724 Varying always have the same definite amount of energy. All, except the action, natural mode of passage, in general require constraint in order that they may be described. Hamilton s grand extension of the subject depended on comparing the actions in a number of natural modes of passage, differing from one another by slight changes in their ter minal configurations, and slight changes in the whole initial energy. In this new form of statement the unintegrated part of the ex pression for 8 A vanishes, since all the modes of passage contem plated are natural. The alteration of the whole energy, however, adds a special term to the equation, and we can at once write, from the expression (A) 202, the equation for the change in the action under the new conditions, viz., 8A = [2i(;cSa; + ySy + z$z}~ + <5H , the part in brackets having to be taken between limits correspond ing to the terminal configurations, and the variations Sx, Si/, Sz at these being subject to the conditions of the system. We cannot here consider this equation in its general form. We content ourselves with the simpler special deductions from it re quired for completing our sketch of Kinetics of a Particle. The last given equation, written in full for a single particle of unit mass, is 8A = [xSx + ySy + 282] - (x tix + y 5y + 2 82 ) + t8ll , where x , y , z is the initial point, and x, y, z any other point, of the path. If the particle be altogether free, the seven variations on the right-hand side are independent of one another ; and thus we have the following remarkable properties of the quantity A, regarded as a function of seven independent variables (the initial and final coordinates of the particle, and its constant energy), viz., A dx / dA dx n . dx J dt <fc, / dt dt dt = t . From these we gather at once that A satisfies the partial differ ential equations fdA dss .(A)=t = 2(H-V ) al2o / 205. The whole circumstances of the motion are thus dependent on the function A, called by Hamilton the "characteristic func tion." The determination of this function is troublesome, even in very simple cases of motion ; but the fact that such a mode of representation is possible is extremely remarkable. 206. More generally, omitting all reference to the initial point, and the equation 204 (2) which belongs to it, let us consider A simply as a function of x, y, z. Then Any function, A, which satisfies the partial differential equation dx) + (dy) + dz) (1), possesses the property tJiat . represent the rectangular dx dy dz components of the velocity of a particle in a motion possible under the forces whosi potential is V. For, by partial differentiation of (1) we have dt dt J dt 2 dx dx dx- dy dxdy dz dxdz with other two equations of the same form. But we have also three equations of the form d /dA dx d 2 A dy d?A dt dx J dt dx* dt dxdy Comparing, we see that dx _dA dy _dA dz dA dt dx dt dy dt dz satisfy simultaneously the two sets of equations. 207. Also if a, /3 be constants, which, along with IT, are in volved in a complete integral of the above partial differential equation the corresponding path, and the time of its description, are given by dA _ dt where a 1} /3 lf e arc three additional arbitrary constants, For these equations give, by complete differentiation with regard to t, d*A dx^ d-A dy d dxda, dt dyda. dt dzda dt d^A dx d^A dy d^A dz __ , dxd& dt + dydfr dt + dzdp dt ~ d 2 A dx d-A dy d 2 A dz _ dxdR dt + dydH ~dt + dzdli dt = J But, differentiating 206 (1) with respect to a, j8, II respectively, we cet d 2 A dA _^ 2 A_ dA __ dadx dx dady dy dadz dz d"*A dA d 2 A rfA d 2 A dA dA J dx The values of -- , &c., in (a) are evidently equal respectively to dt dA those of -- , &c., in (6). Hence the proposition. 208. " Equiactional surfaces," i.e., those whose common equa tion is A = con st., are cut at right angles by the trajectories. For the direction cosines of the normal are obviously proportional to Surfaces of equal action. dy dz dt dt dt Thus the determination of equiactional surfaces is resolved into the problem of finding the orthogonal trajectories of a set of given curves in space, whenever the conditions of the motion are given, The distance between consecutive equiactional surfaces is, at any point, inversely as the velocity in the corresponding path. This may be seen at once as follows : the element of the action, which is the same at all points, is vSs (where 8s, being an element of the path, is the normal distance between the surfaces). 209. In consequence of the importance of the method we will Plane- take two examples of its application. First a direct example, then tary one depending on the equiactional surfaces. motion. To deduce from the principle, of varying action " the form and mode of description of a planet s orbit. In this case it is obvious that represents the attraction of dr gravity ( -/x/r 2 ). Hence the right-hand member of 206 (1) may be written 2(H + /i/r). Let us take the plane of xy as that of the orbit, then the equation 206 (1) becomes ~ dy J (1). It is not difficult to obtain a satisfactory solution of this equation, but the operation is very much simplified by the use of polar co ordinates. With this change, (1) becomes dr) which is obviously satisfied by = constant = a, r =2 H+-* - a .; / V r J r* (2), (3). Hence A-=ae+/dr^2(R + /J ./r)-a*/r 2 . . The final integrals are therefore, by 207, r dr (5), and (M)= I dr . . . (6). These equations contain the complete solution of the problem, for they involve four constants, a v a, H, e. (5) gives the equation of the or jit, and (6) the time in terms of the radius-vector.
To complete the investigation, let us assume