindrical pipe; and to this mode of 
 integration, considered as a tentative method, no objection can 
 be made ; but it is obvious that no very general or decisive 
 conclusions as to the nature of the laws which govern sonorous 
 vibrations can be arrived at in this manner. 
 
 M. Poisson next shews that according to the above integral 
 the velocity of propagation will be uniform ; and he then pro- 
 ceeds to determine the important point of the direction of the 
 vibration, and it is to this point in particular that I am desirous 
 of directing the attention of my readers. 
 
 If m be the angle which the direction of motion makes with 
 the radius vector, we have 
 
 d$ 
 
 
 \V\dr) \du) r 2 \dco) r 2 (i - w a )J 
 
TRANSVERSAL VIBRATION WITH THE SONOROUS MEDIUM. 147 
 
 Now it is evident that the principal term in the above series, 
 which we have assumed as equivalent to r<t>, is the first, and 
 that when the distance from the origin is very great, we may 
 in fact put that term to represent rfy ; so that we shall have 
 
 = -/O, u> w), 
 
 and if we neglect powers of - above the first, we have 
 
 f 
 
 </0 _ 1 df(x, u, ft)) 
 dr~r dr * 
 
 d$ _ 1 df(x, u, ft>) 
 du r du 
 
 d$ 1 df(x, u, to) 
 dco r dw 
 
 from which it appears that ^ , , are all of the same 
 
 dr du dco 
 
 order of magnitude. But in the denominator of cos m the two 
 
 latter quantities , ^- are divided bv r 2 ; hence when r is 
 du dm 
 
 very great these terms will disappear from the denominator, 
 
 which will reduce itself to -f- , or we shall have cos m = 1, 
 
 dr 
 
 i. e. the direction of vibration will coincide with the radius vector. 
 Such is M. Poisson's proof of this important proposition. 
 
 The result which M. Poisson has here obtained, naturally 
 flows from the particular form assigned to the integral of the 
 equation of motion; so much so, that it might have been 
 predicated d priori that if the integral was of that form, trans- 
 versal vibration would be impossible. 
 
 The integral of the equation is assumed to be 
 
 or, confining ourselves to two dimensions, 
 
 Now it is obvious that if we are to have transversal vibra- 
 tion at all, and at any rate it must be allowed us to begin with, 
 
 L2 
 
148 ON THE SUPPOSED INCOMPATIBILITY OF 
 
 the above must be modified into some such form as the follow- 
 ing: 
 
 -at\ (u-bt)}. 
 
 But if this were the form of the equation, nothing can be more 
 evident than that as the wave advances, i.e. as the time in- 
 creases, the value of u - bt will depend chiefly upon the value 
 of t, since u is confined between the limits + 1 and - 1, whereas 
 t may have any magnitude whatever ; so that, according to this 
 form of the equation, when the distance from the origin is con- 
 siderable, the variations of u will have little or no effect upon 
 the motion, which will depend entirely on the values of r and t. 
 To remedy this defect, we must assume the integral of the 
 form 
 
 r<l>=f{(r-at), (ru - bt)}. 
 
 It will be seen that upon this hypothesis the value of - will 
 
 no longer be of the same order of magnitude as -; for putting 
 
 r - at = x, 
 ru -bt = 5, 
 
 d<h I df u dF 
 
 we have -j 2 - = - + -- , 
 
 ar r ax r as 
 
 d$ _df 
 du ds 
 
 Hence the denominator of cos m will no longer reduce itself to 
 
 simply, and cos (m) will have other values than 1, or the 
 dr 
 
 direction of vibration will not coincide with the radius vector, 
 but may have any other direction. 
 
 Now the form above assumed for r0 cannot, I apprehend, 
 be readily applied in its present generality to satisfy the general 
 equation of fluid motion ; but if we assume 
 
 r0 =/(/ - at) + F(ru - at\ 
 
 this object will be attained, at least when the distance from the 
 origin is considerable. 
 
TRANSVERSAL VIBRATION WITH THE SONOROUS MEDIUM. 149 
 
 The equation of motion in two dimensions reduces to 
 
 df 
 
 and since the first term in the value of r$ is known to satisfy 
 the equation, we may omit the consideration of that term and put 
 
 r$ = F(ru - bt) = F(s) suppose, 
 where s = ru - bt. 
 
 We ha *V* _ rf'JFVefe 
 
 We have __(_ 
 
 dV 
 
 dr* 
 
 dd-u')^ 
 au 
 
 <fo ' dr* 
 
 dF ds ^d* 
 
 = - 2u H- (1 - w 2 ) - - 
 y 2 
 
 dF ^d*F 2 
 
 = - 2w - r + (1 - w 2 ) - r 2 . 
 ds ds* 
 
 Substituting these values in the equation of motion, we get 
 
 2 d z F 2 [ 2 ^ 2 P 2w e^F . 2 . d*F} 
 a hrr = M w ^FT ~ - -j- + (1 - w 2 ) - TT , 
 Js 2 I <& 2 r c?5 x Js 2 ] 
 
 so that the equation will be satisfied provided we neglect 
 
 -- , which is small in comparison with the other terms 
 r ds 
 
 when r is large. 
 
 I consider, therefore, that I have established my point that 
 transversal vibrations may occur in the sonorous medium even 
 when the distance from the origin is very great. 
 
 I believe it will be found that in every instance in which 
 M. Poisson attempts to prove the incompatibility of transversal 
 vibration with the sonorous medium, the form he assigns to his 
 integral is such as to preclude the possibility of that kind of 
 motion when the distance from the origin is considerable. 
 
150 ON THE SUPPOSED INCOMPATIBILITY OF 
 
 I now propose to return to the consideration of M. Poisson's 
 proof of the uniform velocity of sound in space. Multiplying 
 all the terms of the transformed equation (1) by du, da), and 
 then integrating with respect to u and o>, from w = lto = -l, 
 and from w = to CD = 2?r, we shall have 
 
 where R denotes /0 du dco. In fact it is easy to see that the 
 second term of the right-hand member of the equation, when 
 integrated with respect to u, vanishes between the limits u = I , 
 u = - 1 : and the third term vanishes between the limits o> = 2?r, 
 o> = 0, when integrated with respect to . 
 The integral of (2) is 
 
 p _ 
 
 /and F denoting arbitrary functions. Hence 
 
 ^|? = r f (r - at) - f(r - at) + r F' (r + at) - F(r + at), 
 
 ? 
 
 ar F (r + at) - arf (r - at). 
 
 dt 
 
 If the velocity of each particle be resolved parallel to its radius 
 vector, - will represent the sum of such resolved velocities 
 for all the particles of the sphere whose radius is r ; and 
 
 - will be the sum of the condensations of the same par- 
 a 2 dt 
 
 tides. If then in the origin of the motion it is confined within 
 
 the sphere of which the radius is a, -=- and must = when 
 
 dr dt 
 
 t - 0, for all values of r > a ; which will be the ease if Fr 
 and fr vanish for all such values of r. Thus we shall have 
 f(r - at) = whenever r > at + a and F(r + at) = when 
 r > a - at. But, and which is of more consequence than this 
 latter circumstance, we shall have f(r - at) = when r < at. 
 For if we suppose the original motion to be confined within the 
 spherical shell whose greatest and least radii are a and a 
 
TRANSVERSAL VIBRATION WITH THE SONOROUS MEDIUM. 151 
 
 respectively, it is evident that and must vanish when 
 
 t = for all values of r < a, therefore f(r) will be zero for the 
 values of r < a as well as for the values of r > a : therefore 
 f(r - at) = when r < at + a ; and this, however small a' may 
 be ; therefore it will be true when a = 0. Hence it is evident 
 
 that and - will be zero when r > at + a ; at the instant 
 dr dt 
 
 when r = at + a they will take other values, and they will 
 again become zero when r = at. The molecules therefore at 
 the distance r will remain at rest until r = at + a, and they will 
 remain in motion during a time t' such that a (t + t') *= at + a, 
 
 i. e. during a time = - : they will therefore all move almost at 
 
 the same instant, since the duration of their motions will be 
 very short, a being very small with respect to a. Hence, 
 whatever the nature of the original motion, " la vitesse sera la 
 meme sur tous les rayons sonores, 1'onde sonore conservera 
 to uj ours une figure sphe'rique, et la vitesse de cette onde sera 
 constante et egale a a." 
 
 The above demonstration takes for granted the exact point 
 
 which it has for its object to prove. The quantities ^ , &c. 
 
 du 
 
 are evidently functions of the motion. If therefore the dis- 
 turbance be discontinuous, unless its form be spherical, we 
 
 cannot take the integral of such a function as 
 
 r 2 (l - u z ) dm* 
 
 between the limits u = + I, u= - I, o> = 0, o> = 2ir: since that 
 would imply that the motion extended throughout the spherical 
 shell whose radius is r : and therefore, unless we assume the 
 moving mass to be resolvable into spherical concentric elements, 
 the simplification of equation (1), which results from the above 
 method, cannot have place. As some of my readers may be 
 reluctant to believe that so eminent a mathematical writer as 
 Poisson could fall into so decisive an error, and as there may be 
 some vague notion entertained that the above method may be 
 justified upon the principles of discontinuous functions, I shall 
 now proceed to shew how M. Poisson's method may be made 
 to conduct us to an absurd result. 
 
SUPPOSED INCOMPATIBILITY OF TRANSVERSAL VIBRATION. 
 
 M. Poisson (p. 337) admits that it is to be feared that the 
 
 functions may vanish without the condensations and 
 
 dr dt 
 
 velocities at the distance r vanishing, which, says he, would 
 destroy the demonstration. " Mais il est evident que cela ne 
 peut avoir lieu qu'en faisant une supposition particuliere sur la 
 nature de 1'ebranlement primitif, et en placant 1'origine des 
 coordonnees d'une maniere convenable : d'ou il suit qu'en 
 transportant cette origine en un autre point de la portion d* air 
 
 dR dR 
 
 ebranlee, les fonctions et - cesseront de pouvoir etre 
 
 dr at 
 
 nulles, sans que les vitesses et les condensations le soient aussi." 
 Now as the origin may be assumed without reference to the 
 nature of the disturbance, suppose that the latter is in the first 
 instance confined between two- concentric spheres of the respec- 
 tive radii a and a', where it may occupy either the whole or 
 only a small portion of the intermediate space. It is evident 
 from what has gone before that we shall have f(r - at) = 0, 
 except when r lies between at + a and at + a', and we shall 
 have F(r + at) = unless r lies between a - at and a - at; 
 from which it is obvious that the whole motion at any instant 
 must be confined between two spherical shells of equal thick- 
 ness (viz. a - a') : of which the outer surface of the one will be 
 at the distance at + a, and that of the other at a - at from the 
 origin. As soon as we have at a this latter will have ceased 
 to exist, and therefore the whole motion will be represented by 
 the function f t and will be confined between the radii at + a 
 and at + a. If therefore the form of the original disturbance 
 be a sphere of the diameter a - a',, and a is the radius of 
 a sphere having for its centre the origin of coordinates, and 
 which touches- the former at its more remote edge, according to 
 
 the above theory, after the time - has elapsed, we shall have 
 
 a 
 
 the whole motion comprised between the concentric spherical 
 surfaces whose radii are o and a' respectively : which as a 
 general fact may safely be pronounced to be impossible. 
 
 London, April 3, 1848. 
 
( 153 ) 
 
 REVIEW 
 
 OF PROFESSOR CHALLIS'S MORE RECENT SPECULATIONS, 
 
 AND OF OTHERS BY THE SAME AUTHOR 
 
 NOT PREVIOUSLY COMMENTED UPON. 
 
 THE following letter was submitted to Professor Thomson 
 with a view to its insertion in the Cambridge and Dublin 
 Mathematical Journal, but that gentleman declined to pub- 
 lish it, for reasons upon which it is unnecessary here to 
 enter. I have no reason to find fault with a decision com- 
 municated to me with much courtesy, and resting as I 
 conceive upon fair grounds of support. The letter is given 
 as originally submitted (with the exception of one or two 
 unimportant verbal alterations, and one more material one 
 which I have pointed out); as there did not appear to be 
 any adequate object requiring me to undergo the labour of 
 remodelling it. 
 
 To the Editor of the Cambridge and Dublin Mathematical Journal. 
 
 SIR, It is doubtless within the knowledge of many of 
 your readers, that within the last few months some very 
 singular and, if true, very important views have been pro- 
 posed by the present Plumian Professor of Mathematics at 
 Cambridge, with regard to the Theory of Undulations. They 
 are contained in a series of papers which appeared in the 
 Philosophical Magazine, commencing in April last, and for 
 the most part may be classed under the following heads : 
 
 1. That undulations may be propagated in thin cylin- 
 drical filaments, in the direction of the axis of the filament, 
 without extending themselves laterally. 
 
 2. That within such filaments there will be lateral motion, 
 but it will be confined to the limits of the filament. 
 
 3. That the velocity of propagation obtained by this theory 
 
154 PROFESSOR CHALLIS'S SVE /ULATIONS. 
 
 will be greater than that ordinarily assigned to the undu- 
 lations of an elastic fluid. 
 
 4. That under no circumstances can there be plane waves 
 or spherical waves. 
 
 5. That to obtain true and consistent results in hydro- 
 dynamics, another general equation, distinct from those com- 
 monly recognized, is absolutely necessary. 
 
 Views of such a nature, which profess to supply what 
 has been regarded as the great desideratum of mathematical 
 science for the last thirty years, and at the same time 
 to overthrow every preceding speculation upon the subject, 
 emanating as they do from the successor of Newton, cannot 
 but command attention. Few persons are capable of entering 
 into a critical examination of investigations of this nature : 
 and of the limited number so competent, some of the most 
 able have been too much occupied with theories of their 
 own to give much attention to those of their neighbours, 
 while others have felt little inclined to subject to very re- 
 fined or searching criticism speculations of whose general 
 object they approved, and of the general ability of whose 
 authors they entertained not the slightest doubt. I hold it 
 to be equally undoubted and lamentable, that, whether owing 
 to these or other causes, a very great amount of error has 
 within the last few years been put forth by some of the 
 most eminent mathematicians of this country with regard to 
 the subjects of molecular action and fluid motion ; and this 
 evil, so far from being on the decline, gives every symptom 
 of still further augmenting. Were I to assert that Professor 
 Challis's theory is only one of many which have recently 
 appeared which are equally erroneous, I should simply ex- 
 press my individual conviction as to a matter of fact: but 
 it is impossible, within such limits as a periodical can afford, 
 to substantiate such a position in its fullest extent. I single 
 out, therefore, as the object of my remarks the theory of 
 Professor Challis, partly from the necessity of the case ; partly 
 also from its recency, its extraordinary nature, and the con- 
 spicuous position of its author. The fact of its having attracted 
 the opposition of the Astronomer Jioyal in a contest from 
 
PROFESSOR CHAILIS'S SPECULATIONS. 155 
 
 which that eminent mathematician is by many considered to 
 have retired with discomfiture, may be thought an additional 
 motive to urge us to a thorough examination of the subject. 
 
 Professor Challis starts from the ordinary equations of 
 motion, (Phil. Mag, No. 215) 
 
 ^ ds_ du = o ^ ds dv 2 ds dw _ 
 
 dx dt dy + di ~ l dz + ~dt = ' 
 
 ds du dv die 
 
 l i | A 
 
 dt dx dy dz 
 
 where s measures the condensation, and u, v, w are the velo- 
 cities parallel to the axes. 
 
 Assuming - cffsdt = $ (xy) x f(zt\ he thence deduces the 
 following equation, 
 
 which, by equating the coefficient of to 2 , a constant, he 
 splits into two others, viz. 
 
 / *\ 2 ^* 72 
 
 (*) CL + b (b 
 
 dy + d?f + v_ Q 
 
 dx z dy z a* 
 From the former of these he deduces the integral, 
 
 = m cos <z - at */l I + - ) + c 
 
 A I V \ 7T ) 
 
 where A is the length of a wave propagated along the axis of z, 
 and e = - . Professor Challis appears to consider that the 
 
 only form of which " gives a definite solution of the problem" 
 is one consisting of a series of terms similar to the above equi- 
 valent for $ (see p. 280). It will not be necessary for us to 
 enter into the discussion of this point, but it may be observed, 
 by the way, that this assumption is entirely gratuitous. 
 
 " As equation (2) does not contain t, there is no propagation 
 of motion in any direction parallel to the plane of xy : or the 
 
156 PROFESSOR CHALLTS'S SPECULATIONS. 
 
 propagation in the direction of z takes place without lateral 
 spreading." 
 
 Having arrived at this important conclusion, Professor Challis 
 assumes " that f is a function of r the distance from the axis 
 of z; according to which the condensation and transverse 
 velocity are the same at the same distance from the axes in 
 all transverse directions. The above supposition respecting / 
 converts equation (2) into the following, 
 
 r rar 
 the integral of which in a series is 
 
 (3) /='-- 2 +S- + &C -' 
 
 assuming that /= 1 and - = 0, when r = 0. 
 dr 
 
 " It is easily shewn from this result, that there are an 
 unlimited number of possible values of r for which f vanishes. 
 But we have no right to conclude that (3) gives the expression 
 for f we are seeking for, unless at some [given] distance there 
 be neither condensation nor variation of condensation, for other- 
 wise there will be transverse propagation." 
 
 It appears however that, " equation (3) does not satisfy this 
 condition," so that instead of equation (2) Professor Challis has 
 recourse to another which he calls an exact equation, of which 
 more hereafter. The rather equivocal proceeding of deducing 
 certain results from exact, and others from approximate equa- 
 tions, was naturally animadverted upon by Mr. Airy (Phil. Mag. 
 vol. xxxn. p. 340> I believe however that Professor Challis 
 considers himself to have obviated this objection, by shewing 
 that the adoption of the exact equations throughout will not 
 affect his result. Into this question however it will be un- 
 necessary for us to enter. 
 
 " The equation alluded to is 
 
 d 2 1 d 2 1 1 
 
 ^~a 7 + T~2 - s ~ * e * = > 
 dz z f dy z f f 
 
 from which, assuming /to be a function of/*, we obtain 
 
PROFESSOR CHALLIS'S SPECULATIONS. 157 
 
 whence it is clear that if / = 0. J- also vanishes " 
 
 dr 
 
 The integral of (4) Professor Challis takes to be 
 
 whence /= l - V + - + &c. 
 
 4 36 
 
 "The least value of r corresponding to f= 0, as given by 
 this last equation, is the radius of a cylindrical surface within 
 which the motion of the fluid filament under consideration is 
 contained." 
 
 From speaking of the " least value of r corresponding to/= 
 given by this last equation," it may be inferred that Professor 
 Challis considered that that equation would give more such 
 values of r than one, and it may also be surmised from the 
 general purport of his remarks that this value of r is expected 
 to be very small. It is obvious, however, that the equation 
 f- 0, where f is given by equation (5), will have but one root, 
 
 and that root will be r = GO ! For it is clear from (5) that - can 
 never change its sign (e = - being necessarily positive), but as 
 
 r increases from to oo, - will increase from 1 up to oo, and 
 
 f will diminish from 1 to 0, but will never vanish at any finite 
 distance from the axis of z, or, to speak more accurately, 
 will never vanish at all. So much for the theory of thin fila- 
 ments. 
 
 The absurdity of the entire theory might have been ex- 
 hibited from an earlier point. If f is a function of r } " the 
 condensation and transverse velocity are the same at the same 
 distance from the axis in all directions." But if the conden- 
 sation be arranged symmetrically about the axis of z, it is 
 obvious that no force can be acting on the particles in that 
 axis in a direction perpendicular to it; from which it is clear 
 that the motion of those particles must be the same as that in 
 a cylindrical tube when motion is propagated in the direction 
 
158 PROFESSOR CHALLTS'S SPECULATIONS. 
 
 of the axis. Hence for particles in the axis of z the equation 
 (l) which represents the motion parallel to z must reduce itself 
 to the ordinary equation of sound, or we must have b 2 = 0. 
 Thus the velocity of propagation along the axis of z will not be 
 different from that ordinarily given, as Professor Challis sup- 
 poses. Indeed Professor Challis admits that the result b* = is 
 entirely incompatible with his theory. 
 
 Having thus disposed of the positive part of Professor Challis's 
 theory, I now come to the not less important negative branch 
 of his views, in which he denies, and in fact assumes to dis- 
 prove, the possibility of plane or spherical waves. When a 
 writer professes to arrive at the conclusion that results are 
 erroneous, which have been known, and as I may say familiarly 
 handled for half a century, without a suspicion of their accuracy, 
 we can hardly repress an involuntary feeling that the zeal of 
 the author has made him the dupe of some egregious fallacy, 
 and in the case under consideration it will be seen that there 
 is good ground for such a sentiment. 
 
 In the Philosophical Magazine, vol. xxxn. p. 496, we find 
 the following : 
 
 "If the motion be parallel to the axis of ,. . ..the exact 
 equation applicable to plane waves is 
 
 which, as is known, is satisfied by the equation 
 
 (7) w = m sin {2 - (a 4- w) t}. 
 
 X 
 
 This equation shews that at any time t, we shall have w - 0, 
 at points on the axis of z, for which 
 
 n\ 
 
 z - (a + w) t r = -- , 
 
 n\ 
 or z = at, + . 
 
 At the same time w will have the value m at the points 
 of the axis for which 
 
 z-(a 
 
PROFESSOR HALLIS'S SPECULATIONS. 159 
 
 n\ X 
 
 or z - at + - - H- mt. -- . 
 
 1 2 l 4 
 
 " Here it is observable that no relation exists between the 
 points of no velocity, and points of maximum velocity. As 
 m, t jt and A, are arbitrary constants, we may even have 
 
 mt, - ^ - 0, 
 
 in which case the points of no velocity are also points of 
 maximum velocity. This is a manifest absurdity. No step, 
 however, of the reasoning by which this result has been ob- 
 tained can be controverted. What then is the meaning of 
 it? Clearly the analysis rejects the supposition of plane waves, 
 by giving an integral which admits of no physical interpretation. 
 Plane waves are thus shown to be physically impossible." 
 So far Professor Challis. 
 
 This objection is founded on a mistake. Professor Challis 
 assumes " that at any time t, we shall have w = at [all] 
 points in the axis of z for which 
 
 n\ 
 z -(a 4- *)*,* , 
 
 n\ 
 
 or z = at, + . 
 
 n\ 
 Such is not the case ; for if z, - at l = , it is plain that 
 
 
 and if w = 0, this gives us 
 
 2TT 
 
 
160 
 
 Now if, corresponding to the above values z lt t l of z and , 
 we have w = m, it is clear from (8) that we must have 
 
 1 = sin mt lt 
 
 A, 
 27T T7T 
 
 and T m '< = T' 
 
 r\ 
 or mt. = ; 
 
 4 
 
 whence we get 
 
 r\ 
 
 viously obtained, namely + , for equating the two, we 
 
 But this value of t tl cannot be reconciled with that pre- 
 us 
 have 
 
 * 
 
 or r-TT = + 2, 
 
 which is impossible, since r is an integer. 
 
 Hence it is not true, as asserted by Professor Challis, that 
 when 
 
 rofj - - = 0, 
 
 *' the points of no velocity are also points of maximum velo- 
 city ;" and the objection taken to the theory of plane waves 
 on the assumption of such being the fact, must therefore fall 
 to the ground.* 
 
 I now come to the " impossibility" of spherical waves, of 
 which Professor Challis gives us two proofs. If we " suppose 
 the waves to be spherical," (Phil. Mag. vol. xxxn., p. 497) 
 it maj' be shewn by an integral of the known approximate 
 equation 
 
 * The above answer to this objection is substituted for one contained 
 in my letter as submitted to Professor Thomson, and which was objected 
 to by Mr. Stokes (to whom the letter was referred by Professor Thomson) 
 as inconclusive. 
 
PROFESSOR CHALLIS'S SPECULATIONS. 161 
 
 that the analysis rejects this supposition also. The equation 
 is satisfied if r(f> = f(r - at) ; whence 
 
 dd> f(r - at) 
 
 08=--?- = J - -- ) . 
 
 aat r 
 
 We may therefore have 
 
 m . 27r , 
 
 as = sin (r - af); 
 
 T A/ 
 
 and putting r - at = c, it follows that a given phase of the 
 wave is carried with the uniform velocity a, the condensation s 
 varying inversely as the distance r. This conclusion is generally 
 adopted : but a very simple application of the principle of con- 
 stancy of mass will prove that it is false. For if cr^cr^ be the 
 condensations in spherical shells of indefinitely small thick- 
 ness a, at corresponding parts of the same wave when at 
 different distances r^r 2 , the above-named principle requires that 
 27rr 1 V 1 a should be equal to 27rr 2 2 cr 2 a, so that r*<r l = r 2 V 2 , and 
 the condensation varies inversely as the square of the distance. 
 The inevitable conclusion from this reasoning is, that the analysis 
 rejects the supposition of spherical waves.^' 
 
 Professor Challis's " very simple application of the principle 
 of constancy of mass" is an erroneous application of that prin- 
 ciple. The principle of the constancy of mass, as applied to 
 spherical undulations, requires that when a single pulse is pro- 
 pagated of constant thickness, the quantity of matter in an 
 elementary shell at a given distance a from the front of the 
 wave, and whose undisturbed thickness is a, should vary in- 
 versely as the square of its distance from the origin; but the 
 principle of constancy of mass does not require that the 
 quantity of matter in a shell at a fixed distance from the front, 
 and whose disturbed thickness is a, should vary in the above 
 ratio: in other words, the principle does not give us the 
 equation 
 
 27rr l V 1 a = 27rr 2 V 2 a, 
 
 as Professor Challis assumes, but the equation 
 2 7 rrVa = 
 
 where a x , 3 are the respective thicknesses at given epochs of 
 
 M 
 
162 PROFESSOR CHALLIS'S SPECULATIONS. 
 
 the motion of strata at distances r^ and r z from the origin, and 
 whose undisturbed common thickness was a. 
 
 The second disproof of spherical waves occurs in the Phil. 
 Mag., vol. xxxiu. p. 463, and stands as follows : 
 
 " The pressure p being 2 (1 + <r), the following expression 
 is obtained for the condensation cr at the distance r from the 
 centre at the time t, 
 
 ar 
 
 and it is stated that there is no condensation wherever r is 
 greater than at + e and less than at - t, 2t being the breadth 
 of the sonorous undulation. Hence, supposing 2c very small 
 compared to r, and putting for r outside the function its value 
 at corresponding to the middle of the wave, the quantity of 
 matter existing at any time in the wave beyond what would 
 occupy the same space in the quiescent state of the fluid, is 
 very nearly 
 
 the integral being taken from r = at - e to r = at + t. Calling 
 A the constant value of this integral, the expression for the 
 quantity of matter becomes 4irAt. Hence the matter increases 
 in quantity with the time ! Now the very equations from which 
 this result is derived are founded on the supposition that the 
 quantity of matter is constant. There is consequently no diffi- 
 culty here which any physical considerations can explain, but 
 strictly a reductio ad absurdum, which necessitates the impor- 
 tant conclusion that the hypothesis of spherical waves is inad- 
 missible. The physical impossibility of plane waves was proved 
 by the same kind of reasoning ; and any attempt to reconcile 
 the contradiction in either case is simply illogical. As neither 
 the hypothesis of plane waves nor that of spherical waves is 
 admissible, the theoretical value of the velocity of sound which 
 rests on those hypotheses necessarily fails of support." Pro- 
 fessor Challis " returns" therefore " to the consideration of 
 non-divergent waves, or, as they may also be called, ray- 
 vibrations." 
 
PROFESSOR CHALLIS'S SPECULATIONS. 163 
 
 The conclusion that " the matter increases in quantity with 
 the time !" which has excited Professor Challis's note of ad- 
 miration, is well calculated to do so. It is satisfactory to 
 reflect, however, that this result is not a necessary one, and 
 that spherical waves, equally with their plane companions, may 
 continue to maintain their position in the fields of science 
 undisturbed by any terror of that dread annihilation with 
 which Professor Challis has threatened them. 
 
 Upon former occasions Professor Challis has been liberal 
 in making particular assumptions with regard to the form of 
 the function/. We will now indulge him with one. 
 
 Let f(at - r) = sin ~ (r - at), 
 
 A/ 
 
 ,1 f f., . \ cos 2?r , 
 
 then lf(at-r)dr = (r - at), 
 
 j Zira A, 
 
 which vanishes between the limits r at - e, r - at + e : so that 
 upon this assumption the expression (13) for "the quantity of 
 matter existing at any time in the wave beyond what would 
 occupy the same space in the quiescent state of the fluid" 
 reduces itself to zero, or the matter does not "increase in 
 quantity with the time." 
 
 This last remark was communicated to a meeting of the 
 Cambridge Philosophical Society, held on the llth of December 
 last, upon which occasion Professor Challis stated that " the 
 limits of r = at - E, r = at 4 e were not his, they were Mr. Moon's 
 own, and that he (Prof. C.) on that account did not feel called 
 upon to found any observations upon them." This oblivion as 
 to the contents of a paper dated only eighteen days before, 
 occurred to me as rather remarkable, more particularly as 
 I had caused to be printed on slips of paper, circulated among 
 the members of the Society, the different formulae to which 
 I referred in the course of my remarks, including the state- 
 ment of the limits of integration. I conclude, however, that 
 Professor Challis's forgetfulness must be attributed to the con- 
 fusion of thought which is sometimes attendant upon public 
 viva voce explanations. 
 
 M2 
 
164 PROFESSOR CHALLIS'S SPECULATIONS. 
 
 Upon the same occasion the question was suggested by 
 Mr. Adams, " Why a cosine should not be assumed for f as well 
 as a sine ?" The answer is, if we assume 
 
 Q_ 
 
 f(at-r)=cos-- (r - at\ 
 
 A 
 
 and consider the initial disturbance to be confined to a sphere 
 whose radius is \* in the beginning of the motion at the 
 distance r = X, we should have cr the condensation a maximum, 
 which is contrary to the assumed basis of the investigation, 
 viz. that the variations of density will be continuous. 
 
 I now come to Professor Challis's " third general equation in 
 hydrodynamics," which I find somewhat variously stated. In 
 the Philosophical Magazine, vol. xxvi. p. 426 (May 1845), this 
 equation is stated to be 
 
 dp d.pV / 1 1 
 
 ^ 
 
 (9) 
 
 dt dr \r r 
 
 where p denotes the density, V the velocity, and r and r' the 
 greatest and least radii of curvature of the surface which cuts 
 the direction of motion at right angles. At p. 428 in the paper 
 last referred to, however, " the third general equation" is stated 
 to be 
 
 , 
 
 t dx dy dz 
 where ^ (xyz t) = is the integral of the equation 
 
 u j v 7 w 7 
 
 dx + - dy + dz = 0, 
 
 A/ A< A/ 
 
 X being the integrating factor of the equation 
 
 u dx + v dy + 10 dz = 0, 
 it being asserted that (9) may be obtained by combining (10) 
 
 * Upon this hypothesis the proper equivalent to be assumed for/ would be 
 a discontinuous function which would be equal to 
 
 - (at~r) from at-r=Q to at r=X, 
 cos X 
 
 and which vanishes for all other values of r at. I have not thought it 
 .necessary, however, to adopt this nicety in going through the above reasoning. 
 
165 
 
 with the equation of continuity. Another form of (10) is the 
 following : 
 
 Now with regard to these different equations it may be 
 remarked I. that (9) is true, but it is simply a particular 
 deduction from the equation of continuity, and expresses no 
 new mechanical condition whatever; II. that equation (10) is 
 untrue. 
 
 As to the first point, Professor Challis gives a general proof 
 of (9) (Phil. Mag., vol. xxvi. p. 425), in which it is evident that 
 the only mechanical condition involved is that of constancy of 
 mass. In the Cambridge Transactions, vol. in. p. 373, et seq., 
 he gives a proof of the same equation on the particular hypo- 
 thesis of udx + vdy + ivdz being an exact differential, which is 
 deduced, from the equation of continuity by the aid of geome- 
 trical considerations only : and the equation so obtained amounts 
 to nothing more than an expression for the radii of greatest and 
 least curvature of the surfaces which cut the direction of motion 
 at right angles (assuming that there will be such surfaces). 
 In the Phil. Mag., vol. xxvi. p. 427, Professor Challis asserts 
 that (9) " cannot be identical with" the equation of continuity, 
 
 because "in the differential coefficient -~ of equation (9) the 
 
 ctr 
 
 variation of coordinates is necessarily from one point to another 
 of a line of motion, while the differential coefficients of the 
 equation of continuity are subject to no such limitation." But 
 admitting this to be an argument against the identity of (9) 
 with the equation of continuity, it is none whatever against (9) 
 being a particular deduction from, the latter. If (9) embodies 
 a new mechanical condition, what is it ? Prof. Challis asserts 
 that it embodies the same condition as (10). In vol. xxvi. of 
 the Phil. Mag., p. 428, where Professor Challis states "the 
 third general equation" to be (10), he observes "we might 
 proceed to test the truth of it by employing it with the equa- 
 tion of continuity in the investigation of (9)," and he intimates 
 that he has " already given the mathematical process for that 
 
166 
 
 purpose " in the volume of the Cambridge Transactions last 
 referred to, p. 385. Now in this last investigation Professor 
 Challis commences a series of operations of enormous and most 
 needless complexity by finding an expression for 
 
 du du du 
 dx dy dz 
 
 in the course of which he has recourse to equation (10), or 
 rather its equivalent (11), by means of which he introduces the 
 
 quantity ~ , which quantity he afterwards eliminates by means 
 at 
 
 of the same equation ; from which it is obvious that the result 
 of the operations cannot embody the principle of (10). The 
 fact is Professor Challis might have deduced the equation (9) 
 in the general case where udx 4- vdy + wdz is only integrable 
 by means of a factor, in precisely the same way as in the 
 Cambridge Transactions he deduces it on the hypothesis of 
 udx + vdy + wdz being a complete derivative, by simply putting 
 
 u v w (t dw d\L d\L 
 
 X~~ t \ 9 ~\~ ^or ~T~ > ~3 ? ~r~ y 
 A A dx ay dz 
 
 respectively, instead of u, v, w, respectively. 
 
 The most complete way of proving, however, that (9) does 
 not embody (10) will be to shew that the latter is untrue. This 
 may be done in a moment. 
 
 If udx + vdy + wdz be an exact differential, we have A = 1, 
 and we may put for i//, where $ is the ordinary auxiliary 
 function of hydrodynamics ; (11) will then become 
 
 But when no extraneous forces act, we have 
 
 p 2 \dtJ dt' 
 jfdp.. i(d*\ <fy 
 
PROFESSOR CHALLIS'S SPECULATIONS. 167 
 
 Now the equation of sound is deduced from this last 
 equation and the equation of continuity by neglecting small 
 quantities of the second order ; and as we have 
 
 /cfoV = <! W d _tf_ 
 \dt) ~ dtf + d^*dz* 
 
 it is clear from (12) that ~ and [-] are both of the same 
 
 at \at ] 
 
 order, and may therefore both be neglected. Thus (13) re- 
 duces to 
 
 whence ^ = 0. 
 
 pdt 
 
 But by the equation of continuity 
 
 1 dp du dv dw _ 
 p dt dx dy dz 
 
 Hence substituting for , &c., we have 
 
 _ 
 dx 2 dy* dz* 
 
 The true equation of sound however is 
 
 Hence either Professor Challis's equation (9) or the ordinary 
 theory of sound is erroneous. 
 
 The above, I think, establishes the fallacy of equation (9) : 
 the fallacy of the reasoning by which it is supported may be 
 shewn with equal facility. 
 
 " It is well known that if - be a factor which makes 
 
 A) 
 
 u dx + v dy + w dz 
 integrable (Phil. Mag. vol. xxvi. p. 427), 
 
 u , v -, w , 
 
 - dx + - dy + dz = ; 
 
 A, A. A, 
 
168 
 
 is the general differential equation of surfaces normal to the 
 directions of motion. Let the integral of this equation be 
 ;// (xyzt) = 0. We thus have an equation embracing all such 
 surfaces at all times. It must, however, be observed that 
 the function \f> cannot have this comprehensive character in 
 all cases of motion, unless it may be regarded as an arbitrary 
 or discontinuous function : the same remark applies to the 
 functions that u, v } to are of the coordinates and the time. To 
 represent the velocities at all points and at all times, what- 
 ever motion be given to the fluid, they must be arbitrary 
 or discontinuous functions. As however in the investigation 
 of "the ordinary equations of fluid motion" u, v, w were 
 assumed to be functions of constant form for at least inde- 
 finitely small variations of the coordinates and the time (for 
 they would not otherwise be subject to mathematical reasoning), 
 the same supposition must be made with respect to the function 
 i// Hence &z, By, 2, and $t, being indefinitely small variations,, 
 we have for a given form of the function 
 
 and i// (x -i- &c, y + %, z + &z, t + St) = ; 
 
 whence, by expansion, we readily obtain 
 
 s'i.. + '|^iV^.i+* : fc. ! o, 
 
 ait ax ay az 
 
 and putting Sx = ut, Sy = vt> Sz = w$t f 
 
 d\L d\L d\L diL 
 
 -~+ ^u + -^v + -^w = 0." 
 
 dt dx dy dz 
 
 The above quotation I have given in extenso, though part 
 of the reasoning it contains is irrelevant to the present purpose ; 
 the passage however is so obscure that I have thought it de- 
 sirable to give Professor Challis's ipsissima verba upon the 
 subject. 
 
 The function i in its rudest state will contain an arbitrary 
 function of the time, and one or more arbitrary parameters. 
 Suppose the form of the arbitrary function determined from 
 
PROFESSOR CHALLIS'S SPECULATIONS. 169 
 
 the nature of the motion, and let $ contain a single arbitrary 
 constant a, or let it stand 
 
 i// (xyzta) = ; 
 and let the value of a derived by solution of this equation be 
 
 the time t being fixed, we may, by assigning successive values 
 to a in the equation ^ (xyzta] = 0, obtain all the surfaces normal 
 to the motion at that time. If the surface is to pass through 
 the point x, y, 2, at the time t, then for that surface 
 
 If the surface is to pass through the point whose coordinates- 
 at the time t t 4 St, are x, 4 $x lt y, 4 By l9 z, 4 Sz l9 we have a or 
 
 a , + &*/ = f( x , + fy) y, + Sy, z , + &*.9 *>, + ^) 
 from which we get 
 
 8a, =/,) *. +/(*,) &, +/(y,) 8y, + /(^,) & 
 
 and the second differential of -^ (xyzta} becomes, after equating. 
 to zero and eliminating # 
 
 Hence putting icy^^ for xyzf, t (which last quantities have 
 been introduced simply for the purpose of rendering the rea- 
 soning more distinct) and putting 
 
 x = , y = v, w = 
 we arrive at the equation 
 
 dil d\L d\L d\L, 
 -?-+u^-+v-i-+w-?-- 
 dt ax ay dz 
 
 fy ( d f d f d f d f\ 
 
 = -- T.I^L+ U -f + v-^- + w -J-] 
 
 da \dt dx dy dz j 
 
 instead of (10). It remains for Professor Challis to shew upon 
 what hypothesis with regard to the motion, the second membci 
 of this equation will vanish. 
 
170 PROFESSOR CHALL1S J S SPECULATIONS. 
 
 The considerations I have last suggested are so elementary 
 that I feel almost ashamed of introducing them. When how- 
 ever we find the first recognized mathematical authority in the 
 University of Cambridge overlooking principles so plain, there 
 is no alternative but to shew distinctly their exact bearing. 
 
 Professor Challis assumes to shew " the defective state of 
 analytical hydrodynamics when" only the two ordinary equa- 
 tions " are made use of" by an example. 
 
 Assuming the fluid to be incompressible, the motion parallel 
 to the plane of xy, and iidx + vdy an exact differential, Professor 
 Challis arrives, by means of the ordinary equations, at the con- 
 clusion that " the boundary of the fluid may at all times be a 
 cylindrical surface ; but it is impossible this can be true, because 
 the particles at the surface are all moving with the same velocity 
 in directions making different angles with the surface." The 
 expression " boundary of the fluid" is somewhat ambiguous. 
 If Professor Challis means by it a solid material boundary, 
 his reasoning does not bear out his conclusion. If he means 
 simply a surface of equal pressure, there is no absurdity in sup- 
 posing the particles in the cylindrical surface to " be all moving 
 with the same velocity in directions making different angles 
 with the surface." Would Professor Challis have them all 
 moving with the same velocity in the same direction, and that 
 of the radius? But were the fact otherwise, and the motion 
 indicated by the equation were really impossible, it is still 
 to be observed that the general equations of fluid motion are 
 only approximate (the first of them being founded on the prin- 
 ciple of equality of pressure during motion which can only be 
 approximately true), and the question would then arise, whether 
 the alleged absurdity is not owing to our having travelled beyond 
 the limits within which the approximation is feasible. 
 
 In the paper in the April No. of the Phil. Mag. for the 
 past year, with which I commenced my criticism, Professor 
 Challis " cannot avoid adverting to a difficulty which has long 
 presented itself to him with respect to the explanation usually 
 given to the excess of the velocity of sound above the value a. 
 Admitting that a sudden condensation by developing heat 
 
PROFESSOR CHALLIS'S SPECULATIONS. 171 
 
 produces a higher degree of temperature, and therefore of 
 elastic force, than would exist in the same state of density 
 without such development, does it not thence follow, that a 
 sudden rarefaction, by absorbing heat, produces a lower tem- 
 perature and a less elastic force than would exist in the same 
 state of density without absorption ? " 
 
 The answer to this question is, that it does not follow " that 
 a sudden rarefaction by absorbing heat produces .... a less 
 elastic force than would exist in the same state of density with- 
 out absorption"; and for this simple reason, that the effective 
 moving force on any particle does not depend on the absolute 
 elasticity of the medium in its neighbourhood, but on the 
 variation of elasticity in passing from one to the other of the 
 strata on opposite sides of the particle or lamina whose motion 
 we are considering ; which last will increase with a rarefaction 
 just as much as with a condensation, although the former does 
 not. 
 
 An answer similar to that just given was afforded by Mr- 
 Airy in his notice of Professor Challis's paper (Phil. Mag. 
 vol. xxxn. p. 343), which elicited the following reply. (Vide 
 p. 498 of the same volume.) 
 
 " The difficulty respecting the augmentation of the velocity 
 of sound by the development of heat, cannot be so summarily 
 disposed of as Mr. Airy appears to imagine. I shall perhaps 
 succeed better in conveying my meaning by using symbols. 
 If 6 be the temperature where the pressure is p and density p r 
 and 9 the temperature in the quiescent state of the fluid, we 
 have by a known equation, 
 
 p = a*p{l + a.(0-0,)). 
 Hence 
 
 *__*_ ^f> _ aX e _ 0) .& _ a * a M : 
 
 df pdz pdz pdz dz 
 
 The usual theory explains how the third term of the right-hand 
 side of this equation may be in a given ratio to the first ; but 
 my difficulty is to conceive how the same can be the case also* 
 with the second term, since it charges sign with the change 
 of sign of 9 - 0,." 
 
PROFESSOR CHALLIS'S SPECULATIONS. 
 
 Professor Challis here entirely shifts his ground, and instead 
 of elaborating, as he professes to do, his former position, which 
 was untenable, he starts a new difficulty. His first objection 
 to the velocity ordinarily assigned to sound was grounded on 
 the assumption " that a sudden rarefaction would produce a less 
 elastic force" instead of a greater one which is required, and 
 which assumption is not true, as I have shewn. The second 
 " difficulty" is equally fallacious. The assumption upon which 
 " the usual theory" makes " the third term of the right-hand 
 side of the equation in a given ratio to the first/' is that the 
 variation of temperature will be proportionable to the variation 
 of density, or that 
 
 B-O^ktp- p), 
 
 p, being the density corresponding to the temperature 0, ; sub- 
 stituting this in (14) the latter becomes 
 
 d*z a* dp " 3 7 
 
 df = ~ ~~ dz ^ P ' Pi ^ 
 
 or neglecting the variation of density in the terms depending 
 upon a, 
 
 d*z a 2 dp 
 
 ~~ 
 
 p, being the density of equilibrium ; the term a z ak (p - p t ) 
 vanishing in the approximation. If Professor Challis can 
 understand why we should put 
 
 -**__** 
 
 dz p dz 
 
 I think he may get over his " difficulty" as to " the second 
 term" of (14). 
 
 My attention having been accidentally directed to Professor 
 Challis's papers, I addressed a short communication to the 
 Cambridge Philosophical Society, which was read at a meeting 
 of that body which took place in November last, in which 
 I pointed out the fallacy of this second " difficulty" as to the 
 velocity of sound, much in the manner I have adopted above, 
 This elicited from Professor Challis the following (Phil. Mag. 
 vol. xxxin. p. 466): 
 
PROFESSOR CHALLIS'S SPECULATIONS. 173 
 
 " Let the relation between the pressure and density, in- 
 clusive of the effect of temperature, be expressed by the 
 equation p = cfp 1+k , as is allowable. Then, putting 1 + a for p, 
 and supposing o- small, we have 
 
 If now the terms h<r, &c. be neglected, the equation is of the 
 same form as that derived from the relation p = a z p, and is 
 consistent with the uniform propagation of a given state of 
 velocity and density. But those terms stand in the way of 
 such a result ; and though they are of small amount, yet their 
 effect on the form of the wave must be accumulative) and must 
 in the end entirely alter its character." 
 
 "With regard to this it may be observed that the effect of 
 the rejected terms " on the form of the wave" is not necessarily 
 " accumulative" but may be periodical. If however we are 
 to give up the theory of sound on account of an objection 
 such as this, we must equally abandon the Lunar and Planetary 
 Theories. Professor Challis may be prepared to throw over- 
 board the sublime results of his illustrious predecessor with 
 the same composure he has exhibited with regard to the 
 Theory of Sound ; but whatever may be his sentiments in this 
 respect, I apprehend that the generality of persons will be still 
 inclined to cling both to the one and the other theory, notwith- 
 standing that "the difficulty alluded to does not exist in the 
 case of ray-vibrations." 
 
 I have now, Sir, brought to a close my observations upon 
 Professor Challis's various speculations. I am aware that your 
 pages have not hitherto been burthened by communications 
 of a controversial character ; but at a time when it is almost 
 as easy to publish a mathematical theory as to insert a 
 paragraph in a newspaper, it has appeared to me highly 
 desirable that the crudities which are thus from time to time 
 put forth should be subjected to a more stringent censorship 
 
174 
 
 than any which has hitherto been put in force ; and it has 
 occurred to me that, as conducting the only exclusively Mathe- 
 matical Journal in England, you might not be unwilling to 
 undertake the responsibility of that office. But you will doubt- 
 less form your own judgment as to what is to be the rule of 
 your own conduct, and I have now therefore merely to sub- 
 scribe myself, 
 
 Your obedient Servant, 
 
 ROBERT MOON. 
 
 Liverpool, Jan. 8, 1849. 
 
 THE END 
 
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