40. Slickensides along bedding plane, interrupted by worm tubes. Bala Beds, Constitution Hill, Aberystwith. 41. Slickenside fluted by lumps and grains seen on the specimen. Landslip in Barton Clay, with some surface material dragged in. Barton Cliff, Hampshire. 42. Fault breccia smoothed and scored by movement against the wall of the fault. Cove below Daddy Hole, Torquay. 43. Part of fault face in Silurian. The Grove, Bodfari, near St Asaph. Showing strong parallel fluting. 44. Fluted and slickensided face of grit with film of mineral. Nodale, Stansfield Moor. 45. Fluted and slickensided face of lode-stuff with film of mineral. Derbyshire. 46. Curved slickenside on slate rock. Quarry north of Aberystwith. 47. Fluted structure due to hard beds in rock, affected by imperfect cleavage and joints. Tirgwyn, three miles E.N.E. of St Asaph. 48. Part of a glaciated Silurian boulder from the drift of Maes Mynan near Caerwys in North Wales. The striæ seen on this fragment however are almost all due to the weathering out of softer lines in the rock. 49. Fragment of Devonian sandy shale with the joints weathered out into grooves. Foot of Rough Tor, about a mile from Okehampton Station by the track leading up to Yes Tor, Dartmoor. Finer bed in the conglomerate at the base of the Carboniferous Rocks showing striation across the matrix and included pebbles. Holbeck Gill, between Dent and Garsdale, Yorkshire. 50. 51. 52. 53. Do. do. Scratched stone from do. do. Do. do. 54. Striated pebble in Triassic conglomerate. Portskewet, Monmouthshire. 55. Hard lump included in the shale at the base of the Cambrian 56. Pebble from the conglomerate twisted into the Archæan Gneiss. Obermitweida, Annaberg. 57. Small boulder crushed and broken, and the several parts more or less separated and then cemented together in the conglomerate by mineral matter. Basement bed of the Carboniferous. Holbeck Gill, between Dent and Garsdale, Yorkshire. 58. Do. do. do. 59. Do. do. do. 61. 62. Surface of Coniston Flags smoothed and striated (slickensided) by thrust at base of Conglomerate (Basement Bed Carboniferous). Holbeck Gill, between Dent and Garsdale. 63. Stylolites from the London Clay, Sheppey. 64. Flint which has been forced by earth movement through the enveloping chalk. Fossils thrust through surrounding material. 66. Pebble of quartzite worn by blown sand. Copitz by Pirna. 67. Do. These are good samples of the stones supposed to have been ground on several sides as they occupied different positions under the ice. Similar specimens are exhibited in the Dresden Museum as evidence of glaciation. 68. Glacially striated pebble found by myself below the lignite at Wetzikon. 69. Do. on the moraine above the lignite of Wetzikon. 70. A piece of Haffield conglomerate, Haffield. 71. A slightly worn fragment of close textured mudstone with a shining surface produced by a film of iron oxide. Haffield conglomerate, Haffield. 72. Boulder of a green sandstone with shells and masses of the phosphate bed attached and covering striations caused by weathering along line of weakness due to rock structure. Compare Nos. 37, 38, 39. 73. Rolled lump of older boulder clay found in newer boulder clay, Clwydian Drift. St Asaph, North Wales. 74. Lump of newer boulder clay (Clwydian Drift) rolled on shore. Colwyn, North Wales. 75. Lump of London Clay rolled on shore. Sheppey. 76. Pebble of Bala Limestone with Halysites catenularius projecting on the surface. In conglomerate at base of Carboniferous. Holbeck Gill, between Dent and Garsdale, Yorkshire. November 13, 1893. PROF. SIR G. G. STOKES IN THE CHAIR. The following Communications were made to the Society: (1) The Application of Newton's Polygon to the Theory of the Singular Points of Algebraic Functions. By H. F. BAKER, M.A. (2) On a Class of Definite Integrals connected with Bessel's Functions. By A. B. BASSET, M.A., F.R.S. This paper the form contains a method for reducing double integrals of λη ε :-λ2 Л(\r) dλ to single integrals; also various expressions for the different kinds of Bessel's functions are found in the forms of definite integrals. 1. In many investigations connected with the motion of viscous liquids and the conduction of heat, definite integrals occur which depend upon integrals of the form The ordinary expression for J, in the form of a definite integral is Jn (λr) = (λr)" T.1.3...(2n-1). cos (Ar cos 0) sin2" Ode...(1), and accordingly when m+n is an even integer, the integral V" depends upon one of the form The value of this integral when s=0 is known to be √π/2a. e-b2/a2, from which the value of the integral (2) can be deduced by differentiation with respect to b. If however m+n is an odd integer, V," depends upon integrals of the form This last integral cannot be evaluated in finite terms. There is however another form of Jo, which is given by the equation Jo (λr) = == sin (Ar cosh ) dp (4), and consequently V2+1 depends upon integrals of the form which can be deduced from the known value of (2) by differentiating with respect to b. 1 m The integral (2) enables V," to be integrated with respect to A when m is an odd integer; but when m is even this integral cannot be employed. Now J'(r) = − J1(r), and if it were allowable to differentiate (4) with respect to λr, we should obtain m We shall hereafter prove equation (6) by a different method ; accordingly when m is an even integer equation (6) enables V," to be integrated with respect to X. Since any three Bessel's functions are connected together by the equation 1 it follows that since Vm and V" can be integrated with respect to λ for any integral value of m, the same process can be performed upon V n m The advantages of reducing a double integral to a single integral are obvious; and it often happens that when the integration with respect to λ has been performed, the integration with respect to 0 or can also be effected, and the integral completely evaluated. A good many proofs of (4) have been given, one of which will be found on p. 431 of the fifth volume of these Proceedings. Equation (6) may be established in a precisely similar manner by integrating the definite integral first with respect to x, and then with respect to u and comparing the results. The function Y 2. In many investigations the second solution of Bessel's equation, which will be denoted by Y, is required. In some cases it is better to employ complex quantities throughout and to discard the imaginary part in the final result, whilst in others it is better to employ a function with a real argument. It is easily shown that the definite integral satisfies the same equation as J, (r), as is otherwise obvious from the theory of linear sources of sound-see Rayleigh, Theory of Sound, Vol. II. p. 275. The integral (8) may be written 2 π S cos (r cosh 4) dø – 24 ) sin (r cosh p) dø = Y. (r) – „J。 (r) by (4), whence π which may be regarded as the definition of Yo; also differentiating with respect to r, and recollecting that Y1(r) = - Y'' (r) we obtain Y, (r) = 2 [* cosh & sin (r cosh p) dø. 1 π 0 ......... This result will be obtained by a more satisfactory process later on. It is proved in Lord Rayleigh's Sound, Vol. II. p. 273, that 22.422 2a . 4o −).. The where γ is Euler's constant, and S1 = 1 + 21 + ...n-1. imaginary part of the right-hand side is obviously equal to -Jo (r), and we must now show that the integral (9) is equal to the real part of either side of the above equation. This we shall do by showing that the integral (8) can be expressed by means of the series on the left-hand side of (11). We shall also establish the legitimacy of equation (10), which was deduced from (9) by differentiation with respect to r. In (8) put x = 1+y and it becomes The latter series is therefore equal to Y, (r) — J。 (r); and by realizing and equating the real and imaginary parts we shall obtain the two series for Y, and J, which are given in Rayleigh's Sound, Vol. II. p. 275 and Vol. I. p. 264. Equations (10) and also (6) may be deduced by differentiating the series (12) with respect to r and then summing the two resulting series to which the differentiation leads. 3. The circumstance that two distinct forms of the J functions exist in one of which the limits are 1 and 0, whilst in the other they are ∞ and 1, for equation (4) is equivalent to |
