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 MECHANISM 
 
 OSSICLES OF THE EAR 
 
 MEMBRANA TYMPANI. 
 
 H. HELMHOLTZ, 
 
 PROFESSOR OF PHTSIOLOGY IN THE rKIVEKSITY OP BERLTN, PRrSSIA. 
 
 TRANSLATED FROM THE GERMAN, WITH THE AUTHOR'S PERMISSION, 
 
 BY 
 
 ALBERT H. BUCK AiND NORMAXD SMITH, 
 
 OF NEW-YORK. 
 
 AVITH T-WELVE ILLUSTRATIONS. 
 
 NEW-YORK : 
 WILLIAM WOOD & CO., 27 GREAT JONES STREET. 
 
 1873.
 
 
 3 
 
 NOTICE 
 
 Prof. Helmholtz's essay on The Mechanism of the Ossicles of 
 THE Ear and Membrana Tympani, originally published in the 1st 
 volume of PJluger''s Archiv fiXr Fhysiologie, Bonn, 1809, is the only 
 treatise in any language which enters fully into the anatomical, physio- 
 logical, and mathematical aspects of the question, and will undoubtedly 
 remain for many years to come the authoritative treatise on this 
 subject. 
 
 In view of the great importance of this essay to those interested 
 in the department of otology, the undersigned have attempted to 
 translate it into English. The style of writing of the distinguished 
 physiologist is so exceedingly condensed that some allowance will be 
 made, we trust, for the evident lack of smoothness in the English 
 version. 
 
 A. IT. P.. and N. S. 
 
 •^06816
 
 CONTENTS. 
 
 PAOB 
 
 § 1. Results due to the small Dimensions of the Audiioiy Apparatus 9 
 
 ^ 2. Anatomy of the Membrana Tympani 16 
 
 ^ 3. Attachments of the Hammer 23 
 
 § 4. Attachments of the Anvil 32 
 
 (Translated by Albert H. Buck.) 
 
 § 5. The Movements of the Stirrup 41 
 
 § 6. The Concerted Action of the Bones of the Ear 45 
 
 § 7. Mechanism of the Membrana Tympani 53 
 
 § 8. Mathematical Appendix, having particular Reference to the Mechanism 
 
 of Curved Membranes 03 
 
 (Translated by Normand Smith.)
 
 MECHANISM 
 
 OSSICLES OF THE EAR. 
 
 Fkom a notice found among the papers of the late B. Rie- 
 mann, and recently pnl)lis]ied in tlie Zeitung fur rationelle 
 Medhin^ we learn what views this man of unusual penetra- 
 tion — alas ! too soon lost to science — took, during the last 
 months of his life, regarding the problems of physiological 
 acoustics, and why so few of them had thus far met with a solu- 
 tion. And here, too, we find that he had discovered the true 
 source of all difficnlty, and the one toward which all scientific 
 efforts must henceforth be directed. lie proposes, as the chief 
 task of aural mechanics, to explain how the apparatus of the 
 middle ear can transmit from the air to the fluid of the laby- 
 rinth sucli extraordinarily fine shades of vibration — as we know 
 it actually does. He proves, by calculatioii, that the excursions 
 of the stirrup, in the fainter (though yet clearly to be distin- 
 guished) tones, must be so small as to escape detection, even witli 
 the highest powers of our modern microscopes. To transmit 
 regularly and accurately vibrations of such delicacy, he holds 
 that there must be a corresponding accuracy and precision in 
 the vibrations of the transmitting a})paratus. 
 
 At the same time, he says he will be obliged to oppose in 
 many particulars the theory of the mechanism of hearing as 
 developed by me in the Lehre der Tonempjindunyen. In 
 this connection, I must remark that I myself at the time consid-
 
 8 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 ered the descrijHion of the vibrations of the apparatus of the 
 middle ear, as given in cliapter 1, section 6, of the work in 
 question, to be simply ])reliminary, and gathered from for- 
 eign sources. It was impossible for me at the time to make 
 aiiv investigations of my own into this question, although I fully 
 recognized the necessity for new investigations. In the 
 description which I gave in the same work I adopted, in 
 its most essential features, the theory of Edward "Weber,* 
 which, compared with former theories, is a decided advance. 
 It is, in the main, correct, although wanting in certain details 
 which are indispensable to its completeness. It struck me 
 that the chief difiiculty in this theory lay in the existence of a 
 joint between the hammer and anvil. According to Weber's 
 description, the hannner and anvil constitute an innnovable 
 angular lever, whose axis of rotation is drawn through the pro- 
 cessus Folianus of the hammer and the end of the processus 
 brevis of the anvil. But how was the existence of a joint, 
 surrounded by a weak and loose capsular membrane, allowing 
 motion in all directions, possible in the midst of a lever whose 
 vil)rationsmust needs be of the greatest fineness and accuracy? 
 As soon as the completion of my work on physiological optics 
 afforded me time for other investigations, I took the above ques- 
 tion into consideration and had obtained nearly all of the fol- 
 lowing residts before seeing Riemann's Notizen.f The solution 
 of the ditticulties was obtained by a closer investigation into 
 the mechanics of the joints and attachments of the bones of 
 the ear, and proved, in fact, to be entirely different from the 
 one proposed by the celebrated mathematician. Besides, I 
 must oppose his statement " that it is the task of the ap- 
 paratus of the middle ear to transmit to the fluid of the 
 lal)yrinth the changes in atmospheric pressure at every moment 
 of tiTue, with perfect accuracy and constant relative strength," 
 because I consider this in nowise proven by the facts of the case. 
 Accuracy in perception requires only that every tone of a 
 
 * Bericlite liber die Verhandluiifjen der Konigl. Sachs. Ges. d. Wi.sscni- 
 scliaften zu Leipzig. Math. Phys. Klasse. 1851, Mai 18, S. 29-31. 
 
 f Sliort notice of them iu the lleidelber^er Jahrbiiclier. Jul^- ^Gth and Au- 
 ;fU8t 9tli, 1807.
 
 MECHANISM OF THE OSSICLES OF THE EAR, 9 
 
 given pitch should cause the same sensation, both in kind and 
 intensity, every time tliat it is reproduced. It is a well-known 
 fact that tones of a certain pitch produce an uncommonly 
 strong impression upon the ear. AYe shall mention further on 
 other new examples ofj abnormities. 
 
 §1. 
 
 Results due to the small Dimensions of the Auditory Apparatus. . 
 
 The most important step in advance made by Edward Weber 
 in the theory of the transmission of sound in the ear — a step 
 which has recei\'ed much less consideration than it deserves — 
 seems to me to be the view that, in the transmission of sound- 
 vibrations, the bones of the ear and the petrous portion of the 
 temporal bone are to be considered as solid, incompressible 
 bodies, and the fluid of the labyrinth as an incompressible fluid. 
 He rightly declares that in the case of these bodies and fluids 
 there can be no question as to the transmission of waves of 
 condensation and rarefaction, but that the bones of the ear 
 must be considered as solid levers, and the fluid of the laby- 
 rinth as a mass only to be moved as a whole. 
 
 I shall take the liberty of going more minutely into this 
 special topic, inasmuch as it forms the basis of the subsequent 
 investigations. 
 
 If in an elastic medium, be it a solid, fluid, or gas, whose three 
 dimensions are inflnitely extended, there be produced plane 
 waves answering to a simple tone, these will })ass througli tlie 
 elastic mass with the rajiidity which belongs to that giveii tone, 
 and produce at different points of the mass either displacement 
 of the ultimate ])articles, or even condensation, Avhere it is 
 caused by longitudinal vibrations. If at a given point of the 
 mass tliere are particles in a state of extreme displacement up- 
 ward, at the same moment of time, there will be, at a distance 
 of half a wave's length, particles in a state of extreme displace- 
 ment downward ; and the same is true of all other directions 
 of displacement. Between these upper and lower limits of 
 extreme displacement — which must be at least half a wave's 
 length apart, as we have seen — we shall find in a continuous-
 
 10 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 line of transition tlie lesser degrees of displacement upward, 
 the zero point of this displacement, and the lesser degrees of 
 displacement downward, so that the difference in displacement 
 of two oscillating jparticlcs^ whose distance from, one another 
 is infinitely small compared with the wave-length, is itself 
 infinitely small compared with the entire amplitude of displace- 
 ment. If we limit ourselves in such a case to the consideration 
 of a small portion of the vibrating mass, all of whose dimen- 
 sions shall be infinitely small compared with tlie wave-length, 
 then all the relative displacements of the individual points of 
 this mass, among themselves, will be infinitely small compared 
 with the amplitude of the entire vibrations, Avhich in their turn 
 must be considered as infinitely small compared with the wave- 
 length, where sound vibrations are regularly produced. These 
 relative displacements of the individual particles of the small 
 mass (which we imagine to be taken out from the whole) 
 among themselves are, therefore, infinitely small magnitudes of 
 second order compared with the wave-length, and infinitely 
 •Bmall magnitudes of first order compared with the amj^litudes 
 of vibration, and with the linear dimensions of the small mass 
 to which they belong : that is to say, the small mass acts in the 
 present instance just as an absolutely immovable body would. 
 
 The conditions remain the same when a large number of 
 plane waves, belonging to the same simple tone, pass tlirough 
 the elastic mass ; and also when spherical waves spread them- 
 selves through it, taking their start from any centre whatsoever 
 of excitement within the mass, excepting, however, in the im- 
 mediate neighborhood of punctiform or linear centres of ex- 
 •citement, whose appearance, however, is more a mathematical 
 fiction than a practical reality. 
 
 The same law applies also to solid elastic bodies, provided 
 their substance is not infinitely extended in all directions, but 
 has limits against which the Avaves of sound may strike and be 
 thrown back toward the centre of the mass. It is here, however, 
 presupposed that either no linear dimension of the vibrating 
 mass shall be very small, compared with the wave-length, or that 
 this should be the case with all the dimensions of the vibrating 
 anass at the same time, so that no one of them should be very
 
 MECHANISM OF THE OSSICLES OF THE EAR. 11 
 
 small compared with the others, as is the case, for instance, in 
 disks, membranes, rods, and strings. 
 
 The proof of these laws is easily deduced from the well-known 
 laws respecting the form and mode of vibration of plane waves 
 — so long, of course, as there is only question of plane waves of 
 simple tones in masses of infinite extent. On the other hand, 
 the influence of boundary-planes (Grenzfliichen) and of the last- 
 named conditions has been elucidated by Kirclioff, in his trea- 
 tise on the equilibrium and vibration of an infinitely thin elas- 
 tic rod.* In this treatise, it is true, only the equilibrium of 
 such elastic masses is taken into consideration, and it is there 
 proven that forces, wliich are infinitely small compared with 
 the constant of elasticity of the body, and which are brought 
 to bear partly on the central and partly on the superficial por- 
 tion of the elastic mass, cause only infinitely small relative dis- 
 placements of such particles as lie within finite distance of each 
 other, so that the diflferential quotients of the displacements, 
 as taken from the co ordinates, also remain finite. On this 
 last point the question chiefly hinges. For, if these differential 
 quotients are finite magnitudes, then, in masses of infinitely 
 small linear dimensions, the relative displacements of the indi- 
 vidual particles are infinitely small, compared with the total 
 absolute displacements which such masses experience. 
 
 The above-mentioned law, wliich Kirehoft* has demonstrated 
 for the condition of equilibrium, the forces involved being infi- 
 nitely small, may also, by means of d'Alembert's rule, be ap- 
 plied to the condition of motion, provided the accelerations 
 which the particles of the mass experience during motion are 
 considered as the forces which disturb the elastic body. These 
 now, when they belong to vibrations whose amplitude, com- 
 pared with the length of tlie wave, is infinitely small, are them- 
 selves infinitely small, and answer, therefore, to Kirchoff's ac- 
 ceptation of infinitely small f disturbing forces. 
 
 * Borcliardt's Journal fiir reine und angewandte Matliematik LVI., in g 1 of 
 tlie treatise in question. 
 
 f If A be the amplitude of vibration, n the number of vibrations of a sim- 
 ple tone, t the time, and c a constant determining the phasis, then has s, the 
 variable departure from the position of equilibrium, the following value : 
 
 S = A sin I 2 TTut -t- c [
 
 12 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 The law demonstrated by Kjrcliofi', and applied to the pres- 
 ent case, might be thus expressed : 
 
 In immovahU elastic bodies, all of whose linear dimensions 
 are not infinitely small compared with the wave-length, or at 
 least none of which are infinitely small compared with the 
 res,t, vibrations of a simple ^o;ie, whose amplitude is infinitely 
 small compared with the wave-length of the same kind of vibra- 
 tions in masses of infinite \\\a\X.%, -produce upon two poi7its of the 
 elastic body, whose distance from one another is infinitely small 
 compared with the same wave-length, relative displacements, 
 which are themselves infinitely small compared with the entire 
 amplitude of the vibrations. 
 
 Tiiat is to say, then, that, under the restrictions mentioned, 
 masses whose linear dimensions are all small compared with the 
 wave-length act exactly like absolutely solid bodies ; or, that 
 the changes in form which they undergo can be disregarded when 
 compared with the entire amplitude of their vibrations. 
 
 If we now take into account that in air the wave-lengths of 
 the tones constituting our musical scale — that is, from C, witli 
 33 vibrations to c ^ with 4224 vibrations — vary from 8 to 1000 
 cm. ; that in water the same waves are more than 4 times, in 
 brass about 11 times, in copper 12 times, in steel and glass 
 more than 15 times greater than in air; that, on the other 
 hand, tbe dimensions of the bones of the ear and of the laby- 
 rinth are only small fractions of a centimetre, the important 
 
 If // re])resent tlu; volunu' (if the small part, tlien k, the power used to ac- 
 celerate the same, is equal to 
 
 k = li I'i = — 4 tt' n* A sin \ 2 -nt + c \ 
 
 If now A be the wave-lenjrth, and a the rate of progress for this kind of vibra 
 tioos in masses of unlimited extent, then 
 
 __« 
 
 and for the maximum of k, which appears as often as the sinus of the for- 
 mula given for it equals ± 1 : 
 
 k , , , A 
 
 a // 
 
 k is therefore infinitely small compared with a', provided A is infinitely small 
 compared with X; and «' multiplied by the density is equal to the constant of 
 the elastic resistance, which in this kind of comparison assumes a value.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 13 
 
 conclusion follows' that the dimensions of the elastic solid and flnid 
 masses, constituting the organ of hearing are all at best only 
 very small fractions of the wave-lengths of those tones which 
 we commonly hear, and which our ear can readily appreciate. 
 
 We are, moreover, to conclude from what has already been said 
 that, in the vibrations of the auditory apparatus, of the bones 
 of the ear and of the petrous bone, caused by the tones ordi- 
 narily appreciable by the ear, the particles of each of these 
 small masses undergo displacements among one another, which 
 are infinitely small compared with the amplitude of the sound- 
 vibrations producing tliem ; that is to say, that they act very 
 nearly like absolutely solid bodies. 
 
 The final reason for this peculiarity of motion is to be found 
 in the very great rapidity with which the influence of every 
 shock, communicated to one of these small solid masses, is trans- 
 mitted through it. This rapidity is so great that the time 
 required for transn)ission of the shock may, as a rule, be con- 
 sidered as infinitely small wlien compared with the duration of 
 the individual sound-vibrations, and its action as instantane- 
 ously conveyed throughout the entire mass. 
 
 An incomj)ressib]e fluid, inclosed within solid w^alls, differs 
 from one that is compressible in the fact that here, too, every 
 shock communicated to one part of its superficies is instantlj'' 
 transmitted through the entire fluid, and sets every portion of it 
 instantaneously in motion ; while in a compressible fluid one 
 wave starts from its point of origin, runs its course with a cer- 
 tain speed, and sets alternately the different portions of the fluid 
 in motion. If, therefore, in the case of the fluid of the labyrinth, 
 the dimensions of the whole mass are infinitely small compared 
 with the wave-length, and the walls of the petrous bone inclos- 
 ing the fluid are strong enough to be considered as absolutely 
 immovable beneath the small pressure exerted in this instance 
 against tlieni, then the transmission of tlie shock throughout 
 the entire mass is practically instantaneous, and the fluid of 
 the labyrinth may be said to act under the influence of vibra- 
 tions of sound precisely as a fluid absolutely incompressible, 
 and therefore incapable of transmitting the vibrations of sound, 
 would do under the same circumstances.
 
 14 MECHANISM OF THE OSSICLES OF THE EAE. 
 
 Finally, it is necessary, at least for the deeper and middle 
 tones of tlie scale, that there shonld be an equality of pressure 
 between the air contained in the middle ear and that of the ex- 
 ternal auditory canal. In tlie case of very high tones, those 
 corresponding to the highest octave of the piano, the length of 
 the auditory canal is very nearly equal to a quarter of a wave's 
 length, to which circumstance is due the occurrence of those 
 phenomena of resonance described by me in the Lelire von den 
 To7iemj[>Jiiulunyen.^ At all events, the diameter of the exter- 
 nal auditory canal is too small to permit of difterent phases of 
 pressure or of speed at difterent points of the membrana tym- 
 pani at the same moment, and we can, therefore, without liesita- 
 tion consider the pressure as equal at all times over all parts of 
 the membrane. This circumstance is likewise of great impor- 
 tance in the mechanism of the ear, for it excludes all possibility 
 of one part of the membrana tympani being excited, while 
 the rest is not ; the part excited being dependent on the 
 locality of the sound-giving body. Hence we have no otlier 
 means of localizing sound except by noting the diflierent degrees 
 of intensity obtained by changing the position of the head and 
 comparing the impressions made upon both eai^s. 
 
 The above-mentioned rule applies, as already stated, to bodies 
 none of whose linear dimensions are infinitely small compared 
 ■viath the rest, consequently not to strings, membranes, rods, and 
 disks. It is also liable to exceptions, as where the middle por- 
 tion of the body in question is contracted and very narrow. 
 Among the component parts of the auditory apparatus, the mem- 
 brana tympani is the only one which falls under the head of 
 exceptions. In point of fact, those bodies which are very thin 
 at one spot, or in one direction, are capable of performing com- 
 paratively slow vibrations ; for, owing to their slight tliickness, 
 they ofter but a feeble elastic resistance, return slowly to a state 
 of equilibrium, and vibrate at a much slower rate than is the 
 case with oscillations in thick masses of the same nature. 
 
 That the bones of the ear do not come under the head of ex- 
 ceptions, is easily shown by comparing them with the metallic 
 rods or tongues which are used in the i)rodu(;tion of high tones. 
 * Pages 175, 176.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 15 
 
 The tongues eni ployed to produce the highest tones of the musi- 
 cal scale in a harmonium are relatively very long and thin when 
 compared with the dimensions of the hones of the ear, and no 
 one, who has any experience in the tones which belong to, or 
 can be produced by such solid bodies, could for a moment doubt 
 that, were it possible to put into regular vibration such small 
 masses as the bones of the ear, including the relatively slender 
 stirrup, these would give forth tones of such enormous height 
 that to our ear they would probably no longer be perceptible — 
 tones lying far beyond the limits of our musical scale. 
 
 The relation sustained by the bones of the ear to the vibra- 
 tions of sound is practically the same as in an iron rod, when 
 hung up and caused to vibrate as a pendulum. Such a 
 rod is elastic and yielding, and is capable of several kinds of 
 vibration ; but these vibrations take place at tlie rate of several 
 hundred per second, while as a pendulum it swings perhaps 
 only once in a second. If sueli a pendulum is caused to vibrate 
 by a force exercised periodically, the periods amounting to one 
 or more seconds, or to larger fractions of a second, each blow 
 communicated by this power to one point of the rod can traverse 
 the same hither and thither several hundred times before the 
 blow belonging to the next period is given, and thus the effect 
 of the blow can be transmitted tlioroughly to every part of the 
 mass before even a small fraction of the period of a vibration 
 has passed. Under these circumstances the pendulum vibrates 
 practically as an absolutely solid body, that is, its real motion 
 is not to be distinguished from the motion of such a body, 
 not even by means of the most delicate methods of observation. 
 Entirely different is the action of the pendulum when we cause 
 it to vibrate by means of a tone whose pitch approximates that 
 of the rod. Then it vibrates, no longer according to the laws of 
 a pendulum, but as a vibrating elastic rod. 
 
 The same is true of the bones of the ear. As long as the 
 periods of vibration of the tones Avhich these must transmit 
 are very great compared with those of the bones themselves, 
 60 long will the latter act, practically, as absolutely solid 
 bodies.
 
 16 
 
 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 §2. 
 
 Anatomy of the Membrana Tympani. 
 
 Before passinj^ on to the discussion of the mechanism of the 
 apparatus of the middle ear. I must make one or two anatomi- 
 cal remarks, not with tlie view of bringing forward any thing 
 materially new, but simply to give prominence to a number of 
 small points, which as a rule are merely noticed and then pass- 
 ed over by the anatomist, but which gain importance in a more 
 thorough investigation of their physiological bearings. 
 
 The opening in which the membrana tympani is set is formed 
 from the squamous portion of tlie temporal bone and from what 
 was once the annulus tympanicus, both of which in adults are 
 firmly connected by a bony union ; not so firmly, however, but 
 that in chiseling out a preparation of the ear, a l^reak is likely 
 to occur at this very spot of union — a circumstance which I 
 found very annoying in exposing to view the connections of the 
 upper part ot the membrana tympani. Even on the dried adult 
 bone this line of separation is still pretty clearly marked by two 
 prominent bony spurs, which rise up, before and behind, on the 
 boundary between both parts ; these separate a lower part, 
 
 which is nearly oval in form and 
 contains a rim for the attachment 
 of the membrana tympani, from 
 an upper part, whicli is irregular 
 in outline and more strongly con- 
 cave. The former belongs to the 
 OS tympanicum, the latter to 
 the OS squamosum. Fig. 1 re- 
 presents the upper and anterior 
 wall of the bony portion of the 
 external auditory canal. The 
 line of section was drawn parallel 
 to this wall, ah \'S, the surface of 
 section of the anterior wall, which 
 separates the auditory canal from the joint of the jaw; C6?is 
 the line of section through the posterior wall ; h d is the outer 
 opening of the auditory canal ; a slight furrow li /, which in the 
 
 Fig. 1.
 
 MECHANISM OF THE OSSICLES OF THE EAE. 17 
 
 engraving is more strongly marked than is actually the case in 
 nature, represents the line of attachment of the membrana 
 tympani. Traces of the fissure, which in the foetus divides 
 the anterior upper border of the annulus tympanicus from the 
 squamous portion, may still be seen running from the point/' 
 in the direction of g. Between a and h the same fissure (fissura 
 Glaseri) is recognizable. The projecting point at /", which, 
 plays an important part in the attachment of the hammer, is 
 called by Henle the spina tympanica posterior, in contradistinc- 
 tion to another more distinctly marked point in the foetus, on 
 the anterior end of the annulus, at its outer anterior angle,, 
 which he calls the spina tympanica anterior, and which, on the 
 much broader os tympanicum of the adult, answers to the 
 point g. The latter, however, lies flat upon the corresponding 
 surface of the squamous bone and no longer stands out as a spur. 
 On the posterior end of the above-mentioned recess, and corre- 
 sponding to a point in Fig. 1 between c and ^, there can be seen 
 a blunt and less prominent projection of the rim, in wliicli the. 
 membrana tympani is inserted, which we shall frequently have 
 occasion to speak of in describing its attachments. In order to- 
 avoid errors which might arise through my giving Ilenle's name 
 of spina tympanica posterior to the anterior point/", I shall take 
 the liberty of applying to it the name of spina tympanica major, 
 and to the posterior point at i that of spina tympanica minor .. 
 The neck of the hammer fits into the recess lying between/" 
 and c in such a manner that the point at / almost touches it.. 
 The line of attachment of the membrana tympani also shows a 
 slight and ill-defined depression where it passes near the j)oints 
 /and i. And just here, moreover, the line is less sharply defined 
 than lower down on tlie part formed from the os tynq)anicum ; 
 and here, too, slight pressure with a blunt instrument will loosen 
 the membrana tympani from its attachments. In fact, it is more • 
 truly attached to the cutis than to the bone. 
 
 This recess in the upper border we shall call the Rivinian re- 
 cess, as it includes the opening described by Rivini, an opening 
 which represents the last trace of the original visceral cleft, but 
 which in the majority of normal adults does not exist. 
 
 Although normally no opening exists there, still the Rivinian. 
 2
 
 18 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 I'ecess is filled with a loose part of tlie drum-head, which, as it 
 iippears beneath the thin cutis, is seen to consist of bundles of 
 loosely interwoven connective tissue, which give passage to ves- 
 sels and nerves and are easily separated. (Membrana flaccida 
 ^hrapnell.) For this reason, abscesses are wont to perforate at 
 this point, and here too, in making preparations of the cutis 
 layer, artificial openings are readily made. The diiference in 
 tension and consistency between this upper part of the drum- 
 head and the rest of the membrane can easily be felt by pass- 
 ing the blunt end of a sewing-needle over the surface of the 
 membrane, in a preparation where the attachments of the bones 
 and of the membrana tympani are still undisturbed. It is then 
 readily perceived that between the spina tympanica major 
 and minor there is a pretty tense cord of fibres, into which the 
 processus brevis of the hammer is inserted in a direction toward 
 the anterior border. This cord forms the upper border of the 
 lower and firmer part of the membrane. As soon as the explor- 
 ing needle passes beyond it, it sinks suddenly into the Rivinian 
 recess, while pressing before it the loose cutis and mass of con- 
 nective tissue. And if, moreover, we examine carefully the 
 vaulting of the outer side of the membrana tympani, in a suit- 
 able preparation and with oblique light, we can generally make 
 out this cord running from the processus brevis mallei toward 
 the spina tympanica minor. As far as I could ascertain, this 
 cord is formed from the peculiar tendinous fibres of the mem- 
 brana tympani. "We shall call it the upper cord of attachment 
 of the membrana tympani. It forms the boundary for that part 
 of the membrane which has to be taken into consideration in 
 vibrations of sound. 
 
 On the inner side, the membrana flaccida is continued on from 
 its line of insertion into the tissue of the fold of mucous mem- 
 brane which forms what has been described by Troltsch as the 
 posterior pocket of the membrana tympani, and in whose lower 
 free border lies the chorda tympani. The line of insertion of 
 the membrana tympani unites with that of the above-mentioned 
 fold at the bottom of the Itivinian recess ; here their attachment 
 to one another is stronger than to the bone ; posteriorly, how- 
 t€ver, the line of insertion of the fold of mucous membrane does
 
 MECHANISM OF THE OSSICLES OF THE EAR. 19 
 
 not run parallel with that of the merabrana tympani, but pur- 
 sues its course along the sharp border of the wedge-shaped bony 
 process represented by c in Fig. 1. The outer surface of this 
 process lies parallel with the membrana tympani and a sliort 
 distance to the inside of it, and can even be seen from the out- 
 side as a whitish object shining through the semi-transparent 
 membrane. Lower down on tlie border of this process is the 
 opening wliich gives egress to the chorda tympani. The smaller 
 recess, which can be seen behind the sharp border near c in Fig. 
 1, represents a section of the funnel-shaped projection of the ca- 
 nal of the chorda. The fold of mucous membrane forming the 
 posterior pocket of the membrana tympani reaches down as far 
 as the exit of the nerve, which itself fnrms the border of the 
 pocket. 
 
 The line where the fold of mucous membrane comes in con- 
 tact with the membrana tympani runs from the highest point of 
 the Rivinian recess forward toward the processus brevis of the 
 hammer. This portion of the fold separates the smaller anterior 
 from the larger posterior pocket. Its line of attachment on the 
 hammer we shall describe hereafter. 
 
 Tlie Rivinian recess lies above and a little in front of the 
 membrana tympani. Its greatest diameter extends in a nearly 
 perpendicular line downward from the posterior end of the re- 
 cess, above the spina tympanica minor. I have measured its 
 length in a number of specimens, and find it agrees with that 
 given by Troltsch — 9 to 10 mm. The smallest diameter is in a 
 nearly horizontal direction, and begins somewhat under the spina 
 tympanica major. Its length I found to be from 7^ to 9 mm. 
 These measurements are, as a general thing, the same in infan- 
 tile skulls as in those of adxilts. 
 
 As is well known, the inner end of the external auditory canal 
 is pointed inward and a little downward ; and, besides, the 
 plane that passes through the groove in which the membrana 
 tympani is inserted is inclined at an angle of 55° to the axis of 
 the external auditory canal, while the membranes of l)oth sides 
 form with each other an obtuse angle, opened upward, of 130° 
 to 135°. 
 
 The membrana tympani is not stretched out flat in the ring
 
 20 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 to whicli it is attaclied, but its centre or Jiavel is strongly drawn 
 inward by tlie liandle of the hammer, with -which it is united; 
 for this reason the membrane has the shape of a funnel whose 
 point or end corresponds to the tip of the handle of the hammer, 
 and whose meridian lines are convexed toward the hollow of 
 the funnel. In order to represent this form of the membrana 
 tympani, a point of great importance in tlie mechanics of the 
 conduction of sound, I took a cast with stearin of the upper wall 
 of the external auditory canal and of the outer surface of the 
 membrane, after having first removed the lower wall of the 
 canal, without, however, disturbing any of the connections of the 
 
 membrana tympani. Its outlines are 
 represented inFig. 2, just as I copied 
 them in the camera clara ; ab is 
 Fig. 2. '^^''''^^^ the upper wall of the external au- 
 ditory canal, b c the vertical outline of the membrana tympani. 
 From the figure it is very clear that the radii drawm on the 
 surface of the membrana tympani are convexed outward 
 toward the external auditory canal. At the same time, it can 
 be seen that, as a result of this drawing in of the navel, the 
 upper half of the membrane is made to lie in almost the same 
 direction as the upper wall of the canal, while the lower half 
 stands almost at a right angle with the axis of this canal. This 
 last circumstance is of importance in the examination of the ear 
 with the reflector, inasmuch as this perpendicular portion of the 
 membrana tympani, which is situated, as a rule, just below the 
 tip of the manubrium, reflects back through the external audi- 
 tory passage the light that is thrown in upon it, and thus gives 
 rise to the triangular " bright spot." 
 
 The outer surface of tlie membrana tympani, which is cov- 
 ered with an epithelial layer, the continuation of the horny epi- 
 dermis of the skin of the external auditory passage, owes its pro- 
 perty of reflecting light to the fat which it contains. In a very 
 fresh specimen of the ear, drops of water can be seen running off 
 from this fatty surface as from oiled paj^er. 
 
 The convexity of the meridians of the membrana tympani is 
 least at that meridian in which the handle of the hammer lies.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 21 
 
 The outline of the stearin cast answering to this part is repre- 
 sented in Fig. 3, the position of the hammer 
 being marked by dotted Knes. At the same 
 time, it can be seen in this drawing that the 
 navel lies somewhat nnderthe true centre of "^gTs. 
 
 the membrane. 
 
 The meridian in which the handle of the hammer lies extends 
 upward and forward from the navel toward the anterior limit 
 of the Rivinian recess, so that the processus brevis of the ham- 
 mer, which forms the upper limit of the handle, comes to lie 
 nearly back of the &p\iv which is situated on the outer side of 
 the line of attachment of the membrana tympani, and Mdiich 
 answers to the inwardly pointing spina tympanica major. To 
 this the hammer is attached partly by means of a compact 
 ligament (ligamentum mallei anterius,) and partly by its so- 
 called long process, (processus Folianus.) The latter, so long as 
 it exists, lies in a furrow on the inner border of the process. 
 
 While, on the one hand, the tip of the manubrium draws the 
 navel of the membrana tympani inward, on the other, the pro- 
 cessus brevis at the base of the manubrium tends somewhat to 
 press it outward. 
 
 The membrana tympani consists essentially of a peculiar 
 tendinous membrane, which, although only one-twentieth of a 
 millimetre thick, is yet comparatively very strong. Externally 
 it is clothed with a thin continuation of the skin of the external 
 auditory canal, internally by a thin continuation of the mucous 
 membrane of the middle ear. Taken together, these layers have 
 a thickness of 0.1 mm. The outer skin layer consists principally 
 of a continuation of the epidermis, supported by a thin layer of 
 loosely-woven bundles of connective tissue. It can be removed 
 entire from the greater portion of the surface of the membrane, 
 excepting at the Rivinian recess and along the handle of the 
 hammer,* where it is more closely united with the thickened and 
 cartilage-like tissue of the membrane. From the lii vinian recess 
 along the upper wall of the external auditory canal there runs 
 
 * Gruber's obliquely descending fibres of the membrana tympani unite at tluB 
 point with the fibres of the cutis, forming in a mechanical sense — although 
 perhaps they must be separated histologically — the deepest layer of the same.
 
 22 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 a line, along wliicli the skin is more strongly attached to the 
 bone. The fibres of the cutis dip down here into the fissura 
 Glaseri, Avhich at an earlier period was a cleft dividing the 
 squamous portion from the os tympanicum, (Fig. 1, fg^ 
 
 The middle and stronger layer of the membrana tympani is 
 fibrous, and consists ])artly of radiating, partly of circular fibres. 
 The radiating fibres lie on the outer side, the circular on the 
 inner side of the layer. In the anterior half of the membrane 
 the radiating fibres proceed from the tip of the manubrium 
 mallei as a centre. On the posterior half, however, they run 
 nearly parallel to each other from the entire length of the 
 manubrium. Their thickness is least along the border of the 
 membrane and grows gradually thicker on approaching the 
 end of the manubrium, where they are packed more closely 
 together. 
 
 In the centre of the membrane the circular fibres form a very 
 thin layer which gradually increases in thickness toward the peri- 
 phery ; at the extreme periphery, however, they disappear alto- 
 gether (according to Gerlach), or at least (according to J. Gruber) 
 form a very much thinner layer than in the centre. In tlieHivinian 
 recess the circular fibres are strongly developed and of a satin- 
 like appearance ; they form here a cord-like boundary for the 
 upper side of the firmer part of the membrana tympani and 
 i ntersect at a very small acute angle the radiating fibres, which 
 at this point radiate, not from the navel, but from the processus 
 brevis of the hammer. Here, too, they are intermingled with 
 straggling fibres of the cutis. 
 
 Tlie tendinous fibres of these layers are very dense and un- 
 yielding, they lie close to one another, and offer very great 
 resistance to any distending force. Through their great power 
 of elastic resistance they differ materially from the very much 
 more yielding yellow elastic tissue. The substance of the 
 membrana tympani swells in acetic acid and solutions of potash, 
 as is the case with tendinous tissue, but not with the elastic 
 tissue. I found that, like tendinous tissue, it would soon dis- 
 solve completely in a boiling potash solution, leaving behind 
 mere traces of elastic tissue, consisting partly of vessels, and 
 partly of a very thin continuous membrane — probably the base-
 
 MECHANISM OF THE OSSICLES OF THE EAR. 23 
 
 ment membrane of tlie mucous layer on the inner side of tlie 
 merabrana tympani. 
 
 This peculiarity of construction of the membrana tympani 
 is a very important element in the mechanical working of this 
 membrane, as we shall see further on. It is not to be considered 
 as an elastic, yielding membrane, but as an almost inextensible 
 one.. Its want of capacity for yielding can be appreciated when 
 one tries to tear it with needles, either after it has been removed 
 and spread out upon a glass slide, or while it still remains attach- 
 ed in its natural position. It cannot be drawn out like a piece 
 of rubber, or a softened animal bladder, but offers very power- 
 ful resistance to tension, and forms folds about the spot that is 
 being stretched, as in a collodion membrane. 
 
 §3. 
 Attachments of the Hammer. 
 
 The manner in which the hammer is attached to the mem- 
 brana tympani has been thoroughly described by J. Gruber in 
 a monograph published recently by him. The part of the 
 membrana tympani corresponding to the attachment of the 
 hammer is thickened, partly by strong fibres of the cutis layer 
 extending from the Rivinian recess along the manubrium, partly 
 by an accumulation of fibro-cartilaginous tissue. The perios- 
 teum of the hammer, along both surfaces of the manubrium, 
 is continuous with the fibro-cartilaginous layer, whose borders 
 are thus closely united to the hammer. Near the lower end of 
 the manubrium the union between the bone and the thickened 
 tissue of the membrana tympani is very close; near the pro- 
 cessus brevis, however, a looser layer intervenes between the 
 bone and the membrane, or there may even be a kind of incom- 
 plete joint-space, which is limited on both sides by the closer 
 union between the periosteum of the hammer and the borders 
 of the cartilao-inous laver, tog-ether with the fibrous tissue of the 
 membrana tympani. 
 
 Tlie hammer by means of its handle draws the navel of the 
 membrana tympani inward ; to maintain a close union between 
 these two parts the connection between them should be strongest 
 at this point. At the processus brevis the hanmier simply presses
 
 24 
 
 MECHANISM OF THE OSSICLES OF THE EAR, 
 
 against tlie membrana tjmpani; consequently here a less 
 intimate union suffices, while at the same time it aiFords a 
 possibility for slight motions of the hammer upon the mem- 
 brane, a necessity whose conditions we shall investigate more 
 thoroughly further on. 
 
 The second and relatively strongest attachment of the ham- 
 mer is to the spina tympanica major. The end of this spur 
 extends close up to the neck of the hammer, into the hollow at 
 <?, (Fig. 4,) just above the root of the processus Folianus. The 
 hammer, in this sketch, is seen from the outside ; cjp is the 
 head, h the processus brevis, m the handle, * the articular sur- 
 face for the reception of the anvil. The processus Folianus lies 
 along tlie inner border — that turned toward the tympanum — of 
 the spina, (the marginya of Fig. 1,) so that between Z, the end 
 of the processus Folianus, and the hollow at rZ, the border of the 
 spina and the corresponding border of the processus Folianus run 
 almost parallel to each other, with a space intervening between 
 them of about |-mm. From the upper sur- 
 face of tlies pina a bony margin extends uj)- 
 ward in such a manner as to fit pretty 
 closely to the side of the hammer between 
 d andy, the intervening space between the 
 hammer and the bony margin being con- 
 tinuous with the space previously mentioned 
 and of the same width with it. This entire 
 fissure is bridged over with short and tense 
 ligamentous fibres ; longer fibres of the same 
 kind start from the surface of the spina and 
 from its downward projecting border, and, 
 converging toward the point d of the ham- 
 mer, envelop both the lower border and the outer surface of tlie 
 processus Folianus, so that it lies completely hidden in this mass 
 of tendinous fibres which forms the ligamentum anterius mal- 
 lei and is covered by a fold of mucous membrane. 
 
 As for the processus Folianus of the hammer, I must remark 
 that in children it is a long elastic blade of bone extending as 
 far as to the fissura Glaseri. As regards, however, its condition 
 in adults, I must side with those wlio describe it as having
 
 MECHANISM OF THE OSSICLES OF THE EAR. 25 
 
 dwindled down to a short stump. I would state that, in prepar- 
 ing a number of temporal bones, I took particular care to no- 
 tice whether this process was not perhaps broken oif in the 
 attempt to loosen the hammer. To ascertain this, before the 
 hammer had been at all disturbed in its natural position, I 
 searched for the processus Folianus with the point of a fine 
 needle, inserting it as a probe between the fibres of the ligamen- 
 tum anterius mallei. In this way I could follow it distinctly 
 for a short distance, when it came abruptly to an end in 
 the midst of the anterior ligament, and 1 could feel nothing 
 like a continuation of the bony blade, which would have been 
 present if the process had simply been broken. 
 
 I must remark, moreover, that this remaining stump of the 
 processus Folianus does not lie in immediate contact with the 
 bony mass of the spina; it is attached to it throughout by 
 means of short fibrous bands. Hence in a specimen where the 
 hammer retains its natural position, and all of its attachments 
 are preserved entire, it is possible, by pressing upon the base of 
 the processus Folianus with the point of a needle, to move this 
 part of the hammer not only upward and downward but also 
 backward and forward, as far as the short fibrous bands of 
 the ligamentum anterius will permit. The contact of bone with 
 bone, if it existed here, would ofier an obstacle to any one of 
 these motions. 
 
 If we, therefore, leave out of the account the longer supei*fi- 
 cial fibres, the ligamentum anterius mallei appears in the main 
 to be a very short and broad band whose line of insertion ex- 
 tends from I tof on the hammer (Fig. 4) ; from Itod it lies 
 opposite the inner border of the spina tympanica major, but 
 near the point d it is nearly opposite to a bony ridge which ex- 
 tends from d on the spina upward toward f. I must add, 
 moreover, that, above and below, this band becomes lost in 
 a fold of mucous membrane. Above, tlie fold of raucous 
 membrane follows pretty closely the contour of the bone 
 (see Fig. 4) ; it is here sickle-shaped and quite thin, because 
 here the outer wall of the tympanum is everywhere in close 
 proximity to the head of the hammer. This fold of mucous 
 membrane finally terminates on the upper surface of the head
 
 26 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 of tlie liamnier, and in its border lies the short, round ligamen- 
 tum mallei superius, which descends obliquely downward and 
 outward upon the head of the hammer. Its office is, there- 
 fore, to restrain all motions of the latter outward. 
 
 The ligaraentum anterius is prolonged downward from I into 
 two folds of mucous membrane, one of which runs from the 
 base of the processus Folianus toward tlie point h of the process- 
 us brevis. Its opposite line of insertion lies on the membrana 
 tympani. This is the fold wliich separates the anterior from 
 the posterior pocket of the membrana tympani, dividing them 
 in such a way that the space above the processus brevis belongs 
 chiefly to the posterior pocket.* The second prolongation of 
 the liganientum anterius downward is a thin fold with a free 
 margin, which follows the lower border of ?, (after having en- 
 veloped this process,) and then continues down along the con- 
 tour of the bone as far as to the tendon of the tensor tympani 
 muscle (Fig. 4). In the drawing this spot corresponds to where 
 the curved line at 5 meets the contour line of the bone. This 
 last-named fold is a limit between the anterior pocket and the 
 cavity of tlie tympanum. 
 
 From the present group of ligaments and mucous folds, 
 which in Fig. 4 follows always the contour line of the bone 
 from h to c^, and at d is composed of the shortest and strongest 
 fibres, a second group branches off at d, to which I shall give 
 the name of liganientum mallei externum. It springs from 
 a ratlier prominent bony ridge on the hammer, which may be 
 seen extending from d to c in Fig. 4, and is inserted into the 
 sharp border of the Rivinian recess ; at the same time it follows 
 posteriorly the line of attachment of the posterior pocket of the 
 membrana tympani (hence in Fig. 1, from/" to c along the con- 
 tour line of the drawing). This ligament consists of a number 
 
 *In tlie "Archiv fiir Olirenlieilkunde," vol. iii., pages 255-266, Dr. Prussack 
 has given a description of the pockets of the membrana tympani, which dif- 
 fers from this. He describes the space above the processus brevis of the ham- 
 mer as a special upper pocket, separated from the posterior pocket by a parti- 
 tion wall ; I have never been able to find such an one. The supposed entrance 
 into this pocket, above and in front of the top of the head of the hammer, 
 leads into the space above the liganientum mallei externum, and therefore not 
 at all to the membrana tympani.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 27 
 
 of distinct, satin-like, tendinous fibres, whicli radiate from the 
 short crest of the hammer (lying between d and c) toward 
 the much broader curved line of attachment on the temporal 
 bone. 
 
 In Fig. 5, this ligament is represented as seen from above ; 
 e g \s> its line of attachment to the temporal bone. The tympa- 
 num in this preparation was opened from above, and its upper 
 and outer wall suificiently chiseled away to permit of a free 
 view between this wall and the surface of the bones facing it. 
 
 m is the head of the hammer, i the body of the anvil, hi the 
 end of its short process, and Tu the entrance to the tube. Deep 
 down a jjart of the stirrup St can be seen, and the tendon of its 
 muscle M.st^ and further on the tendon of the tensor tympani 
 with the funnel-shaped osseous canal from which it issues. 
 Gh.Th the chorda tympani, whicli marks the free border of the 
 folds of mucous membrane limiting the ])ockets ; at / are the 
 upper fibres of the ligamentum mallei anterius, which arise 
 above the spina tympanica major SjJ.t. The prominent crest 
 on the neck of the hammer, from which the fibres of the liga- 
 mentum externum radiate, is here distinctly visible. 
 
 The stronofest and most tense bundle of fibres of this last- 
 named ligament is the posterior one, which is inserted at g. 
 The line of the direction which it follows, when continued, 
 passes through the end of the spina. It is also this bundle 
 which represents chiefiy the axis of rotation of the hammer. 
 I prefer, therefore, to call this posterior group of fibres of the
 
 28 MKCHANISM OF THE OSSICLES OF THE EAR. 
 
 ligamentum exteniutn by the special name of ligamentum 
 mallei postieiim, because in a mechanical sense it has indeed a 
 special importance. In a specimen where all the attachments 
 of the ossicles remain undisturbed, the great tension of these 
 posterior fibres can be distinctly felt by pressing upon them with 
 the point of a needle ; at the same time the border of the fold 
 of mucous membrane, in which the chorda tympani lies, will 
 always be found in a relaxed condition ; and moreover tlie an- 
 terior bundle of fibres of the ligamentum externum (at e in 
 Fig. 5) is never very tense unless the tensor tympani is in a 
 state of contraction, or the membrana tympani is being forced 
 outward. By pressing harder with the needle upon the chord 
 of the ligamentum posticum, the hammer can be made to in- 
 cline appreciably. In forcing the membrana tympani inward 
 or outward, it is this very group of fibres of the ligamentum 
 externum which moves the least of all the attachments of the 
 hammer. The I'eason for this slight displacement will appear 
 further on. 
 
 If we suppose the line of direction of the ligamentum posti- 
 cum continued on through tlie liammer, it will be found to meet 
 and run in the same direction with the middle and strongest fibres 
 of the ligamentum anterius which take tlieir origin from the 
 spina tympanica major. These two sets of fibres, which, al- 
 though separated by the intervening body of the hammer, are 
 still in a mechanical sense one band, we may call the axis-band 
 of the hammer. This band alone is sufficient to hold the ham- 
 mer in its natural position, even after the anvil has been care- 
 fully separated from it ; but if the tendon of the tensor tym- 
 pani be still tense, this position will then be really quite firm. 
 In Fig. 4, the approximate position of the axis of the hammer 
 is marked by a dotted line a a. 
 
 The cords forming the anterior portion of the ligamentum 
 externum (Fig. 5, e) are made up of shorter fibres which are di- 
 rected outward toward the edge of tlie membrana tympani, 
 where it is attached at the bottom of the Ilivinian recess. As 
 they lie above the axis of the hammer, they oppose any move- 
 ment of the manubrium mallei or of the membrana tympani 
 outward toward the external auditory canal. Hence their es-
 
 I 
 
 MECHANISM OF THE OSSICLES OF THE EAR. 29 
 
 sential office is to restrain the rotation of the manubrium out- 
 ward. In a suitable preparation, like that of Fig. 5, this fact 
 can be readily verified. When the membrana tympani is press- 
 ed inward, or the head of the hammer outward, these fibres 
 become relaxed. They allow only a slight rotation of the ma- 
 nubrium outward, even after the tendon of the tensor tym- 
 pani, the stapedo-incudal joint, and the ligamentum superius 
 mallei have been divided. By pressing upon these same fibres 
 from above with the blunt end of a needle, and thereby putting 
 them on the stretch, the inclination of the membrana tympani 
 inward will be increased. Finally, it must be remarked that, 
 whenever the tensor tympani is strongly stretched, the manu- 
 brium mallei is prevented from being drawn further inward 
 by the tense condition of the membrana tympani, while the 
 axis-band of the hammer is prevented from being pulled inward 
 beyond a certain limit by the above-mentioned set of fibres of 
 the ligamentum externum ; when these become tense the limit 
 has been reached. At ,this point, the traction of the tensor 
 tympani will be transferred to these fibres, and can no longer 
 affect the axis-band. 
 
 While the ligamentum externum, on the one hand, protects 
 the axis-band of the hammer from being pulled too strongly 
 inward, the upper and lower fibres of the ligamentum anterius, 
 on the other hand, prevent it from being drawn too strongly up- 
 ward or downward. If the hammer were to rotate about its 
 attachment to the spina as a centre, with its head backward 
 and the end of the manubrium forward, the upper fibres of the 
 ligamentum anterius would be put upon the stretch ; and if in 
 the opposite direction, the lower fibres. Hence it happens 
 that, even after the anvil has been removed, the ligaments 
 hitherto described remain unaffected, the hammer is still able to 
 resist any such inclinations, and remains jjretty steady in its natu- 
 ral position. The uppermost fibres of the ligamentum anterius 
 usually approach the head of the hammer in an inward direc- 
 tion, (as can be seen in Fig. 5 at/",) and hence, like the liga- 
 menta superius and externum, they become tense when the 
 membrana tympani is pushed outward. 
 
 The tension of these ligaments, in the natural state of things, 
 
 ki i
 
 30 MECHANISM OF THE OSSICIiES OF THE EAR. 
 
 is increased by tlie elastic force of the comparatively strong 
 miisculus tensor tympani, whose tendon is attached to the ham- 
 mer on the anterior half of its inner side facing the tube, at the 
 commencement of the manubrium, and a little lower down than 
 where the processus brevis projects from the side. In Fig. 9, 
 the line of insertion of this tendon is shown extending from 
 above obliquely downward and backward. The muscle, as'is 
 well known, lies in a special bony canal whose course runs pa 
 rallel with and above the Eustachian tube, by means of which 
 the cavity of the tympanum communicates with the pharynx. 
 The further end of the muscle arises outside of this canal, from 
 the under surface of the pyramidal portion of the petrous bone 
 and from the cartilaginous portion of the Eustachian tube. It 
 then proceeds through its appropriate canal, whose open end, 
 toward the cavity of the tympanum, terminates in a spoon-like 
 process ; around this the tendon of the muscle passes, and then 
 finally crosses the cavity of the tympanum obliquely {Tt^ Fig. 5) 
 toward its point of insertion on the hammer. The direction 
 followed by the tendon is nearly perpendicular to the plane 
 drawn through the border of the membrana tympani, so that 
 its line of traction varies only a little downward and forward 
 from such a perpendicular. On the other hand, it forms a mo- 
 derately acute angle with the lower portion of the handle of 
 the hammer and with the anterior portion of its axis of rotation. 
 The tensor tympani is a penniform muscle; it originates 
 from the periosteum of the upper wall of the canal in which it 
 lies ; its tendon lies next to the under surface of the canal, and 
 presents a free, smooth surface to tlie smooth periosteum. The 
 muscular fibres are rather short, and hence the tendon extends 
 back to the very end of the canal. The periosteal tube which 
 sheathes the muscle is continued orer the tendon in its course 
 through the cavity of the tympanum ; its outer surface is there 
 covered with the mucous membrane of that cavity. Toynbee 
 calls this free part of the sheath the tensor ligament of the 
 membrana tympani. The separation of the tendon from its 
 sheath — if we compare the descriptions of difterent observers — 
 seems to be more or less complete ; in an anatomical collection 
 of this city I have seen a specimen where the perfectly smooth
 
 MECHANISM OF THE OSSICLES OF THE EAR. 31 
 
 tendon was surrounded by a perfectly free sheatli, just as 
 Toynbee describes it ; in microscopic sections, however, Ilenle 
 has seen the two united by pretty strong bands of connective 
 tissue. As the hammer, however, requires exceedingly little 
 space for its excursions, there is no need whatever that the ten- 
 don should have much room for motion. 
 
 The tensor tympani draws the handle of the hammer, and 
 with it the membrana tympani, inward, thereby putting the 
 latter on the stretch. This action can be readily seen on a spe- 
 cimen where the canal of the muscle and the cavity of the tym- 
 panum have been opened from above. By pulling upon the 
 tendinous fibres of the muscle within the canal, the membrana 
 tympani becomes tense. As the point of insertion of the 
 muscle lies but a trifle lower down than the axis-band of the 
 hammer, the band itself will also at the same time be stretched 
 inward, and especially the posterior portion of it, the ligamen- 
 tum mallei posticum, which lies very nearly in the same line 
 with the line of traction of the tensor tympani. The position 
 of the hammer is thus made quite firm, even though the tendon 
 be but moderately tense. We must remember here that a 
 slight traction, when made at right angles to the length upon 
 an inextensible cord which is already tense, can very materially 
 increase its tension ; and also that during life a muscle in a 
 state of rest must be considered as a very yielding, though al- 
 ways slightly tense elastic band, whose tension can be very con- 
 siderably increased by active contraction. Aside from the fact 
 that the tensor tympani, on account of its penniform construc- 
 tion, is, mechanically speaking, equivalent to a muscle of much 
 greater diameter and shorter length of fibre, we can consider 
 its simple elastic traction, without the occurrence of any ac- 
 tive contraction whatever, as a pretty important power. 
 
 In this way it is clear that the hammer, so long as it retains 
 its natural attachments — although these be but yielding bands 
 — and even after the division of the stapedo-incudal joint, is 
 capable of only a limited motion in the direction of rotation 
 about the above-mentioned axis; and that any attempts to 
 move it in a different direction meet with very strong resis- 
 tance. Anteriorly its axis is held securely by the ligamentum
 
 32 MECHA^■ISM OF THE OSSICLES OF THE EAR. 
 
 antcrius, and the processus Foliaiius, which is embedded in its 
 meshes ; posteriorly by the posterior fibres of the ligamentum 
 externum : the two togetlier we liave called the axis-band of 
 the hammer. Tliis is always pretty tense, even after the tendon 
 of the tensor tympani has been divided ; but if the latter 
 draws upon the axis-band at a right angle, its tension then be- 
 comes very great. 
 
 The hammer thus fastened, possesses, besides the tendon of 
 the tensor tympani, the following bands, capable of restraining 
 any rotation of the handle outward : 1. the middle and ante- 
 rior fibres of the ligamentum externum, 2. the ligamentum 
 superius, 3. the upper fibres of the ligamentum anterius. The 
 inembrana tympani itself acts as a band of restraint against 
 too strong rotation of the handle of the hammer inward. 
 
 As far as the slight yielding capacity of the axis-band 
 and of the upper (relatively lower) fibres of the ligamentum 
 anterius will allow, the head of the liammer can incline 
 forward and backward, or rotate about a vertical axis. 
 Nevertheless, when the hammer is connected with the anvil 
 these motions are thereby still further limited. Still, we shall 
 see that the motion of the hammer and anvil together requires 
 a certain amount of yielding capacity on the part of the axis- 
 band. 
 
 Attachments of the Anvil. 
 The body of the anvil is united to the hammer through the 
 medium of a joint. Its long process extends downward, and 
 has at the end (which is bent somewhat inward) a small articu- 
 lar surface for the stirrup. The short process extends back- 
 ward, and its extremity, on whose lower aspect there is a small 
 incomplete articular surface, lies in an appropriate hollow cut 
 out of the bony wall of the tympanum, at a point where the 
 cavity of the latter merges into that of the cells of the mastoid 
 process. The capsule of this joint, on it supper surface at 
 least, is composed of strong tendinous fibres which extend in- 
 ward, backward, and outward from the short process (see Fig, 
 5, hi). In the same figure, i represents the body of the anvil, 
 and * the capsular ligament of the malleo-incudal joint.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 83 
 
 The shape of the hist-nanied articiihar surface is usually de- 
 scribed as resembling a saddle ; it must be remarked, however, 
 that, unlike the saddle, not only the convex sides come together 
 to form a ridge that is almost sharp, but also the concave ; and 
 the union of the two forms a continuous and almost flat surface 
 on either side of the ridge. In order to gain a clear idea of 
 the mechanism of this joint, it is better, I believe, to make use 
 of a different comparison than that of the surface of a saddle. 
 It is, in fact, like the joint used in certain watch-keys, where the 
 handle cannot be turned in one direction without carrying the 
 steel shell with it, while in the opposite direction it meets with 
 only slight resistance. As in the watch-key, so here the joint 
 between hammer and anvil admits of a slight rotation about 
 an axis drawn transversely through the head of the hammer 
 toward the end of the short process of the anvil ; a pair of co^s 
 oppose the rotation of the manubrium inward, but it can be 
 driven outward without carrying the anvil with it. 
 
 If such a joint had to be constructed of metal, we should 
 make use of screw-surfaces. A hollow cylinder, cut as A is 
 represented in Fig. 0, and upon which the y-;!.-^-^'""'^--''::;-^.. 
 
 piece B (marked in dotted lines) fits, would ( \. B ) ] 
 represent the normal shape of such a joint. | f ""-■■ ^-■■""^ "- "f I 
 It is clear that A and B, revolving in the \"\ I \ 
 
 direction of tlieir res})ective arrows, must I i \ \ 
 
 necessarily strike against each other with 
 their cogs a and h ,' hence, in this direction 
 tlieir rotation is limited. In the opposite 
 direction, however, their rotation is free, and 
 is accompanied by a slowly increasing sepa- 
 ration of the two cylinders. The mechanic, 
 in making such a joint, usually employs a 
 hollow cylinder, because in the neighbor- 
 hood of the axis, were the cylinder solid, 
 the screw surface would incline uj)ward at a 
 pretty steep angle, like the inner border of a 
 winding staircase, and hence would be difficult to execute. 
 The articular ends of the bones, which are covered with a layer 
 of elastic yielding cartilage, filling out all the irregularities of 
 3 
 
 FJK 0.
 
 34 
 
 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 surface, show, as a (general rule, only a modified form of the above 
 geometrical outlines," one where the margins are rounded oif, etc. 
 The periphery of the m all eo-in cud al joint is not a regularly form- 
 ed screw outline. If we imagine it rolled out upon a plane from 
 the cylindrical circumference of the joint, it would have more 
 or less the form represented in Fig. 7, where the ends «„ «, are 
 naturally continuous. Near the axis of the joint the surface 
 assumes, not exactly the shape of a screw as in a winding stair- 
 case, but rather more that of a cone. If we suppose straight 
 lines to be drawn from a point in the axis of the cylinder to all 
 points of the line of circumference «„ «,, we shall obtain an ap- 
 proximate idea of 
 the shape of the 
 joint in question. 
 In the case of the 
 hammer, we would 
 have to place the 
 apex of the cone 
 thus formed somewhat lower down than the straight part of the 
 line J„ «„ a^ &„ so that this portion of the surface of the cone 
 would be concave on the hammer, while on the anvil the cor- 
 responding part is convex. Such a joint may therefore be said 
 to consist of four nearly plane surfaces, which come together 
 
 at its centre, and along its l^order 
 show the following margins : 1. 
 ^0 c„ 2. c„ J„, 3. c. h„ 4. \ a, a, i„ 
 while the upward-turned surface 
 of the joint, like a saddle, shows 
 two salient borders c„ and J„ and 
 two reentering Z»„ and c,. 
 
 In Fig. 8, the hammer is rep- 
 resented as it appears from above 
 and inside. The letters aJ„ c„ J, c, 
 have the same meaning as in Flo;. 
 7. The fiat portion of the arti- 
 cular surface is seen foreshortened. P.F. is the stump of the 
 processus Folianus ; Cr., the commencement of tlie bony crest 
 from which the ligamentum mallei posticum arises ; T.t.h the 
 
 I 
 
 Fiff. 8.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 35 
 
 tendon of the tensor tympani. As can be seen, the part a 
 «, lies on the upper side of the malleo-inciidal joint, while the 
 line of junction of the two cogs c„c, lies lower down between 
 the handle of the hammer and the .long process of the anvil ; 
 the cog <?„ of the hammer lies on that side of the joint wliicli 
 is turned toward the membrana tympani ; the cog of the 
 anvil, on tlie median side. The handle of the hammer cannot 
 therefore rotate inward without carrying the anvil with it ; the 
 limit of rotation outward is governed by the flexibility of the 
 ligaments and cartilaginous covering of the articular surfaces. 
 
 In the articular surfaces of both ossicles the point a^ «, lies 
 at a greater distance from the axis than the border of the cog 
 c„ (?, ; ill other words, that portion of the surfaces which is nearly 
 plane is greater than the surfaces of the cogs, and consequently 
 the entire surface of the joint is elliptical in shape, with the 
 longer axis running vertically. It should be stated, furthermore, 
 that the apex of the cone-shaped articular surface (we have 
 assumed the cone to be the type of this joint surface) does not 
 come to a point, but is rounded off, as in a saddle. 
 
 A conical surface, such as may be made by aid of Fig. 7, 
 cannot be made to revolve upon its exact counterpart without 
 loss of contact at some point ; for when the two screw 
 lines h^ c„ and 5, c, glide upon each other at the periphery of the 
 joint, the central portion of the joint and the more level portion 
 will separate each from the corresponding portion, leaving the 
 two bones in contact only at the above-mentioned screw lines. 
 Since the working space in such a motion is very small, there- 
 fore we may readily perceive that in a fresh specimen a com- 
 pressible cartilage may quite fill it. 
 
 This peculiar mechanism of the joint can be readily seen 
 in dry specimens of the ossicles; one match may be fastened 
 with sealing-wax to the head of the hammer, just above and ini 
 the same direction with the processus Folianiis, and another 
 to the anvil, at the end of its processus brevis, and in the same 
 direction with it; by using these matches as handles, and 
 bringing the two articular surfaces together, the hammer may 
 be rotated about its match as an axis, while the anvil is simply 
 pressed lightly against it. If the hammer is rotated in the
 
 36 MECHANISM OF THE OSSICLES OF THE EAR, 
 
 direction from the liead toward the short process and liandle, it 
 will grasp the anvil very tinnly, and compel it to follow. When 
 the hammer is rotated in the opposite direction, the two articular 
 sm'faces at once separate from each other and the anvil remains 
 stationary. 
 
 The two articular surfaces are kept in contact at their peri- 
 pheries by a capsular band which is inserted into grooves in the 
 bone at various points surrounding the joint. This capsular 
 •band is not very strong ; it tears when the ossicles are exposed 
 to comparatively slight strains. Of this band the strongest 
 iibres are those which proceed from the cog of the hammer ; 
 -at this point, too, a few fibres of the ligamentum externum of the 
 hammer pass over to the anvil. The length of an excursion of 
 the malleo-incudal joint, measured at the lower end of the long 
 process of the anvil, amounts to about half a millimetre; and 
 this point being nearly 6 millimetres distant from the axis of 
 the joint, the rotation of the two ossicles upon each other will 
 hardly reach 5'^. 
 
 So long as the hammer and anvil retain their natural con- 
 nections with each other, and with the petrous bone, (the anvil, 
 however, being separated fi'oin the stirru}),) they may rotate 
 •conjointly in such a manner that the handle of the hammer, the 
 long process of the anvil, and the membrana tympani will all at 
 the same time move either inward or outward. The hammer 
 by itself would rotate about its axis-band as axis, but owing to 
 its connection with the anvil, this mode of rotation is somewhat 
 modified. It will be noticed, in the first place, that the short pro- 
 cess of the anvil (Fig. 5, hi) is attached to the petrous bone at a 
 point considerably to the inside of the prolonged axis band of 
 the hammer. In a simple rotation about a fixed axis, only those 
 ipoints are at rest which are in the very axis of rotation. The 
 distance, furthermore, of any individual points in the rotating 
 body, that are situated outside of the axis of rotation, from an 
 external fixed point, cannot remain unchanged during rotation; 
 in the present case the fixed point is where the short process of 
 the anvil is attached to the petrous bone. The only exception 
 io this, however, is in the case of infinitely small rotations, where 
 ihose points of the rotating body, that lie in a plane carried
 
 MECHANISM OF THE OSSICLES OF THE EAR. 37 
 
 through the axis of rotation and the external fixed point, remain 
 at the same distance from that lixed point. This is not the 
 case with the head of the hammer, Avhich is situated above the 
 axis of rotation ; hence, when the handle of the hammer is rotated 
 inward, the head will be lifted away from the point where the 
 short process of the anvil is bound down. Now, inasmuch as 
 the anvil is, so to speak, suspended by pretty short and rather 
 unyielding bands between the head of the hammer and the 
 above-mentioned point of attachment, and inasmuch as it 
 maintains an immovable position, the rotation of the handle of 
 the hammer inward must be accompanied by a slight inclina- 
 tion of the head backward toward the anvil, simultaneously 
 with a motion of the handle forward. That such an inclina- 
 tion of the head backward actually does take place, may be in- 
 ferred from the tension that is visible during rotation in the 
 capsular ligament at the upper side of the malleo-incudal joint, 
 in the uppermost libres of the ligamentum mallei anterius, and 
 in the bands which give firmness to the tympano-ineudal joint. 
 By pressing the membrana tympani inward with the head of a 
 pin, we can see, with an ordinary magnifying-glass, that these 
 two capsular ligaments become tense the moment the pressure 
 is exerted. Furthermore, if we press Avith a needle upon the 
 .short process of the anvil, while the membrana tympani is thus 
 being pushed inward, we shall feel and see that this process 
 does not lie in contact with the bottom of tlie shallow groove in 
 which it fits, but is lifted slightly above it, and that, when we 
 bring this process into contact with the groove by pressure from 
 above, the upper, satin-like accessory ligaments of .the joint 
 become relaxed, and are thrown into folds. On the other hand, 
 the tip of this short process lies in contact witli the wall of the 
 tympanum, which rises up on its outer side ; its connections, 
 moreover, with this outer wall are such that it can glide a little 
 upon the latter in a vertical direction. At the same time, the 
 anvil is held suspended free in the air by the hammer, so that 
 in its normal position it comes in contact with bone only at the 
 outer side of the end of the short process. If, however, the 
 handle of the hammer and the membrana tympani are forced 
 outward, the tip of the sliort process of the anvil will glide
 
 38 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 down into its appropriate groove in the bone and will press upon 
 it with its under surface. 
 
 With a joint of this nature, a slii^-lit displacement must neces- 
 sarily take ])lace between the hannner and anvil whenever the end 
 of the handle of the hammer is driven inward. If we imagine 
 for a moment the anvil and hammer immovably joined together, 
 and the latter made to revolve about its axis-band, the end 
 of the short process of the anvil, lying out of the line of this 
 axis, will necessarily be lifted from its bed wdienever the mem- 
 brane of the drum is driven inward. To lower the end of the 
 short process again and bring it back to its place, the anvil 
 must rotate slightly on the hammer. A limited degree of such 
 motion is possible from the saddle-shape of the malleo-incu- 
 dal joint. At the same time the long process of the anvil 
 approaches slightly the handle of the hammer. This latter 
 motion can be observed on specimens where the stapedo-incudal 
 joint has been severed, while all the other connections remain 
 undisturbed. It is just in this position of the two ossicles, more- 
 over, that the lower parts of the joint surfaces, which are here 
 armed witli teeth, press most upon one another (see Fig. 8). 
 This can be ascertained to be a fact by placing the two dry os- 
 sicles together in the manner described above and noticing 
 in which position they lit the closest. 
 
 The nature of this joint further requires a slight displacement 
 on the part of the hammer. If its head must incline toward 
 the anvil, it cannot do so without drawing the axis-band 
 away from a straight line. The anterior side of the neck with 
 the ])rocessus Folianus and ligamentum anterius would have to 
 be lifted up, while the posterior side of the neck Avith the pos- 
 terior fibres of the ligamentum externum were being drawn 
 down. The first of those motions, however, could hardly take 
 place on account of the spina tympanica posterior, which lies 
 immediately above the processus Folianus, and against which the 
 latter would at once strike. All the more marked, then, would 
 have to be the sinking of the posterior side of the neck, and, with 
 it, of the entire hammer, thus causing a greater tension of the 
 fibres of the lig. mallei post, which run from the hammer in a 
 backward and somewhat upward direction.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 39 
 
 These views are in harmony with a short notice recently pub- 
 lished by Politzer.* He attached line glass rods as levers to the 
 ossicles in order that he might determine more accurately their 
 individual axes of rotation. He put the ineml)rana tympani in 
 motion hy compressing the air in the external auditory canal. 
 In this way he ascertained that the axis of rotation of the ham- 
 mer runs through the end of the processus Folianus, and that of 
 the anvil through the extremity of its short process, but that 
 both of these axes are movable. 
 
 It appears to me that it is in a great measure owing to the 
 slight changes in the axis of the hammer, caused by the pecu- 
 liar nature of the anvil's attachments, that the navel of the mem- 
 brana tympani always moves in a normal direction relatively to 
 the plane of insertion of this membrane. For inasmuch as the 
 axis band of the hammer is situated obliquely to the])lane of in- 
 sertion of the membrana tympani, every motion of the manu- 
 brium mallei inward would also at the same time produce a 
 slight displacement of the navel of the membrane backward. 
 But, through the medium of the anvil, the head of the hammer 
 is at the same time drawn backward, thereby causing a motion 
 of the handle forward in the opposite direction. 
 
 Again, the navel of the membrana tympani lies at a greater 
 distance from the plane of insertion of this membrane than tlie 
 axis of rotation of the hammer, (excepting, possibly, its extreme 
 anterior end at the spina tympanica,) so that every motion in- 
 ward of the manubrium would also carry the navel of the mem- 
 brana tympani a little upward (that is, in the direction of the 
 head of the hammer). This upward motion is counteracted, as we 
 have just described above, by the circumstance that the hammer 
 as a whole is drawn somewhat downward by the anvil (in a rota- 
 tion inward of the manubrium). 
 
 In this manner, therefore, both these deviations are corrected 
 in the motion of the navel of the membrana tympani ; the oidy 
 kind of motion that remains to it, then, is that which takes place 
 in a direction at right angles to the plane of insertion of the 
 membrana tympani. 
 
 At the same time it will be apparent that in these displace- 
 
 * Wochenblatt der K. K. Gesellscliaft der Aerzte. Wien, 18G8, Januar 8.
 
 40 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 ments of tlie liainmer its short process nnist glide a little upon 
 the inembrana tyinpani. Such a gliding motion is rendered 
 possible by the peculiar manner (described by J. Gruber) in 
 which these two parts are attached the one to the other. 
 
 I would like, furthermore, to call attention to the fact that, 
 by the contraction of the tensor tyrapani muscle, all the bands 
 wliicli give firmness to the position of the ossicles are rendered 
 tense. This muscle, in the first place, draws the handle of the 
 hammer inward, and with it the membrana tympani. At the 
 same time it pulls upon the axis-band of the hammer, drawing 
 it inward and putting it upon the stretch. Another effect, as 
 we have shown, is to draw the head of the hammer away from 
 the tympano-incudal joint, to tighten all the ligaments of the 
 anvil, those toward the hammer as well as those at the end of 
 its short process, and to lift the latter up from its bony bed. In 
 this way the anvil is brought into the position where the cogs of 
 the malleo-incudal joint fit into one another the tightest. Fi- 
 nally, the long process of the anvil is compelled to perform a ro- 
 tation inward in company with the handle of the hammer ; 
 in so doing, as we shall see further on, it presses upon the stir- 
 rup and drives it into the oval window against the fluid of the 
 labyrinth. 
 
 In this respect the construction of the ear is very remarkable. 
 By the contraction of the single mass of elastic fibres constitut- 
 ing the tensor tympani (whose tension, besides, is variable and 
 may be adapted to the wants of the ear) all the inelastic tendi- 
 nous ligaments of the ossicles are simultaneously put upon the 
 stretch. 
 
 The only ligament that is thereby relaxed is the ligamentum 
 mallei superins, whose influence as a ligament is essentially in 
 the same direction as that of the tendon of the tensor tympani. 
 
 Hence, if we examine a freshly prepared specimen of the ear 
 in which the rigor moi'tis is still manifest in the tensor tympani 
 muscle, we shall find every thing in the tympanum stiff and un- 
 yielding; whereas later, if we attempt to separate the different 
 parts, we shall find that almost all the bands and ligaments of 
 the ossicles are loose and relaxed; in fact, without a careful in-
 
 MECHANISM OF THE OSSICLES OF THE EAR. 41 
 
 vestigation of the relations of these parts, we should be at a loss 
 how to harmonize the two conditions.* 
 
 The Movements of tJie Stirrwp. 
 
 The joint of the anvil and stirrup resembles the fiat segment 
 of a sphere which is convex toward the stirrup. The capsule 
 is soft, and interwoven more with elastic fibres than the two 
 other articulations ; on- its inferior side there are compact fibres 
 which, when the anvil is drawn upward, are put on the stretch 
 and carry the anvil with them, but when the reverse movement 
 takes place, are folded together so tliat the anvil does not follow 
 so closely as before. 
 
 The base of the stirrup is surrounded by a lip of elastic fibro- 
 cartilage resemljling the li])s of cotyles and of larger articula- 
 tions : it has a breadth of 0.7 mm. The union between the 
 base of the stirrup and the wall of the labyrinth appears to be 
 formed by means of the periosteum of the vestibule, which perios- 
 teum is extended over the base of the stirrup. The fibrous lip 
 of the stirrup is not attached to the sides of the fenestra 
 ovalis. 
 
 The mucous membrane of the cavity of the tympanum ex- 
 tends also over the outer side of the joint. The attachments of 
 the base of the stirrup along its straight edge are more tense on 
 the inferior than on the superior surface, and most compact at 
 the posterior end. Now, if you apply a pin to that side of the 
 
 * In rejrard to the question wlien the tensor tympani contracts, I would con- 
 firm in this place the recently published observation of Politzer, that it occurs 
 durin<if the act of yawnint^f. Before hearing of his experiments, I had already 
 noticed that whenever I attempted to restrain the motion of the jaws 
 during the act of yawning, I would first hear the well-known hnapping noise^ 
 indicative of the opening of the tube ; then, at the acme of the yawn, I would 
 notice, in addition to the sense of tension in the ear, a loud muscular noise, 
 greater even in intensity than that produced by the most powerful contractions 
 of the masticator muscles during closure of the meati, and certainly greater tliau 
 when tlie meati are open. All sounds from without appeared at the same timo 
 to be muffled. From these observations I concluded that contraction must have 
 been excited in a muscle whose oscillations are communicated to the organ of 
 hearing with far greater distinctness than those of any other muscle : I refer 
 to the tensor tympani.
 
 42 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 base of the stirrup wliicli is toward the vestibule and press it 
 outward, it will, tliough separated from the anvil, make at the 
 same time a lever-like movement by which its head is pushed 
 downM'ard and backward. If you employ a fine scAving-nee- 
 dle as a lever of sensation and press it into the base of the stir- 
 rup, the lever-like movement can be still better observed. In 
 other respects the mobility of the base of the stirrup is very 
 slight. I have calculated it partly by direct observation and 
 partly from the movement of the water of the labyrinth. For 
 the purpose of direct calculation I employed a preparation in 
 which the cavity of the tymi)anuni and the vestibule had been 
 opened from above. 
 
 The })reparation was held firmly in a vice in such a position 
 that the base of the stirrup looked downward. The point of 
 a fine sewing-needle was then inserted into the membrana ob- 
 turatoria, near the anterior limb of the stirrup ; the needle had 
 for its second fulcrum the sharply cut edge of \vhat remained of 
 the osseous wall between the cavity of the tympanum and the 
 labyrinth. This ]>oint, 3.8 mm. removed from the point of the 
 needle which was lixcd in the ligainentum obturatorium, served 
 as the centre of motion for the lever-like movements. The free 
 portion of the needle, M'liich was horizontal, formed the second 
 longer arm of the lever, (length, 23 mm.). The point of this 
 longer arm moved backward and forward 0.20 mm., when the 
 stirrup was pressed inward and outward by means of a needle 
 applied to its base ; and .15 mm., when the pressure was made 
 by alternately condensing and rarefying the air in th€ external 
 meatus, whereby the movement of the membrana tympani was 
 transmitted through the other bones of the car to the stir- 
 rup. 
 
 Now, since the movements of the stirrup seemed magnified, 
 at the free end of the needle, /f , therefore the displacements 
 of the stirrup itself amount in these cases only to 0.033 and 
 0.025 mm. After frequent repetition of the experiment, by 
 which the ligaments were thoroughly stretched, the amplitude 
 of displacement increased to 0.050 mm. In another preparation 
 only the superior semi-circular canal of the labyrinth was open- 
 ed, according to Politzer's plan, from the upper side of the tem-
 
 I 
 
 MECHANISM OF THE OSSICLES OF THE EAR. 43 
 
 poral bone. Into the opening tlins made Avas inserted a slen- 
 der glass tnbe, whose transverse section was found, by calibration 
 with quicksilver, to be 0.228 of a square millimetre. The vesti- 
 bule and a portion of the tube were filled with water.* 
 
 The movements of the bones of the ear produced by forcing 
 air into the external meatus caused the fluid in the tube to rise 
 0.9 mm. Now, since the diameters of the fenestra ovalis were 
 found equal to 1.2, and 3 mm., therefore the surf ace of the fe- 
 nestra ovalis is nearly 12.4 times as large as the transverse section 
 of the glass tube. The mean amplitude of the excursion of the 
 base of the stirrup must then be j-j^ of that of the fluid in the 
 tube, which is 0.0726 mm. According to the highest calcula- 
 tion, the excursions of the stirrup amount to y\ and j\ mm. 
 
 The relation of the stirrup to the anvil is such that if the 
 handle of the hammer be drawn inward, the long process of 
 the anvil presses firmly against the knob of the stirrup ; the 
 same takes place if the capsular ligament between both be cut 
 through. 
 
 If the manubrium be moved outward as far as the ligaments 
 of the hammer will allow, then, in case the capsular ligament be 
 cut, the long process of the anvil will recede from ^ to ^ mm. 
 from the stirrup. "With this position of the hammer, if the han- 
 dle of the anvil be pressed back against the stirrup, it will re- 
 main in this position without springing back ; at the same time 
 the cogs of the joint of the hammer and anvil become separated 
 entirely, and there is no force present sufficiently powerful to 
 draw the anvil back. In the normal condition of the arti- 
 culation of the anvil and stirrup, the })oint of the handle of 
 the anvil remains always attached to the stirrup; but it follows, 
 from the already-mentioned facts, that the anvil exercises no 
 strain upon the stirrup when the handle of the hammer is driven 
 
 * In order to render it air-tight, I first dried the bone as much as possible witli 
 blottino-paper ; then I applied a red-hot iron wire to the margin of the opening 
 in the semi-circular canal, and upon this spot I placed immediately a drop of hot 
 cement made of wax and resin ; in this the glass tube was fastened. Finally, 
 the preparation was set in a bowl of water of sufficient depth to cover the free 
 end of the glass tube, and then the whole was placed under the air-])ump. On 
 exhausting the air in the chamber, the air iu the vestibule escaped through the 
 tube, and then water took its place.
 
 44 MECHAJ^ISM OF THE OSSICLES OF THE EAR. 
 
 outward, since tlie handle of the anvil, even when the articu- 
 lation is severed, can remain in contact with the stirrup with- 
 out being drawn outward with the handle of the hammer. 
 
 This arrangement has this important result, namely, that by 
 means of an increase of pressure in the cavity of the drum or 
 diminution of pressure in the meatus, the membrana tympani 
 and the hammer can be perceptibly driven outward without the 
 stirrup incurring the danger of being torn from the fenestra 
 ovalis. Tlie membrana tympani serves as a very powerful re- 
 straint to the reverse movement of the hammer. 
 
 Since the point of the long process of the anvil, seen from the 
 axis-band, is inclined still further backward than the point of 
 the handle of the hammei', therefore the former rises wlien press- 
 ed inward more than the latter, and the elevation is not entire- 
 ly compensated for by the slight depression of the hammer al- 
 ready mentioned ; in short, the driving of the membrana tym- 
 pani inward causes the point of the head of the hammer to be 
 driven outward and at the same time slightly elevated. 
 
 This agrees witli the corresponding movement of the tirrup, 
 whose knob-like head rises slightly when the stirrup is pressed 
 inward in consequence of its unequal attachment to the upper 
 and lower border of the fenestra ovalis. This lever-like move- 
 ment has already been observed and described by Ilenke,* 
 Lucae,t and Politzer.ij: In reply to the first, I will only remark 
 that the lever-like movement of the stirrup is by no means its 
 only one ; that " perhaps" one border of the plate of the stirrup 
 is not moved inward while the other moves outward. Look- 
 ing at the base of the stirrup from the vestibule, we can much 
 more easily recognize the fact that both borders are driven out- 
 ward and inward simultaneously ; the superior, however, more 
 than the inferior. 
 
 The apparent discrepancies between the observations of 
 Lucae and Politzer, in respect to the effect which an increased 
 pressure of air in the cavity of the tympanum has upon the 
 
 * Der Meclianisinus der GehOrkiiuchelclien, in der Zeitschrift fur rationelle 
 Medicin. 1868. 
 f .\rcUiv fiir OhrenUeilkunde. Vol. iv. pp. 36, 37. 
 } Woclienblatt der K. K. Gesellscliaft der Aerzte. Wien, 1868.
 
 MECHANISM OF THE OSSICLES OF THE EAE. 
 
 45 
 
 stirrup and the water of tlie labyrintli, are to be explained by 
 supposing that Lucae observed the lever-like movement of the 
 stirrup, and Politzer the oscillation which takes place in the wa- 
 ter of the labyrinth when tlie stirrup is pressed inward. Neither 
 is it always necessarily in the same degree ; at least not 
 in this case, because the pressure of the air through the fenestra 
 rotunda can also cause increased pressure in the labyrinth. 
 
 §.6. 
 The Concerted Action of the Bones of the Ear. 
 If we suppose the hammer and anvil so united that their 
 cogs press against one another and both move like one com- 
 pact body, exerting a pressure upon tlie point of the handle 
 of the hammer, which is continued inward and transmitted 
 from the anvil upon the stirrup, then the system of the two 
 ossicles can be considered as a one-armed lever whose fulcrum 
 lies where the point of tlie short process of the anvil presses 
 outward against the wall of the cavity of the tympanum. The 
 tip of the handle of the 
 hammer represents the 
 point of i^ressure, and the 
 point of the handle of the 
 anvil the other point which 
 resists this pressure. These 
 three points lie in fact very 
 nearly in a straight line, so 
 that the point of the anvil 
 and stirrup recedes only 
 very slightly inward from 
 the straight line of union 
 between the tip of the han- 
 dle of the hammer and the 
 outer side of the articula- ^'S- 9- 
 
 tion of the anvil with the tympanum. This can easily be seen 
 in preparations where the natural union of the bones is still pre- 
 served. Fig. y. represents both bones in tlie position where 
 their cogs are attached to one another ; a « is the straight 
 line which passes through the three points mentioned above ;
 
 46 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 P . F., the stmnp of the processus Foliaims; T.t. the tendon of 
 the tensor tympani ; and at h we have the cog of the anvil. 
 In this ])reparation I found the entire length of the lever to be 
 9^ mm., the shorter arm between the two points of the anvil 
 6J mm., so that the latter is f of the length of the longer arm. 
 Hence, when the hanniier and anvil lie close together, the excur- 
 sion of the tip of the handle of the anvil will amount only to f 
 of that of the handle of the hammer, but the amount of the 
 pressure wliicli tlie former exerts upon the stirrup will be 1^ 
 times as great as the force wliich is exerted against the handle 
 of the hammer. 
 
 Since the three points of the lever lie in a straight line, there- 
 fore the pressure is quite independent of the position of the 
 remaining parts of the ossicles — presupposing only that tlie 
 latter maintain a position in which the articular surfaces press 
 firmly against each other. 
 
 This result will be obtained in the following manner: While 
 the membrana tympani is being driven inward, the hammer 
 will rotate about an axis which is inclined obliquely (30°) to- 
 ward the plane of insertion of the memln'ana tympani. and its 
 head will recede from the tympano-incudal joint, thus putting 
 the malleo-incudal capsular ligament on the stretch. Now, since 
 every attempt to rotate the ossicles in such a manner that the 
 cogs will press against each other, produces instantly a conside- 
 rable separation of the articular surfaces, therefore the already 
 tense fibres of the capsular ligament serve to resist any force 
 capable of producing this result. On the other hand, if the 
 membrana tympani be driven outward, the capsular ligament 
 becomes relaxed and yields to such an extent as to allow of a 
 slight separation of the articular surfaces, like that which 
 happens when the cogs are separated. The remaining displace- 
 ments, which the malleo-incudal joint admits of, do not cause 
 the above-mentioned three points of the lever to depart from 
 the straight line. One of the axes of rotation of this saddle- 
 shaped joint passes through the tip of the handle of the ham- 
 mer ; the other is perpendicular to the plane which passes through 
 the three points and tlie articuhition, and consequently sepa- 
 rates the long process of the anvil from the handle of the hammer.
 
 MECHANISM OF THE OSSICLES OF THE EAE. 47 
 
 In tlie above-mentioned experiments, Avliere, as a general thing 
 the stirrup has been set in motion by exerting pressure upon the 
 hammer or upon tlie membrana tympani, we have seen that tlie 
 amplitude of the excursion becomes sliglitlj diminished, that is, 
 to about f of its magnitude. In order to determine the firm 
 ness of the mechanism I tried the reverse experiment and 
 endeavored to measure the extent of the excursion of the 
 liammer by pressing the base of the stirrup outward and thus 
 setting the liammer in motion. In this case, of course, the 
 onl}' movements of the hammer to be taken into consideration 
 are those which cause it and the anvil to remain in close con- 
 tiguity at the point of contact. (The preparation used in this 
 experiment has been already described as having a tube inserted 
 into the vestibule.) Having cemented a glass thread 59 mm. 
 long to the head of the hammer, I endeavored to find how 
 mnch motion could be produced in the hammer by alternately 
 forcing fluid through the glass tube and then withdrawing it 
 from the same. The excursion at the tip of the glass thread 
 amounted only to about ^ mm. If 4 mm. be the distance 
 from the axis of rotation to the point where the glass thread 
 is fastened to the head of the hammer, then the length of the 
 lever is 63 mm. and the above-mentioned excnrsion of ^ mm. 
 corresponds to a rotation of about half a degree. For the tip of 
 the handle of the liammer, whose distance from the axis-band 
 amounts to 4|- mm., this gives, on the other hand, an excursion 
 of only 2V mm,, an amount which is about the same as the 
 mean value of the excursion of the base of the stirrup. Tlieo- 
 retically we should expect a somewhat greater value for the 
 excursion of the handle of the hammer. Taking into consider- 
 ation the diminished tension of animal tissue after death, and 
 the want of elasticity, namely, of the tensor tympani, we cannot 
 well expect in the nnited action of the bones of the ear the 
 same precision which we find in the living subject. In this way 
 the transmission of the light movements of the anvil to the 
 hammer may be impaired. * 
 
 * I will remark in this connection tliat the transmission of the movements of 
 the membrana tympani to tlie water of the vestibule was also perceptibly im- 
 paired at the time when I made the above described experiment. I obtained
 
 48 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 Tliese different attempts at'measurement coincide tlius far in 
 showing tliat the disph^cenients of the stirrup and hannner, as 
 long as tlie two remain tirmly connected, are limited to ampli- 
 tudes each of which is smaller than a tenth part of a millimetre. 
 
 On tlie other hand, if we put the hammer in motion by 
 forcing air into the external meatus and then withdrawing it? 
 the glass thread attached to the ossicle indicates much greater 
 excursions ; its point moves backward and forward 5 mm., 
 while before (as already mentioned) it experienced, in a direction 
 from the stirrup outward, only a displacement of ^ mm. 
 
 The excursion which the hammer can make without the an- 
 vil is nearly nine times as great as that which the two together 
 can accomplish. This kind of movement is not transmitted to 
 the water of the lal)yrinth, excepting of course the slight changes 
 in pressure which the changed tension of the articular liga- 
 ments, or the rubbing of the articular surfaces of the malleo- 
 incudal joint upon one another, are perhaps sufficient to pro- 
 duce in the water of the labyrinth when the cogs of the articu- 
 lation are no longer in contact. If you force air into the ca- 
 vity of the tympanum of your own ear, you will hear feeble 
 tones issuing from the middle and upper portions of the scala, 
 almost if not quite as distinctly as usual ; on the other hand, it 
 is very apparent that we hear the same tones, when they are 
 given forcibly, much more distinctly when the pressure in the 
 cavity of the tympanum is uniform than when it is increased. 
 This, I think, shows that the articular surfaces of the hammer 
 and anvil can adhere together and be firmly united by means 
 of motion on one another, similar to that which takes place in 
 anatomical preparations when the joint of the anvil and stirrup 
 lias been cut through aiid the rarefaction of the air in the mea- 
 tus auditorius has caused the liammer to be drawn outward. 
 The anvil then is also drawn outward ; but if we turn it by 
 means of a needle, so that its long process again touches the 
 stirrup, it M'ill, as mentioned above, still remain fixed in this 
 
 only 0.4 mm. ele\ation in the manometer, whereas, on the day before, when I 
 had filled the vestibule with water under the air-pump, the elevation amount- 
 ed to 0.9 mm. It is to be hoped that some anatomist, who has an abundance 
 of suitable preparations at his command, will repeat these experiments. Of 
 course the specimens should be as fresh as possible.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 49 
 
 position. Friction will also cause tlie anvil to adhere iirnily to 
 the hammer in the position already given, in opposition to the 
 tension of the ligaments or any other mild forces, and that too 
 when the vibrations of sonnds are feeble. More powerful forces 
 or concussions will of necessity cause the two bones to slide 
 upon one another, and strong vibrations of sound in such a posi- 
 tion of the bones will be very percejjtible. 
 
 I have used in these experiments a watch and a tuning-fork ; 
 striking the latter lightly, I held it so far from the ear 
 that the beats, which the rotation of the fork upon its long 
 axis produced, were still perceptible. We hear them just as 
 distinctly when the membrana tympani is distended, provided 
 they belong to the upper octaves of the scale, and very 
 nearly as distinctly in the middle octaves. The deeper tones 
 are, of course, considerably weaker. On the other hand, a tun- 
 ing-fork of a higher pitch, when struck forcibly and held be- 
 fore the ear while the membrana tympani was distended, 
 showed a very perceptible crescendo, just as we restored the 
 equilibrium of the air by the motion of swallowing. 
 
 I wish to call attention, in this connection, to another pheno- 
 menon whose explanation, I think, can be deduced from the 
 mechanism already described. If we take a tuning-fork which 
 consists of a single piece of steel, and which therefore has noth- 
 ing about it which can give a rattling sound, and, after having 
 struck it forcibly, hold it near the ear so that the sound can be 
 heard very distinctly, the character of the tone becomes sharp, 
 and we hear distinctly jarring sounds similar to what is heard 
 in musical instruments when something is loose, or from a tun- 
 ing-fork when pressed rather lightly upon a sounding board. 
 These iarrino; sonnds result from the slight shocks which a vi- 
 brating body makes upon a body at rest or vibrating in a diffe- 
 rent manner. These blows are repeated regularly and produce 
 sound ; but inasmuch as they correspond to an interrupted pe- 
 riodical movement, the sound possesses very many overtones 
 and is harsh in character. Such tones occur, as is well known, 
 in the ear itself as the result ot very loud sounds. We can hear 
 also from a B tuning-fork of 116 vibrations a jarring sound so 
 distinctly that it resembles a buzzing in the ear. This jarring 
 4
 
 50 MECHANISM OF THE OSSICI,ES OF THE EAR. 
 
 tone is very distinct and strong when the pressure of the air in 
 the cavity of the tympanum is equal to or less than that of the 
 atmosphere, and when the cogs of the hammer and anvil are 
 closely united; but it disappears when the air is driven into the 
 cavity of the tympanum and the cogs are consequently sepa- 
 rated. I think, therefore, that we are justified in concluding 
 that this jarring tone is caused by the cogs. 
 
 When tlie excursions of the membrana tympani are very 
 great, and during the outward phase of the vibration, the anvil 
 is not driven outward with any considerable force, and cannot 
 therefore follow perfectly the excursions of the hammer ; the 
 result of wliich is that they are separated, and that during the 
 next vibration inward the anvil receives a blow from the re- 
 turning hammer. This mechanism is also well adapted to the 
 production of combination tones,* and the peculiar sensation of 
 buzzing in the ear resulting from tlie combination tones of two 
 strong soprano voices, when thirds are sung, can, I think, be re- 
 ferred to this jarring which takes place between the hammer 
 and anvil. 
 
 This phenomenon is also of great importance in its relation 
 to the sensation which liarmony produces in the ear, since strong 
 tones whieli take place outside of the ear, and without over- 
 tones, must of necessity develop liarmonious overtones in the 
 ear. In this way sounds with harmonious overtones, which 
 correspond to a regular periodical movement of the air, acquire 
 a natural preference over those with unliarmonious overtones, 
 especially as the whole doctrine of conferences becomes, through 
 this circumstance, independent of the overtones connected with 
 external sound. The jarring tones can be much deeper than 
 the exciting tone, if the vibrating body falls back only after the 
 expiration of the vibrations, and hence receives another blow. 
 
 To this class, I believe, belong also certain deep, harsh sounds 
 which we liear when the shrill high notes of the upper octave 
 (a^ — g^) are sounded very distinctly. It is probable that an 
 unusually strong vibration is produced at the same time in the 
 surface of the membrana tympani, judging from a certain buz- 
 zing, tickling sensation which is felt in the deep parts of the 
 * VideLebre von den Tonempfindungen, pages 233-236.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 51 
 
 ear. The apparatus described further on (Fig. 11) is particularly 
 adapted to the production of such tones, 
 
 I will mention here that I have made an enlarged model of 
 the apparatus of the cavity of the drum, in order to prove the 
 comi3leteness and correctness of the explanation just given. 
 The bones of the ear are made of wood, the membrana tym- 
 pani of glove-leather, cut in such a way that a seam shall run 
 along the handle of the hammer where the leather is attached 
 to that ossicle. By this means we can give to it its conical 
 form. An opening of suitable form, cut in a board, and having 
 beveled edges to which the edges of the artificial membrana 
 tympani are fastened, represents the inner end of the external 
 meatus. On the outside of the board a tin ring is fastened,, 
 which surrounds the already mentioned opening. To this,, 
 finally, a tin cover having a gutta-percha edge is fitted, like the 
 covers of hermetically sealed fruit-jars. Now, if we place this 
 cover so that a portion of the gutta-percha rim remains be- 
 tween it and the tin ring, condensation of the air may be pro- 
 duced on the outer side of the artificial membrana tympani, 
 which will act upon the bones of the ear. 
 
 On the inside, near the upper and anterior edge of the open- 
 ing, a thin piece of wood with a projecting point is fastened, 
 which latter represents the spina tympanica major. A hemp 
 string, which is attached to the latter, penetrates the hammer 
 and passes around it, then passes through the board at the pos- 
 terior superior edge of the opening. This string, which is intend- 
 ed to represent the axis-band, can be rendered tense by an ordina- 
 ry screw-eye. The tendons of the ligamentum externum and the 
 ligamentum anterius mallei, which pass from the spina upward,, 
 can be represented by other strings, which of course must be ac 
 curately applied and provided with screw-eyes to vary their 
 tension. Finally, the tendon of the tensor tympani can be re- 
 presented by a silk thread which passes through an iron ring 
 made fast to a small wooden pillar, and then is connected with 
 a tense gatta-percha band. I first spread warm sealing-wax 
 upon the articular surf ace of the hammer, and endeavored as far 
 as possiI)le to give to the former, before it had grown cold, the 
 corresponding form ; then I laid soft, hot sealing-wax upon the
 
 52 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 articular surface of the anvil, and, after having covered the articu- 
 lar surfiice of the hammer with tin-foil, I pressed the two to- 
 gether, Tlie tin-foil then adheres to the anvil. Now, before the 
 sealing-wax had become quite cold, I made a twisting movement 
 witli the anvil, similar totliat which occurs between these bones 
 in the ear, in order to render these surfaces capable of sliding 
 upon one another. After the articular surface of the anvil had 
 become cold, it served as a form upon which to mould the sur- 
 face of the hammer (which must be heated and covered with 
 tin-loil) and render it capable of gliding over the articular sur- 
 face of the anvil. This experiment was repeated alternately 
 with one and the other articular surfece until the two moved 
 sufficiently easily upon one another. It was, of course, neces- 
 sary not to make any sliding movements strong enough to dis- 
 turb the cogs. In this way I succeeded finally in obtaining a 
 good articulation. The capsular ligament was constructed of 
 loops of thin elastic India-rubber cord, which were fastened to 
 the anvil and could be drawn over and attached to the hammer 
 by means of small hooks made of pins, thus holding the two 
 hones together by a very slight elastic pressure. 
 
 The articular ligament of the short process of the anvil was 
 represented by a loop of silk threads which passed through a 
 hole in the anvil. This band can be loose, but it is of impor- 
 tance that the point of support of this part of the anvil upon the 
 outer wall of the cavity of the drum should be represented in 
 the model. 
 
 Simple contact is sufficient to represent the union between 
 the long process of the anvil and the stirrup, or a loop of silk 
 threads can be employed. The former is quite sufficient for 
 giving direction to the blows above described. 
 
 The fenestra ovalis was cut in a thin piece of board, which 
 was held parallel to the larger board by means of small wood- 
 en pillars. This hoard consisted of two plates screwed together, 
 between which was a thin layer of gutta-percha, representing 
 the membrane of the fenestra ovalis. The foot-plate of the 
 artificial stirrup was likewise double and had an interposed 
 layer of India-rubber, and the whole was fastened together by 
 screws.
 
 MECHANISM OF THE OSSICLES OF THE EAR. 63 
 
 Such a model is very useful, partly in demonstrations and 
 partly to show clearly what part the individual ligaments and 
 also the articulations play in connection with the attachment 
 of the bones of the ear ; for all these different parts can be se- 
 parated, and each ligament can be made tighter or looser. 
 Moreover, this model transmits with great facility to the stirrup 
 the small blows which are directed on the outside, immediately 
 upon tlie manubrium or upon the already mentioned air-tight 
 cover ; this can be felt when the linger is placed over the base 
 of the stirrup and over the plate in wdiich it rests, and is also 
 recognizable in the bounding up of light bodies which have 
 been laid upon it. 
 
 The diameters of the artificial membrana tympani are 80 and 
 120 mm. The remaining parts are constructed according to 
 this measurement. All the statements contained in the foregoing 
 description, in regard to the mobility and mode of attachment 
 of the parts, I have tested and found confirmed in this model. 
 
 Mechanism of the Menibrana Tympani. 
 
 The membrana tympani is to be considered as a tense mem- 
 brane, which, however, ditiers essentially from those which 
 have been hitherto studied in acoustics, in the fact that it is 
 curved. Its tension is modified by the handle of the liammer 
 which draws it inward, and which is itself retained in tliis 
 position by means of ligaments of attachment, and by the elas- 
 ticity of the tensor tympani. If the radial fibres of the mem- 
 brana tympani were not united by transverse ones, they would 
 be stretched in a straight line. In point of fact, however, tliey 
 maintain a curved shape with the convexity looking toward 
 the meatus ; hence we conclude that the radial fibres are drawn 
 toward one another by circular fibres, and that the latter are 
 also made tense at the same time. There is, in fact, in the mem- 
 brana tympani at rest no other force capable of holding the 
 radial fibres in a curved position, except tlie tension of tlie 
 circular fibres. 
 
 In the concussions which sound produces, the pressure of the 
 air acts sometimes upon the convex, sometimes upon the con-
 
 54 MECHANISM OF THE OSSICLES OF THE EAE. 
 
 cave siirfiice of tlie inerabrana tympani, according as tliis pres- 
 sure is alternately greater or less in the meatus thaii in the 
 cavity of the tympani ; in every case the pressure of the air 
 acts perpendicularly upon the membrane, also perpendicularly 
 upon the curve formed by the radial fibres, which curve it at 
 one time increases and at another diminishes. Since the 
 curves formed by the radial fibres of the membrana tympani 
 are only slight, therefore, as will bo shown afterward, the 
 mechanical operation is the same as if the pressure of the air 
 were exerted at the end of a very long lever-arm, while the tip 
 of the manubrium represents the end of a very short lever-arm. 
 A relatively great displacement of the surface of the mem- 
 brana tympani in the same direction as the pressure of the air 
 necessitates a comparatively small displacement of the point of 
 the hammer, and vice versa. Hence, in accordance Avith the 
 well-known law of virtual velocities, a relatively sniall amount 
 of pressure of the air wnll counterbalance a comparatively 
 strons: force actini>; at the handle of the hammer — in other 
 words, it will supply an equivalent force. 
 
 In order to understand this, we can limit ourselves to the 
 examination of a single curved radial fibre, which we can su])- 
 pose changed by the pressure of the air into circular arcs of 
 constant lengths but of differing curves, and hence having 
 differing radii. If, then, I represents the length of the fibre, 
 r the radius of the circle to wliich the arc belongs, and A the 
 
 1 /L 
 
 chord which belongs to the arc I, then is -^j- -^ the sine of half 
 
 of the angle at the centre, which belongs to the curve I ; there- 
 fore, 
 
 Z = 2 r. Arc. sinf n".) 
 or 
 
 and the difference between the chord and the curve 
 
 '-^=^Hi - ^^"Qf 
 
 Now, if the curve is very slight — that is, r xery large compar- 
 ed with I — then we can suppose the sine of this formula devel-
 
 MECHANISM OF THE OSSICLES OF THE EAR. 55 
 
 oped according to the involution of its arc, and limit onrselves 
 to tlie first of the two divisions of this development, since the 
 divisions become small ru])idl)\ 
 
 ^'" K2r) = 27- - 6 • (sT-) . 
 This gives 
 
 1 P i 
 
 l-^= ~ -, 1. 
 
 24 r- ) 
 
 The degree of curvature of the are, or the distance s of its 
 centre from the centre of the chord, is given by the equation 
 r - s fl\ 
 
 — cos ( — ) 
 
 or 
 
 If we make here the progressive evolution for the cosine, we 
 have 
 
 '=^^ ■■•■h 
 
 or, eliminating ?' from 1 and 2, 
 i - A _ - _ 
 
 !Now, the difference Z —A represents the shortening of the 
 chord which is caused by the increase in the curve of the arc, 
 or the extent to which the two ends of the fibre are drawn to- 
 gether. On the other hand, s is the displacement of the middle 
 of the fibre. If, now, s be infinitely small in comparison Avith I, 
 the length of the fibre, therefore the magnitude l — X in the last 
 formula is an infinitely small magnitude of the second order 
 in comparison with s. The reverse is clear, namely, if we may 
 be permitted to consider the fibre as inextensible, the very small 
 lengtliening of the fibre to the amount Z —A cannot happen 
 in any other way except that the fibre becomes straightened 
 and its centre experiences the relatively much greater displace- 
 ment 8. 
 
 On the other hand, in respect to the estimation of the rela- 
 tion of forces, there is a well-known formula in mechanics, that
 
 56 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 the tension t of the fibres, if p represents the pressure upon its 
 unit of length, is represented by the following equation : 
 
 t — ip r. 
 The correctness of this formula can be most easily understood 
 when we suj^pose each fibre everywhere (from end to end) 
 equally curved and perfectly parallel to its similarly curved 
 neighbors, and in this way lengthened to a half circle. Then 
 the forces which draw upon the two ends of the fibre — that is, 
 2 t — must counterbalance the pressure which acts upon the 
 whole diameter of the semi-circle throughout a width equal to 
 that of the fibre — that is, the amount 2 /'. j9 ; and hence the cor- 
 responding equation 
 
 3 i = 2 7- p. 
 
 Therefore the greater r is, tliat is to say, the less the curve is 
 under the operation of the pressure of the air, the greater will 
 be the change in the tension produced in the fibre by the pres- 
 sure of the air. 
 
 These changes in the amount of tension of the radial fibres 
 of the membrana tyrapani are the very ones, however, which 
 the concussions of sound transmit to the handle of the hammer. 
 The amount of tension can increase very considerably under 
 the influence of comparatively slight changes in the pressure of 
 the air, even when the radial fibres of the membrane are stretch- 
 ed out in a very flat curve. It is self-evident that in propor- 
 tion as the action of this force increases, so the excursions of 
 the handle of the hammer, which can be caused by this force, 
 grow smaller, similarly to that which happens when the inten- 
 sity of a force is increased by means of a lever. 
 
 On the other hand, it is to be remarked that the changes in 
 tension wdiich the pressure of the air induces, can always ap- 
 pear as the increase or diminution of the tension which is 
 maintained by means of the elastic attachments of the mem- 
 brana tyrnpani and the elasticity of its own radial fibres. A 
 considerable increase in tension, through the pressure of the 
 air from within outward, can only produce a slight eflect upon 
 the stirrup, because the articulation of the hammer and anvil 
 yields. Again, on the other hand, the pressure of the air from 
 without can, at the most, only force the handle of the hammer
 
 MECHANISM OF THE OSSICLES OF THE EAR. 57 
 
 inward until tlie radial fibres of the menibrana tynipani be- 
 come straight; should the pressure be still greater, tlien it 
 would curve them again, shorten their chord, and draw tlie 
 manubrium again outward, provided the circular fibres of the 
 membrana tymi^ani could actually yield so much without 
 breaking, which latter I consider very improbable. 
 
 The labyrinth is likewise protected from extremes of pressure, 
 while at the same time the effect of slight variations in pressure 
 can be rendered extremely powerful through the peculiarities 
 of the mechanism already described. 
 
 By introducing a manometer into the external meatus, ac- 
 cording to the plan proposed by Politzer, it may be shown that 
 the excursion of the parts of the membrana tympani situated 
 in the middle, between the handle of the hammer and the bord- 
 er of attachment, is considerably greater than that of the ma- 
 nubrium itself. In the ordinary anatomical preparations I 
 found it better to fill the meatus entirely with water than to 
 shut up the air contained in the meatus by means of a drop of 
 water in the tube of the manometer. A drop of water so plac- 
 ed resists small displacing forces, since it adheres to the glass 
 tube and does not move when most desirable. If we, however, 
 fill the entire meatus with water, and then introduce the mano- 
 meter-tube (after having attached to it a suitable plug of seal- 
 ing-wax) in such a manner that at the same time a certain 
 amoimt of water will enter it, then the surface of the fluid in 
 the tube will indicate very accurately the displacements of the 
 membrana tympani. As already mentioned, a tube was intro- 
 duced into the vestibule of the labyrinth in the same prepara- 
 tion, and thus, by forcing in the fluid or withdrawing it, the 
 stirrup and the hammer could be moved. 
 
 It has already been stated that in tliis experiment the excur- 
 sion of the tip of the handle of the hammer was only ^\- mm. The 
 height of the fluid, however, in the manometer varied 1 mm. 
 By calibration with quicksilver, the inside diameter of the tube 
 was found to be 1.37 mm. ; the diameters of the membrana 
 tympani were 7^ and 9 mm. From this we can calculate a 
 mean displacement of the membrana tympani of somewhat 
 more than ^ mm. ; that is, 3 times as great as the synchronous
 
 58 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 iriovement of tlie tip of inanubriuni. Now since the outside 
 border of the ineinbrana tympani is firm, it follows that the 
 middle free parts of the membrane must have experienced a 
 relatively much greater displacement than the amount of the 
 mean displacement given above and therefore more than three 
 times stronger than the tip of the handle of the hammer. 
 
 In the foregoing elementary examination of this mechanism 
 we have not taken into consideration the following facts, viz. : 
 that the respective meridional curves of the membrana tympani 
 are closely united ; that their distance from one another 
 increases in the direction of the firm border of the membrane ; 
 that they are bound together by circular fibres, and that they 
 cannot move without stretching these ; in fact, that the natural- 
 ly curved form of the membrana tympani cannot exist without 
 its circular fibres being extended and made tense by every force 
 which draws the handle of the hammer inward. 
 
 The form of the membrana tympani being so irregular, a 
 perfect analysis of the mechanical action of the parts cannot be 
 given. It would be necessary first to know the tension and the 
 measure of the elasti(;ity of the circular fibres. We can, how- 
 ever, make a nuithcmatical representation which would better 
 correspond to the actual relation of the parts, if, instead of the 
 real membrana tympani, we imagine an ideal one, which is 
 conical in the centre, but toward the periphery is curved 
 and symmetrical, and represents therefore a surface of ro- 
 tation. The radial fibres, which follow the direction of 
 the meridians of such a surface, can be considered as incapable 
 of extension ; the circular fibres, however, must possess a certain 
 degree of elasticity in order to remain always tense. In the 
 appendix, I have developed the theoretical question regarding 
 the mechanical workings of such a membrane, and the most 
 advantageous form to give it. It will be sufficient here to remark 
 that the pressure of the air will produce the strongest effect 
 upon a slightly curved membrane, when it has, by means of its 
 own elasticity, taken the form which the pressure of the air tends 
 to give it. This form is one where with unchanged length of 
 the radial fibres and unchanged position of its centre, the volume 
 of its concave side, that is, the volume of the cavity of the
 
 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 59 
 
 drum becomes a maxiinum, and tliat on its convex side is re- 
 duced to a minimum. If the membrane had not originally 
 possessed such a form, still the pressure of the air would have 
 produced such a result by changing the tension of the circular 
 fibres, before it could have excited its entire force upon the 
 centre. 
 
 The form here required of a circular membrane, can be calcu- 
 lated — the transverse section of such an one, in some degree cor- 
 responding to the relation of the membrana tympani, is given 
 in Fiii'. 10. This form will be seen to coincide well with the 
 
 relatively free lower portion of the membrana tympani. 
 
 Let a represent the angle which the tangent of the membraTie 
 drawn through its apex (umbo) in the meridian-plane makes av ith 
 the axis; /3, that which the corresponding tangent of a point in 
 the periphery of the membrane makes with the axis ; B^ the ra- 
 dius of the circle at the periphery ; /, the pressure of the air ; 
 then will k be the force which must be applied at the centre 
 of the membrane to counterbalance the pressure of the air : — 
 
 _ P"^ ^' ^^^ ® 
 
 cos a — cos (3 ' 
 
 In this equation we see once more that the smaller the differ- 
 ence between the two angles a and /3, that is, the shallower tlio 
 curve made by the the tense radial fibres of the membrane, tho
 
 60 
 
 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 stronger will be the force. Further, the force increases as tlie 
 cos a, if the angles a and jQ become smaller, while the difference 
 cos a — cos 13 remains the same, that is, if the apex of the mem- 
 brane be drawn in more strongly. 
 
 Thus far the acoustic action of such curved membranes has 
 not yet been practically studied. It may be proper to state 
 here, that in the Tunis Cafe at the Paris industrial exhibition, 
 I saw a curved piece of leather employed as a sounding board 
 in an Arabian stringed instrument. A membrane similar to 
 the membrana tympani can be made by stretching a wet piece 
 of a pig's bladder over the upper end of a glass cylinder : the 
 cylinder should be placed in an upright position, then place 
 a rod loaded with metal perpendicularly to the centre of the 
 membrane, so that its lower end presses the centre of the blad- 
 der downward. In this position the bladder must be allowed 
 to dry. It will then retain permanently a form similar to that 
 of the membrana tympani, with its retracted navel and its curved 
 meridian lines whose convexity looks outward. 
 
 In order to test the acoustic action of such a membrane under 
 relations similar to those in which the membrana tympani is 
 placed, I fastened the cylinder, whose inside diauieter amounts 
 to 44 mm., to a strong wooden board (A, Fig. 11.) (In the 
 figure the cylinder is situated between e and/", and has been 
 
 
 
 
 Jj 
 
 jt 
 
 iJ 
 
 
 
 A 
 
 
 
 
 a 
 
 0? 
 
 c 
 
 
 J. 
 
 
 H — ^ 
 
 <P 
 
 A 
 
 
 f 
 
 d 
 
 6 
 
 Fig. 11. -si 
 
 represented as cut in two. The cylinder was then tied to the 
 board, its open end at 6, having iirst been supported by a piece 
 of wood, properly cut out to receive it, by which arrangement
 
 MECHANISM OF THE OSSICLES OF THE EAR. 61 
 
 any sliding movement in the direction of e was prevented. A 
 small, light wooden rod was then placed upon the retracted 
 centre of the membrane and this served as a bridge for a string 
 stretched between two pegs at a and h. At c the string runs 
 over a bridge placed on the centre of a block of lead, be_yond 
 which, of course, the string cannot vibrate. Another block of 
 lead was placed at fZ, and between it and the rod/", a thin board 
 like the bridge of a violin was inserted parallel to the string. This 
 board supported the rod, but offered no obstruction to the shocks 
 which the rod received in the direction of its own length, from 
 the string. 
 
 The blocks of lead serve to weaken the transmission of the 
 vibration of the string to the board and so through it to the 
 atmosphere, so that when we draw a violin bow across tlie string 
 and at the same time lift up the rod/" from the curved mem- 
 brane, or grasp the string with the lingers at a point near/", be- 
 tween this point and c the resulting sound will be very much 
 muffled. As soon, however, as the vibrations of the string can 
 pass through the rod to the curved membrane, the latter, not- 
 withstanding its suddeness, gives forth almost as powerful a 
 tone as the violin itself. The string can easily be shortened by 
 holding it between two fingers of the left hand, while with the 
 right hand we draw the bow, which should be placed near the 
 fingers of the left hand. It is evident then, that this powerful 
 resonance extends over the greater portion of the scale, and in 
 the case of the high tones in the middle of the octave c^ . . c^ 
 it reaches such an intensity that they can hardly be endured. 
 
 This process resembles that which takes place in the membrana 
 tympani, in so far as the curved membrane serves to transmit 
 the vibrations from the air to a solid body of moderate weight 
 and relatively small amplitude of vibration, as for instance 
 the labyrintli-water on the one hand, and the ends of the 
 strings on the other. If, however, sound be easily transmitted 
 from the string to the air, then the reverse must also easily take 
 place, according to the general law of reciprocal forces for oscil- 
 lations of sound in perfectly elastic bodies.' 
 
 ' The law for masses of air inclosed in firm walls is stated and proven in an 
 essay published by me in the "Journal fiir reine and an^rewandtc Matliematik." 
 Bd. LVII. p. 29. Equation 93. The title is, " Theorie der Luftschwingungen 
 in Rohren niit offenen Enden."
 
 62 MECHANISM OF THE OSSICLES OF THE EAR. ' 
 
 I. 
 
 Tlic proof of tins can easily be found by experiment on the \ 
 aboveinentioned apparatus. If we place small paper riders (or 
 thin fibres of wood) upon the string and sing the tone which 
 belongs to it over the mouth of the tube, the paper figures w'ill 
 begin to dance. The tone of a tuning fork placed upon a sound- 
 ing board will also cause a string of the same pitch with it 
 to resound, and the paper figures will dance. The same effect 
 will be produced when the tuning fork is held at a distance of 
 several feet from the string. The pitch, moreover, of the glass 
 tube exerts an influence similar to that which takes place when 
 the ear is armed with a resonator. If we make the string of 
 such a lensTth that its fundamental tone agrees with the tone of 
 the tube, then the tone of the string will be particularly full and 
 powerful. 
 
 §8. 
 
 MATHEMATICAL APPENDIX, 
 
 Tiaving particular reference to the rnechanisin of curved 
 rnemhranes. 
 
 In the following we presuppose a membrane of circular form, 
 having inextensible meridian lines, and tense elastic fibres ; we 
 assume also that^ the pressure of the air acts upon one of the 
 surfaces of this membrane, and that, on the other hand, there is 
 present a force g acting upon its centre in the direction of sym- 
 metry. 
 
 Then in order to understand, as shown in the preceding 
 section, how the pressure of the air produces the strongest resul- 
 tants in the centre of a feebly curved membrane (provided the 
 membrane, through the action of its elastic circular fibres, lias 
 assumed the same form which the pressure of the air would give 
 to it, were the elastic tension of the circular fibres wanting), let 
 us take into consideration the following propositions. 
 
 In Fig. 10, page 50, let ah be a diameter, c the middle point 
 of the firm border of the membrane, and y the centre of the 
 membrane, which centre the force fg draws in the direction of 
 the axis. The meml)i-ane may have assumed the form indicated 
 by the curved line merely through the influence of the tension 
 of its elastic circular fibres and be thus in permanent equilibrium.
 
 MECHANISM OF THE OSSICLES OF THE EAR, 63 
 
 We will next presuppose that the air presses equally upon both 
 sides of the membrane. 
 
 Now it is a well known general law in mechanics, that in 
 every case, where the law of the conservation of forces comes 
 into play, permanent eqnilibrium takes place only when, among 
 all the positions which the movable system can assume, the con- 
 dition of equilibrium is that one in which the measure of inter- 
 nal and external forces acting upon it is a maximum. 
 
 This law is applicable to the membrane in question and it 
 follows that in its condition of equilibrium the total amount of 
 force exerted by the contraction of the elastic circular fibres must 
 be a maximum in comparison with that which may be exerted 
 in any of the other forms into which the membrane could pass 
 while the position of the point/" remains unchanged. 
 
 Should then any other force whatever give the membrane an- 
 other form, while the position of the point h remains unchanged, 
 the work accomplished by this force must necessarily be of a 
 positive character, since the quantity of the power of the tension 
 exerted by the membrane must be increased by this transition. 
 
 The same would hold true if the membrane were brought into 
 the position afh, not by means of its elasticity but through the 
 pressure of the more condensed air above it, exerting a ioreafg 
 upon its centre/". In this case the membrane must necessarily 
 assume such a form that the force, exerted by the expansion of 
 the more condensed air above, would be a maximum. This latter 
 would however take place, if the volume of the air contained 
 above the membrane and the prolonged plane a h became a 
 maximum. It follows again, that if another force were em- 
 ployed in order to change the form of the meuibrane in any 
 way whatever, the volume of the more condensed air above 
 would necessarily become less and the additional force would 
 have to produce positive results. 
 
 Now, if the form afh^ produced by the elastic force, be 
 exactly the same as that which the pressure of the air produces 
 and if the former counterbalances the force g and the latter the 
 force y, then will the membrano without change of form be in 
 equilibrium, through the simultaneous action of the elastic circu- 
 lar fibres and the pressure of the air, aTid will counterbalance the
 
 64 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 force g and y which acts at tlie point f. If the position which 
 the membrane assumes, when in a position of eqnilibriuiu 
 through the action of elastic forces (the centre of the membrane 
 being at/"), differs from tliat wliich the pressure of the air pro- 
 duces (the position of the centre being still the same), then the 
 membrane under the united influence of both forces will come 
 to a state of rest in a position varying from either of the other 
 two positions. In this position, neither the elastic forces nor 
 the pressure of the air will have exerted the maximum of their 
 power, that is, what they are capable of doing when the centre 
 is aty. 
 
 Taking chen, as a starting-point, that form which the mem- 
 brane receives, when the force g is infinitely great and when, as 
 a result, the radial fibres must be extended in a straight line, 
 and then supposing the force g to be gradually diminished until 
 the centre of the membrane has advanced to the point y, we 
 shall find that the membrane exerts a force which increases in 
 value from zero to a value of G, and that this value is dependent 
 upon the position of the point /". 
 
 Let then G^ represent the ft)rce exerted in this case, when 
 the elasticity alone acts, G^ when the pressure of the air alone 
 acts, and G.2 when both forces act simultaneously ; then 
 ^2 < ^0 + ^i5 except in the case where the elasticity and the 
 pressure of the air give the same form to the membrane. 
 
 Starting from the position where these quantities are equal 
 
 to zero, then, if the length g f be represented by A, during a 
 
 first period must 
 
 dO, < ^0 + ^^' 
 d h d h d h ' 
 
 because otherwise from the outset the following would have 
 
 been true : 
 
 Oq_G<, + G\. 
 
 Now, however, the above differential quotients equal the re- 
 sulting forces which tend to draw the centre of the meinbrane 
 toward c. 
 
 The force with which the elasticity of the membrane (taken 
 alone) acts, is represented by g: 
 
 " dh
 
 mechanism: of the ossicles of the ear. 65 
 
 Bj y is represented tlie force with which the pressure ot the 
 air, taken alone, acts : 
 
 ^ dh 
 and the force, with which the pressure of the air and the elas- 
 ticity together act, we will indicate bj ^ + y„ : 
 
 dh 
 It follows from the above equation that in the smaller curves 
 of the membrane 
 
 9 + 7" < g + 7, 
 or 
 
 ro < 7, 
 provided the form of the inembrane, in which the condition of 
 equilibrium exists, is not the same for the simple force of elas- 
 ticity and for the pressure of the air ; Q. E. D. 
 
 To determine the form of a membrane made tense hy the 
 pressure of the air alone^ and containing inextensihle radial 
 fibres. 
 
 Let 3 be a given portion of the axis of the membrane, and r 
 the radius of the circle in which a plane, passing through a va- 
 riable point near the end of z perpendicularly to the axis, cuts 
 the membrane. The volume which lies between two such planes, 
 corresponding to the values z and 3 + d z, whose dilFerence is 
 infinitely small, is therefore 
 
 ■K r^ dz 
 
 The entire volume -y, between the membrane and the plane 
 which passes through its circle of attachment, is therefore 
 
 ^ _ I irr^dz 
 
 if for the centre of the membrane z = o and for tlie circum- 
 ference z — a. 
 
 Let jy represent the excess of the pressure of the air, upon the 
 upper surface of the membrane, over the pressure of the air on 
 the under surface, and G the effects of a force acting upon the 
 centre of the membrane and in a direction parallel to its axis, 
 then the combined effect of this force and the pressure of the air 
 is equal to
 
 66 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 The conditions which permit of equilibrium are that this 
 quantity should be a maximum while the length of the radial 
 fibres remains the same ; the element of this length has been 
 given in the equation 
 
 ds^ = dr'' + dz\ 
 
 Considering then r as an independent variable^ then must 
 
 or, according to the principles of differential calculus, if we dif- 
 ferentiate z : 
 
 dz dSz 
 
 dO , r / ddz dr dr 
 
 dz — rrp / / 2 — _ 
 
 dz (1 dr (dz 
 
 j/l-^(^) 
 
 ?y = 0. 
 
 Partially integrated we have the following result, if we make 
 6 s, at the periphery of the membrane, equal 0, and in its cen- 
 tre equal (5 Sq '- 
 
 dz 
 
 J n dr , 
 
 i^ + ^p A . V fe.. 
 
 dz 
 
 /-(tr 
 
 dz 
 
 -h Trp 
 
 ^f''^'^'-''\^^y'- 
 
 Since then dz^ and dz are absolute quantities independent of 
 each other, it follows that those magnitudes which are multi- 
 plied by them equal zero ; hence, 
 
 1. For the centre of the membrane : 
 
 dQ . dr^ 
 
 + TTOX = 
 
 \/' + {^)
 
 MECHANISM OF THE OSSICLES OF THE EAR. 67 
 
 2. For its surface , 
 
 dz 
 
 |/l +(§)' 
 
 where C is understood to represent a constant. In the cen- 
 tral point of the membrane r = o, and -^= cot. a, a repre- 
 senting the angle so designated in Fig, 10. For this point 
 therefore equation No. 2 becomes reduced to C= — A cos a and 
 the equation No. 1 gives for the same point 
 
 + Tz p A cos a = 0. 
 
 d z 
 
 When we, on the other hand, represent the amount r at the 
 
 border of the membrane by B, and let -£r = tang i3, as in Fig. 
 
 10, then according to equation No. 2 
 
 R^ — X cos p = G = — % cos a 
 
 consequently 
 
 R^ = A {cos j3 — cos a) 
 
 and the force g 
 
 _ dO _ Tz p R^ cos a 
 ' dz ~~ cos (3 — cos a 
 
 as given in the previous section. 
 
 Again, from equation No. 2 follows : 
 
 or 
 
 r' + A cos a , 
 
 dr 
 
 i 
 
 A* — {r" + A cos af 
 
 This is an elliptical integral which we restore to the normal 
 form when we make 
 
 dr = — 
 
 
 2 "k sin —- . cos u 
 
 2 X sin -— . sin u d u
 
 68 MECHANISM OF THE OSSICLES OF THE EAR. 
 
 then will 
 
 1 — 2 siji^ — wi' u 
 2 
 du 
 
 l/^A 
 
 dz = — 1/ i/1 — sin^ — sin^ u 
 
 i 
 
 Or, if we follow Legendre, 
 
 do 
 
 Fu 
 
 Eu — 
 
 and place 
 then is 
 
 i/ 
 
 4/1 — jC ^i<^ <•> 
 
 V 1 — x^sin^ udu 
 
 X — sitv — 
 2 
 
 A ( ) 
 
 -2~^2 Eu — Fo)'r + Const. 
 
 = 2 A / -^ .xcos u 
 
 \' 
 
 At the same time we easily lind the length of the arc of the 
 radial iibres — 
 
 1/"^^" 
 
 By means of Legendre's tables, which give the values of E(^ 
 and i^w for all values of |- and w, which correspond to whole 
 degrees, we can construct the form of this curve in the easiest 
 possible manner. For arbitrary values of a and w the values 
 oi E (ji and F w can be computed according to well-known 
 methods. 
 
 Fig. 12 shows a perfect curve of this kind, drawn from one axis- 
 point to the other, in which the value 180° — 40° = 140° is 
 given to the angle a, corresponding to the form of the membrana 
 tympani. The axis-point may represent the centre of the mem- 
 brane. Each point of the arms of the curve extending from a 
 could correspond to the circumference of the membrane, as far
 
 MECHANISM OF THE OSSICLES OF THE EAR, 
 
 69 
 
 Fig. 12. 
 
 as that point wliere the curve, descending in the direction of h, 
 meets and crosses itself again. The membrana tympani itself 
 corresponds only to a small part of this curve. 
 
 For the present, I shall defer the special description and dis- 
 cussion of my experiments (referred to in foot-note of page 8) 
 on " resonance-tones " in the living ear, because I hope to obtain 
 better means of producing deep and simple tones than I have 
 had thus far, and in order that the experiments may be better 
 performed.