[Not before published. (See page 229.)] SUPPLEMENT TO A PAPER ON THE THEORY OF OSCILLATORY 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, 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 ∞ to +∞, other is x or a quantity which extends with a from 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, y, where $=f(udx+vdy), y=√(udy — vdx), simplicity might be gained in consequence of the immediate relation of these functions to the problem. We know that 4, 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) x= - +Σ(Aeim4/c+Be-imy/c) sin imp/c.........................(9). 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), a will be an odd function of 4, in accordance with what has already been shown at page 212. Corresponding to the above. value of a we have У ป +Σ (A ̧eim/c - Be-im/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 convenient to do, being arbitrary, we may take, as it will be 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, = 0, and we must therefore have for ↓ = 0 g (y + C) S− 1 = 0, (C+ΣA ̧ cos ip) {1 – 2EiA ̧ cos ip + Σï3A ̧2 + 2Σijà ̧A, cos [(i — j)4]} where in the last term within parentheses each different combination of unequal integers i, j is to be taken once. On account of the complicated form of this equation, we can proceed further only by adopting some system of approximation. The most obvious is that adopted in the former paper, namely to proceed according to powers of the coefficient of the term of the first order. If we multiply out in equation (13), and replace products of cosines by cosines of sums and differences, we may arrange the equation in the form where the several B's are series of terms involving the coefficients A. And as the equation has to be satisfied independently of p, we must have separately A slight examination of the process will show that A, is of the order i, and that consequently the product of any number of the A's is of the order marked by the sum of the suffixes, and that B is of the order i. In proceeding therefore to any desired order we can see at once what terms need not be written down, as being of a superior order. Thus in proceeding to the fifth order we must take the six equations B, 0, B1 = 0,... B1 = 0, which when written at length are = These equations may be looked on as giving, the first, the arbitrary constant C, the second, the velocity of propagation, and the succeeding ones taken in order the values of the constants „,  ̧ à ̧‚ à ̧, respectively. I say "may be looked on as giving", for it is only when we restrict ourselves to the terms of the lowest order in each equation that those quantities are actually given in succession; the equations contain also terms of higher orders; and 29 3' to get the complete values of the quantities true to the order to which we are working, we must use the method of successive substitutions. As to the second equation, if we take the terms of lowest order in the first two we get Cg, and then by substitution in the second equation 1 =g, the constant A, dividing out. The equation 1 = g becomes on generalizing the units of space and time cg/m, and accordingly gives the velocity of propagation to the lowest order of approximation. On eliminating the arbitrary constant in the above equations, and writing b for A,, the results become x = − + be sin - (b+b) e2 sin 24+ (b +196) e sin 34 — § b*e1 sin 40 + 125 b3e5 sin 50..................................(15), y = −y+be cos & − (b2 + 1 b1) e2 cos 2 + (3 b3 + 12 b3) e3 cos 34 - & be1 cos 4+ 125 b5e5 cos 50............................(16). The equation (14) gives to the fifth order the square of the velocity of propagation in the wave motion; and (15), (16) give the point where the parameters 4, have given values, and also, by the aid of the formulæ previously given, the components of the velocity, and the pressure, in the steady motion. These same equations (15), (16), if we suppose constant give implicitly the equation of the corresponding stream line, or if we suppose constant the equation of one of the orthogonal trajectories. To find implicitly the equation of the surface, we have only to put = 0 in (15), (16), which gives x=-4+bsin - (b2+ b) sin 24-(b+126) sin 34 y= bcos - (b2 + b) cos 24 - (63 + 1265) cos 36 It is not necessary to form the explicit equation, but we can do so if we please, most conveniently by the aid of Lagrange's theorem. The result, carried to the fourth order only, which will suffice for the object more immediately in view, is y + { b2 + b * = (b + § b3) cos x − (1 b2 + 1 b1) cos 2x + § b3 cos 3x - b1 cos 4x... (19). |