^V *Jwp5rr*Q&&*?- I T I H UNIVERSITY OF PENNSYLVANIA A GENERALIZED EQUATION OF THE VIBRATING MEMBRANE EXPRESSED IN CURVILINEAR COORDINATES BY #ARRY M. SHOEMAKER A THESIS Presented to the Faculty of the Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. 1918 UNIVERSITY OF PENNSYLVANIA A GENERALIZED EQUATION OP THE VIBRATING MEMBRANE EXPRESSED IN CURVILINEAR COORDINATES BY HARRY M. SHOEMAKER A THESIS Presented to the Faculty of the Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. 1918 QMS* S-5 The author wishes to express his sincere thanks to Professor Frederick H. Safford, under whose direction and supervision this work was done. CONTENTS. PAGE. 1. General Equation of the Vibrating Membrane 1 2. Derivation and Discussion of the General Equations of the Families of Confocal Conies used as Coordinate Systems 3 3. Discussion of the General Equation of the Vibrating Membrane 5 4. Solution of the Equation in Rectangular and Circular Coordinates 7 5. Solution of the Equation in Parabolic Coordinates 9 6. Review of the Work Previously done on the Equation in Parabolic and Elliptic Coordinates 11 7. Bibliography 13 in 380309 1. GENERAL EQUATION OF THE VIBRATING MEMBRANE. The problem of the vibrating membrane has been attacked by previous writers from the standpoint of the shape of the boundaries of the membrane, rectangular, circular, parabolic, elliptic. No attempt has been made to derive a single equation which contains all of these. Here we shall derive such an equa- tion by the use of a general system of curvilinear coordinates. These coordinates shall be restricted only by the condition that the resulting equation shall have a solution in the form of a product of functions of the independent variables involved, each function depending only upon a single variable. If a stretched elastic membrane is fastened at the edges and is made to vibrate at right angles to the plane of the membrane, its equation of motion in rectangular coordinates is, dW_ JdW dW\ W dt 2 ~ a V dx 2 + dy 2 ) where a is a physical constant depending upon the mass per unit area and the tension along any line in the surface of the membrane. In (1) put V = T-u, where T is a function of t only and u a function of x and y only. The equation then separates into the two parts, d 2 u d 2 u d T 2 (2) d? + df +K2u=0 > and w +a2K2T==0 ' where K is any constant. Solutions of the latter are T = sin aKt and T = cos aKt. Now suppose x and y are determined as functions of two new unrestricted variables, x — fi(a, j8), y = / 2 (a, 0) and express d 2 ujdx 2 + d 2 ujdy 2 in terms of these new variables, dx 2 ^ dy 2 ~ da 2 l\dx) + \dy ) ] + dp 2 l\dx ) THE VIBRATING MEMBRANE. (3) + (g)']+m»°.% + £#] \vy J J 3a3$\_3x 3x dy dy J )ot\_dx* + dy* \ + dp Uz 2 dy 2 ]' Assume now that (4) * + iy = /(« + i$), x — iy = f(a- i@). This assumption will greatly simplify (3). Putting this simpli- fied result in (2) we obtain, By means of the orthogonal relation {dx'J + \dy~) =1 "L\^J + \te) J ^ (^f^Tj^' this equation becomes v?o&Cj 1/ctM (6) ^+^__j^rf*Y + r«sLY1 - . kVj W d*a + dP Ku \_\da) + {d a ) I TJ^^ The right member of this may be written in a more convenient form. From (4) we may derive the relation, which when put in (6) gives, £ (of 4 ftf $^ *ft 3 u 3 u (8) 3rf + dj2 = - K 2 u-f'(a + tf) •/'(« - tf). . On the assumption that this equation has a solution of the form u = G(ot)-H((3), it is necessary that the relation (9) should hold. (9) / '(a + W •/'(«- #) = £(«) + TO) where £ and F are functions which we shall determine now. By differentiating first with respect to a and then with respect to j3 we obtain the equation (10) /'"(a + $) -/'(a - 10) - f'"(a - #)•/'(« + #) = 0. ^ <3) THE VIBRATING MEMBRANE. 6 This equation has the solutions (11) f'(a + ip) = B sin A(a + i@) and C cos 4 (a + #), A, B, C are constants. These are the functions then which satisfy the assumption in (9). Substituting in (9) we obtain, E(a)+ F(0) = B 2 sin A(a + ifl -sin ^(« - i0) = £ 2 (sin 2 ^4« + sinh 2 Aft and (8) becomes, 0*11 fi 11 23+2 + # 2 £ 2 w(sinh 2 48 + sin 2 ii«) = 0. OCT OjCT Hence (1) may now be written, a 2 r /a 2 r e 2 v\ (12) [£ 2 (sinh 2 40 + sin 2 Act)] -^ = a 2 { ^ + g^ J . That the assumption in (9) really has resulted in changing our curvilinear coordinate system to families of orthogonal confocal ellipses and hyperbolas may be seen from (11) and (4). And the method here employed shows that (12) is the most general equation obtainable for the vibrating membrane which has for its solution the function V = T(t)-G(a)-H(8). 2. DERIVATION AND DISCUSSION OF THE GENERAL EQUATIONS OF THE CONFOCAL CONICS USED AS COORDINATE SYSTEMS. From the equations of the families of the curves of the orthog- onal curvilinear coordinate system in which (12) is expressed, we purpose now to derive the following systems as special cases, (a) System of rectilinear lines. (b) System of concentric circles and radial lines. (c) System of confocal parabolas. In order to do this, however, care must be exercised in selecting a general solution for (10). Take this solution in the form /'(« + i$) = B sin A(a + ip) - C cos A(a + ip). 4 THE VIBEATING MEMBRANE. Then C B f{a + tf)=2 sin {Aa + Aip + iD) + j cos {Aa + Aip + iD). By using the relations C = (L4 sin B and 5 = CA cos 5 and making various trigonometric transformations and dropping the dashes from the constants, we arrive at the form, (13) x + iy = C[cos {Aa + B) cosh (40 + 0) + t sin {Aa + 5) sinh (4/3 + Z>)] in which A, B, C, D are arbitrary constants. By equating real and imaginary parts and combining the resulting equations in the usual way, we obtain je 2 ifi (14) C 2 cosh 2 (4/3 + D) + C 2 sinh 2 (40 + D) = *' 3u W 2 (15) C 2 cos 2 {Aa + 5) ~ C 2 sin 2 (4a + B) = l$ as the equations for the families of the curves of our coordinate system. It is to be noted that the focal distance of the families is the constant C. (a) System of Rectilinear Lines. Put C 2 sinh 2 (4/3 + D) = Jfc 2 in (14). (16) vT&+h =L If C = oo while k remains finite this becomes y = ± k. The relation between the parameter /3 and the new parameter A; is k = ± C sinh (4/3 + D) and this will remain finite and equal to d= F/3 {F a constant), as C = oo if at the same time 4 = in such a way that C-A = F, with D = 0. Put C 2 cos 2 (4a + £) = A 2 in (15). x 2 v 2 ( 17 ) ^2 - (?_ h 2 = !• When C = oo and ^ remains finite this becomes x = db h. The THE VIBKATING MEMBRANE. 5 relation between the parameter a and the new parameter h is h = ± C cos (Act + B) and this will remain finite and equal to =F Fa as C = co if B = 7r/2 and A = in such a way that CA = F. (b) System of Concentric Circles and Radial Lines. Put C 2 cosh 2 (Ap + D) = r 2 in (14) and let C = while r re- mains finite, and we obtain X 2 -\- y 2 = r 2 . Now since r = C cosh ^4/3 cosh D -\- C sinh ^4/3 sinh Z>, if we let D = oo as C = in such a way that both C cosh D = F and C sinh D = F then r 2 = FV and equation (14) becomes (18) x 2 +y 2 = FV^. In (15) let C = with 5 = and let ^4 remain finite and we obtain (19) y = ± x tan ^4a. (c) System of Confocal Parabolas. Put C 2 cosh 2 (Aj3 -\- D) — a 2 in (14) and change the origin to the left focus. Then let a — C = y and 0. From the equations connecting 7 with /3 and 8 with /3 it may be shown that 7 = A^/2 when C = 00 and D = 0, provided that A = in such a way that C--4 2 = ^4 2 and that 5 = 00 under the same conditions. Put C 2 cos 2 (Aa + 5) = a the parameter n is positive and for those in which (Ao?/2) < a the parameter n will be negative. If we had changed the origin of the curves in (14) and (15) to the right focus and then sent the left to infinity in the negative direction, we would have obtained y 2 = Ao?(Ao? + 2x) and y* = A^(A^ - 2x) as the equations of the families of parabolas. From these the families of rectilinear lines may be obtained by changing the origin to the point (— a, 0) and then sending the focus to infinity in the positive direction. 3. DISCUSSION OF THE GENERAL DIFFERENTIAL EQUATION OF THE VIBRATING MEMBRANE. The same values of the constants used for the derivation of the three special systems of curvilinear coordinates from the general system should, when put in the general equation of the THE VIBRATING MEMBRANE. vibrating membrane, with the time element removed, give three special equations expressed in the above mentioned coordinates. With the aid of (13) equation (6) may be written (28) —. j + ^ + tf 2 CMMcosh 2 (A$ + D) d 2 u d 2 u da 2+ dP 2 - cos 2 (Aa + B)] = 0. This separates into, (29) |^ + [K 2 (PA 2 cosh 2 (A$ + Z)) - M 2 ]Y = 0, d 2 X (30) ^- - [K 2 C?A 2 cos 2 (Aa + B) - M 2 ]X = 0. The family of horizontal lines, y = db F/3, were obtained from the family of confocal ellipses by having C = and A = in such a manner that C-A = F and Z) = 0. If these values are put in (29), it becomes d?Y — — M (31) jy+ (K 2 - M 2 )Y = where ^ = y- The family of vertical lines x = =F Fa were obtained from the family of confocal hyperbolas by having C = oo and .4 = in such a way that C-A = F and 5 = tt/2. If these values are put in (30) it becomes J2 y yr (32) -^ + M 2 X = where M = y . And these are the equations into which d 2 u d 2 u dtf + dtf separates. The family of concentric circles, x 2 + y 2 = F 2 e 2AB , were ob- tained from the family of confocal ellipses by having D = oo and C = in such a way that C cosh D = F and C sinh D — F. Moreover dP 2 ~ Ar dr>~*~ AT dr 0+^+^=0 8 THE VIBRATING MEMBRANE. in this instance. If these values are put in (29) it becomes (33) - - M K=KF and M = -f. A The family of radial lines, y = ± x tan Act, were obtained from the family of confocal hyperbolas by having C = and B = 0. If these together with

where * 2 " ^ 2 ^ 2 - THE VIBRATING MEMBRANE. 9 The family of confocal parabolas, y 2 = Ao?{Ac? — 2x) were obtained from the family of confocal hyperbolas by using the same values of the constants as above and B = 0. If these are put in (37) it becomes (39) t4 + (k 2 a 2 + M 2 )X = where k 2 m K 2 -A 2 . 4. SOLUTION OF THE EQUATION IN RECTANGULAR AND CIRCULAR COORDINATES. The complete solution of the equation of a stretched elastic rectangular vibrating membrane is obtained by an extension of Fourier's Theorem and may be found in the works of any of the writers on Partial Differential Equations. The complete solution representing the vibrations of a circular membrane is obtained by the use of Bessel's Functions. If the membrane vibrates so that it has circular symmetry about the axis per- pendicular to the plane of the boundary at the origin, the Jo functions furnish the solution. If the mode of vibration is a function of both the coordinates r and

(l>) 2 —j^- = J n -l(x) ~ Jn+l(x), dJ n (x) n (<0 -T- = J n -i(x) - - J n (x), 2n (d) ~Jn(x) = Jn+l(x) + J„-l(x), («) dx 2 (^ i - 1 ) ViCj ^ ) - 10 THE VIBRATING MEMBRANE. The usual method of obtaining these relations is to consider the functions entirely independent of the equation of which they are solutions. Since the functions are defined by a differential equation it was thought that the equation itself ought to yield these fundamental relations. Starting with J n (x) as a solution of Bessel's Equation, and substituting z = x n v, we obtain v = x~ n J n (x) as a solution of dh 2w + 1 dv Substituting this value or v in (41) (42) ^ (x-»J n x) + -rr^-r ^ (x-*J n x) + x-»J n x = 0. The corresponding equation for J n +iX is ff , 1T , , 2ra + 3 d , (43) £5 O^'W) + -7- Tx (x—V^x) + X-n-Vr+lX = 0. Differentiate (42) with respect to x, add to (43) multiplied by x and arrange terms, ^[i c (x '" J " x)+x ~" J ^ x ] + ( x - 2J ^) • W*~° J " x) + «"***]- °- A solution for this is (45) ^ (ar»J«aO + af*JWi* = 0. A corresponding relation between J n x and J n -\X may be obtained. Put z = aT"i> in (40) THE VIBRATING MEMBRANE. 11 ... dh 2ft — 1 dv dx 2 x dx A solution for this is v = x n J n x. Put this in (46), .._. d 2 (x n J n x) 2ft - 1 d(x n J n x) . (47) —& x dx—+ xJ » x = °- The corresponding equation for J n -\X is ~ * V "-i*J 2ft - 1 d T d . _ • , "1 (49) " —^ £ [ & ( *" J "* } " xnJn ~ lX J A solution for this is (50) -g (x n J n x) - a: n J n _!a; = 0. Relations (a), (6), (c), (d) may be obtained from (45) and (50). Relation (e) may be obtained from (40) by the substitution, z = v/ V x, where v = V xJ n x. 5. SOLUTION OF THE EQUATION IN PARABOLIC COORDINATES. The equation of the vibrating membrane written in parabolic coordinates is (51) (Q?+W _ =0 2^__ + _j. This separates into d 2 T 12 THE VIBRATING MEMBRANE. and (52) ^-+(JfeV + X)X=0, d?Y (53) ^r+(W-X)y = 0. Dr. H. Weber in his Partiellen Differentialgleichungen, Vol. 2, page 256, finds two imaginary solutions for (52) as limiting forms of the hypergeometric series. He also finds two real solutions in definite integral form. Several new solutions for special values of X will now be obtained. Repeated differentiation of (52) will point the way to the construction of equations which when differentiated 1, 2, 3 • • • n times will give (52). The nth equation of this series can be solved if X has the special value X = ik(2n + 1), where n is a positive integer. And from this solution a solution of (52) may be obtained. It is (54) X=C^eY^-^ n (e-^). From this a series of functions depending upon n may be ob- tained. For n — we have the very simple one, (55) X= C(V«)- iW . Weber's two solutions for X = ik become X x = Ci( V e) w • [ 1 + (- iko?) + t^ (- ike?) (56) L *isi<-*y.+ ]■ (57) X 2 = C,( V e)" a, V (- ike?) [ 1 fj~ (- ike?) +rr|r 5 (-^) 2 + r^5T7(-^) 3 ---]- Equation (56) reduces to Zi = C\( V e) - *** 1 and this agrees with (55). The series in (57) may be summed by means of the integral THE VIBRATING MEMBRANE. 13 f e~ x (^x)- 1 dx = 2e _I Va;l 1 + ^3^ + ^7375^+ •*•], which is obtained by repeated integration by parts of the left member. Applying this to (57) we obtain, p-ikcfl (58) Z 2 = ( V e) -****( V-2&) e**da. Jo None of the above results lend themselves to the solution of the problem of the vibrating membrane because they are imaginary. For X = 0, however, two real solutions of (52) are (59) Xx-aojl 3.4+3.7.4.8 3-7-11-4-8-12" 1 " '" J 1 (60) k e a n -1 4-8-12-5-9-13 The corresponding y-functions for (53) may be found from these by replacing a with j8. These functions then may be used in solving the problem of the vibrating membrane with parabolic boundaries when the mode of vibration is such that X = 0. The first two roots of X x = and X 2 = in (59) and (60) have been computed by the writer. They are , (*! = 4.013- -., J:n=5.563..-, for Xt - j^ _ 1Q 246 . . , t for Z 2 = j ^ = n ^ , % The roots for Fi = and F 2 = are identical with the corre- sponding roots of Zi = and X% = 0. These roots are the values of the parameter which give the nodal lines for this par- ticular mode of vibration. Weber's two imaginary solutions for X = are, X, = ( V e) w [ 1 + i(- iko?) + |~~ (- iko?f (61) , 1 - 3 ' 13 < V,4 1 14 THE VIBRATING MEMBRANE. (62) X 2 = ( V e) ikai V ( - iho?) [ 1 + ft- ika 2 ) + peg (" *^ + s^i (" ^) 3 ' • •] • Equation (61) reduces to (59), a real solution, if a = 1. Equa- tion (62) reduces to (60), a real solution, if a x = V— i. These results may be obtained by performing the indicated multipli- cations in each case. 6. REVIEW OF THE WORK PREVIOUSLY DONE ON THE SOLUTION OF THE EQUATION IN PARABOLIC AND ELLIPTIC COORDINATES. Parabolic coordinates : The real solutions of the equation 7)2 x — +(Fa 2 + X)X=0 obtained by Dr. H. Weber* and referred to previously are (63) Xi - jf 1 s~i(l - »)H cos [ Vc?(s *- *) + £ log — ] ds, (64) Xi = f t-»(l -s)-i cos VtfcPis -h)+ Ij; log \ZrA *- The corresponding functions of /3 may be obtained from these by changing a into /3 and X into — X. Using these solutions Weberf discusses the vibration of regions bounded by parabolic nodal lines. He proves the theorem, " If a function of u different from zero, defined by the equation d 2 u . d 2 u , ., is continuous and its first differential quotient also, inside a bounded region but vanishes on the boundary, then k 2 must be real and positive." Such regions are of three kinds, * "Partielle Differentialgleichungen," Vol. 2, page 256. t Mathematische Annalen, Vol. 1, 1868. THE VIBRATING MEMBRANE. 15 (a) Bounded by two parabolas one from each family. (b) Bounded by two parabolas from one family and one from the other. (c) Bounded by two parabolas from one family and two from the other. For (a) the boundary conditions are X = for a = ± a\ and Y = for j3 = ab ft. Under these conditions four particular solutions may be found, u = X x • Y\ t u = X 2 -Y 2 , u = Xi • Y 2 , u = Xi-Yi. The last two however will not satisfy the condition that u shall be single valued within the region. Two separate systems of solutions then may be built from u = X\ • Y x and u = X 2 -Y 2 . In each solution the constants X and k appear and may be determined as the roots of the two transcendental equa- tions involved. For (6) the boundary conditions are X = for a = a\, a = a 2 and Y = for /3 = ± ft. Two types of solutions will fit these conditions, u = Yi{A x X\ + A 2 X 2 ), u = Y 2 (AiXi + A 2 X 2 ). These give rise to two systems of solutions. The constants in each solution are A\JA 2f k, X and they may be determined from the three transcendental equations involved. For (c) the boundary conditions are X = for a = «i, a = a 2 and Y = for = ft, jS = &• One system of solutions only can be used here, u = (^iZi + A 2 X 2 ){BiYx + l^l^) an d there are four transcendental equations to determine the constants AxlAt, B 1 /Bi, X, k. M. Hartenstein* obtains solutions for the equation d 2 u d 2 u 70 in parabolic coordinates for negative values of k 2 by trans- forming the solutions of the equation in circular coordinates (Bessel's Functions) into their corresponding functions expressed in parabolic coordinates. This is a part of a general discussion of the equation expressed in the four systems of coordinates used in the third part of this paper. The results obtained contribute nothing to the discussion of the problem of the vibrating mem- brane. * Archiv der Mathematik und Physik (2), Vol. 14. 16 THE VIBRATING MEMBRANE. Elliptic coordinates: The equation of motion of the vibrating membrane expressed in elliptic coordinates is, dW fdW dW\ (65) (cosh 2 - cos 2 a) -^ = a 2 ^ + ^ J and separates into ¥+« = » and S u d u (66) ^ + ^ + F(cosh 2 - cos 2 a)u = 0. This last equation separates into, (67) -^ - (P cos 2 a - \)X = 0, (68) ^ + (fc 2 cosh 2 18 - X) Y = 0. Dr. Heine* shows that (66) is a limiting form of the differential equation of the Lame Functions in the same way as the differ- ential equation of the Bessel's Function is a limiting form of the differential equation of the Spherical Functions. He solves (67) by assuming that the constant X depends in a definite way (as a root of a known transcendental equation) on the constant k and the solutions periodic functions of a. Four classes of func- tions are found to be solutions of (67) under these assumptions. They are, (a) Series of cosines in ascending even multiples of a. (b) Series of cosines in ascending odd multiples of a. (c) Series of sines in ascending odd multiples of a. (d) Series of sines in ascending even multiples of a. E. Mathieut in discussing the vibratory movement of an elliptic membrane, uses periodic functions as solutions also of the above equations. He shows that the general solution of such linear differential equations of the second order as (67) and (68) is composed of two parts, one zero for the zero value of the argument and the other a maximum. If the constant X has * "Handbuch der Kuglefunctionen," page 401. t Journal de LiouvUle (2), Vol. 13, 1886. THE VIBKATING MEMBRANE. 17 the form X = g 2 + bk A + ck* -f- • * • where g is an integer and b, c, etc. constants determined by the periodic property, the solutions of (67) and (68) may be written as periodic functions of a and |8 in ascending powers of k 2 , and they become zero g times from to w. For the special case when k = these solu- tions become Xi = sin (ga), X% = cos (ga), Yi = sin (gi(3), F 2 = cos (gift). Solutions for these equations are also given in series according to the ascending powers of sin and cos of a and /3. In this case the general solution of (67) is the sum of solutions one even and the other odd in sin a and one even and the other odd in cos a. These results agree with those given by Heine. From these solutions two types of vibration are possible. And some simple cases of hyperbolic nodal lines are, the major axis alone, both together, two asymptotes of the same hyperbola, two asymptotes of the same hyperbola with the major and minor axes. The elliptic nodal lines are obtained from Fi = and F 2 = as follows: If /3 = B on the boundary of the membrane when Y = 0, the constant X may be found from Y(B, X) = 0. If Xi, X 2 , X3, • • • are the roots of this equation in order of increasing magnitude, F(j3, Xn) = will give, through its roots in /3, the parameters of the elliptic nodal lines. This equation is shown to have n — 1 roots less than B, and there- fore the number of elliptic nodal lines is n — 1. F. Pockels* has investigated the function defined by (66) for the region bounded by any two ellipses, j3 = j3i and j3 = jff 2 and any two hyperbolas, a = ai and a = a 2 under the supposition that « = 0on all four sides. He proves the following theorem: " There is for such a region a doubly infinite series of normal functions which satisfy the boundary condition u = on all four sides, and of which a definite number (m — 1) of elliptic and a definite number (n — 1) of hyperbolic nodal lines are characteristic, (m = 1, 2, • • •, 00, n = 1, 2, • • •, 00)." By cut- ting the bounded region along the line connecting the foci, and assuming that u and its first derivative are continuous along this new boundary, the new region thus formed may be regarded as a simply connected Riemann surface. This makes possible the consideration of regions bounded by less than four curves, as * " Ueber die Partielle Differentialgleichung Au + & 2 w = 0." 18 THE VIBRATING MEMBRANE. for example, those bounded by two ellipses and two pieces of the same hyperbola, two hyperbolas and one ellipse, etc. Hartenstein* has solved (66) for ¥ negative. His method is entirely different from that employed by the above writers and consists in extending Bessel's Functions by transformations which will make them solutions of (66). The transformations are many and complex and the results add nothing to what has already been noted. Dr. F. Lindemannf solves (67) and (68), obtaining non- periodic functions of a and /3 for his solutions. He finds four special solutions for each in series form which include those periodic solutions of Heine noted above, as special cases. 7. BIBLIOGRAPHY. E. H. Barton. Textbook on Sound. W. E. Byerly. Fourier's Series and Spherical, Cylindrical and Ellipsoidal Harmonics. Grey and Mathews. Treatise on Bessel Functions. E. Heine. Handbuch der Kugelfunctionen. H. Hartenstein. Archiv der Mathematik und Physik, Vol. 14, 1890. Dissertation, Leipzig, 1887. G. Lame. Coordonnees Curvilignes. Lecons sur Elasticite. F. Lindemann. Mathematische Annalen, Vol. 22, 1883. E. Mathieu. Journal de Liouville, Vol. 13, 1868. Cours de Physique Mathematique, 1873. Michell. Messenger of Mathematics, Vol. 19, 1890. F. Pockels. Ueber die Partielle Differentialgleichung Am + Wu = 0. Lord Rayleigh. Theory of Sound, Vol. 1. B. Riemann. Partiellen Differentialgleichungen und deren An- wendung auf Physikalische Fragen. E. Reinstein. Annalen der Physik, Vol. 35, 1911. H. Weber. Die Partiellen Differentialgleichungen, Vol. 2. Mathematische Annalen, Vol. 1. A. Winkelmann. Handbuch de Physik, Vol. 2. * Archiv der Mathematik und Physik (2), Vol. 14. t Math. Annalen, Vol. 22, 1883. 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