ON THE CRITICAL VALUES OF THE SUMS OF PERIODIC SERIES. 12 312 the surface of discontinuity Oe will be the less the smaller be the notion of the cylinder; and although the expression (119) fails for points very near O, that does not prevent it from being sensibly correct for the remainder of the fluid, so that we may calculate k'2 from (122) without committing a sensible error. In fact, if y be the angle through which the cylinder oscillates, since the extent of the surface of discontinuity depends upon the first power of y, the error we should commit would depend upon y2. I expect, therefore, that the moment of inertia of the fluid which would be determined by experiment would agree with theory nearly, if not quite, as well when a>π as when a<π, care being taken that the oscillations of the cylinder be very small. As an instance of the employment of analytical expressions which give infinite values for physical quantities, I may allude to the distribution of electricity on the surfaces of conducting bodies which have sharp edges. 56. The preceding examples will be sufficient to show the utility of the methods contained in this paper. It may be observed that in all cases in which an arbitrary function is expanded between certain limits in a series of quantities whose form is determined by certain conditions to be satisfied at the limits, the expansion can be performed whether the conditions at the limits. be satisfied or not, since the expanded function is supposed perfectly arbitrary. Analogy would lead us to conclude that the derivatives of the expanded functions could not be found by direct differentiation, but would have to be obtained from formulæ answering to those at the beginning of this Section. If such expansions should be found useful, the requisite formulæ would probably be obtained without difficulty by integration by parts. This is in fact the case with the only expansion of the kind which I have tried, which is that employed in Art. 45. By means of this expansion and the corresponding formulæ we might determine in a double series the permanent temperature in a homogeneous rectangular parallelepiped which radiates into a medium whose which had no radial motion and but little in a perpendicular direction, and the rapidly rushing fluid on the other side. The smaller μ is made, the narrower will this stratum be, but not, so far as I can see, the shorter; and a very narrow stratum in which there is intense molecular rotation passes, or may pass, in the limit to a surface of discontinuity. The above is what was referred to by anticipation in the footnote at p. 99.] temperature varies in any given manner from point to point; or we might determine in a triple series the variable temperature in such a solid, supposing the temperature of the medium to vary in a given manner with the time as well as with the co-ordinates, and supposing the initial temperature of the parallelepiped given as a function of the co-ordinates. This problem, made a little more general by supposing the exterior conductivity different for the six faces, has been solved in another manner by M. Duhamel in the Fourteenth Volume of the Journal de l'École Polytechnique. Of course such a problem is interesting only as an exercise of analysis. 12 ADDITIONAL NOTE. If the series by which r2 is multiplied in (119) had been left without summation, the series which would have been obtained for k” would have been rather simpler in form than the series in (122), although more slowly convergent. One of these series may of course be obtained from the other by means of the development of tan x in a harmonic series. When s is an integer, k'2 can be expressed in finite terms. The result is 12 -2 + (s − 1) ̄1} + 4π ̄2 {2-2 + 4 ̃ ̄2 ... + (s − 1)−2} – †, (s odd,) -2 -2 k'2 = {1 ̃1+3¬1· + ( s − 1)2) — 12. (s even.) Moreover when 2s is an odd integer, or when a = 45°, or = 135°, &c., k'2 can be expressed in finite terms if the sum of the series 12+52 +92 + ... be calculated, and then be regarded as a known transcendental quantity. [Not before published. (See page 229.)] SUPPLEMENT TO A PAPER ON THE THEORY OF OSCILLATORY WAVES. THE labour of the approximation in proceeding to a high order, when conducted according to the method of the former paper whether we employ the function & or y, depends in great measure upon the circumstance that the two equations which have to be satisfied simultaneously at the free surface are both composed in a rather complicated manner of the independent variables, and in the elimination of y the length of the process is still further increased by the necessity of expanding the exponentials in y according to series of powers, giving for each exponential a whole set of terms. This depends upon the circumstance that of the limits of y belonging to the boundaries of the fluid, one instead of being a constant is a function of x, and that too a function which is only known from the solution of the problem. If we convert the wave motion into steady motion, and refer the fluid to two independent variables of which one is the parameter of the stream lines or a function of the parameter, and the other is a or a quantity which extends with a from ∞ to +∞o, we shall ensure constancy of each independent variable at both its limits, but in general the equations obtained will be of great complexity. It occurred to me however that if from among the infinite number of systems of independent variables possessing the above character we were to take the functions 4, 4, where simplicity might be gained in consequence of the immediate relation of these functions to the problem. We know that o, y are conjugate solutions of the equation so that if the form of either be assigned, satisfying of course the equation (1), the other may be deemed known, since it can be obtained by the integration of a perfect differential. If now we take, for the independent variables, of which x and y are regarded as functions, we get by changing the independent variables in differentiation so that x, y are conjugate solutions of the equation The mode of proceeding is the same in principle whether the depth of the fluid be finite or infinite; but as the formulæ are simpler in the latter case, it may be well to consider it separately in the first instance. If c be the velocity of propagation, c will be the horizontal velocity at a great depth when the wave motion is converted into steady motion. The difference between and - cx will be a periodic function of x or of p. We may therefore assume in accordance with equation (5) No cosines are inserted in this equation because if we take, as we may, the origins of x and of at a trough or a crest (suppose a trough), x will be an odd function of p, in accordance with what has already been shown at page 212. Corresponding to the above value of x we have Y C +Σo° (à ̧çîmų¡¢ — Be ̄imp/c) cos imp/c...........(10), the arbitrary constant being omitted, as may be done provided we leave open the origin of y. The origin of being arbitrary, we may take, as it will be convenient to do, y = 0 at the free surface. We see from (10) that increases negatively downwards; and therefore of the two exponentials that with -imy/c for index is the one which must be omitted, as expressing a disturbance that increases indefinitely in descending. We may without loss of generality shorten the formulæ during a rather long approximation by writing 1 for any two of the constants which depend differently on the units of space and time. These constants can easily be reintroduced in the end by rendering the equations homogeneous. We may accordingly put m=1 and c = 1. The expressions for x and y as thus shortened become, on retaining only the exponential which decreases downwards, ΣΑ (C+ΣA ̧ cos ip) {1 – 2ZiA ̧ cos ip + Σï3Ã2+ 2Σijà ̧à ̧ cos [(i −j)ø]} ....(13), |