Equilibrium of Vis Viva, 157 



after a laborious consideration of many special cases*, I can 

 appreciate the value of continually testing general theorems 

 by means of special examples, and will therefore take up one 

 of the particular cases suggested as a test-case by Lord Kelvin, 

 namely the one he mentions last ; because it is the simplest, 

 and because I respect the sentence of De Morgan, quoted once 

 by Prof. Tait, to the effect that "formula?, if too long, are 

 often not read." 



Let a material point of unit mass move in the plane of x,y. 

 Let x, y be its coordinates, q its velocity, u, v the components 

 of the velocity in the directions of the axes of coordinates, 

 and 6 the angle which the direction of motion makes with 

 the positive direction of the axis of abscissse, the angle being 

 counted from to 2tt. Suppose the potential V to be any 

 function of the coordinates. We assume that the motion 

 neither becomes infinite nor asymptotically approaches a fixed 

 limit ; and that all possible sets of values of x, y, and 6 which 

 are consistent with the equation of vis viva are obtained with 

 any required degree of approximation, provided the motion 

 continues for a sufficiently long time T. 



Let us take a ^-coordinate perpendicular to the plane of 

 x, y, and define any condition of the moving point by marking 

 off as ^-coordinate over the point where it happens to be, the 

 angle 6 which its velocity makes with the positive direction of 

 abscissae. We shall call the point of space with the coordi- 

 nates x, y, 6 the point which characterizes the condition of 

 the moving body, or briefly the instantaneous condition-point. 



We can. then define our assumption thus : — In the course 

 of an interval of time T the condition-point occupies all 

 positions in a finite cylinder (the condition-cylinder) which 

 has a height 2tt in the direction of the axis of z. The 

 condition-point passes suddenly from the base to the summit 

 of this cylinder, and vice versa ; with this exception its motion 

 is continuous. 



Suppose the moving point to be at the point x, y at any 

 time t, and let its velocity make an angle 6 with the axis of 

 abscissa? and have the components u, v. The condition-point 

 at time t will then be a point A of space with coordinates 

 x,y,0. 



After the lapse of a very short time St, that is at time t + St. 

 let the moving point be at x' ,y' . Let#' be the angle between 

 the direction of motion and the axis of abscissa^, and u f , v' the 



* "On the Equilibrium of Vis Viva, 1 ' Wien. Sitz.-Ber. vol. lviii. 8th 

 October 1868.—" Solution of a Mechanical Problem,'' ibid. 1 7th December, 

 1868. — " Some General Theoreu.s on Equilibrium of Heat " (end of part '2 ) 

 (/. c). — " Remarks on some Problems in the Mechanical Theory of II cat .'" 

 Wien. Sitz.-Ber. vol. lxxv. 11th January, 1877 (end of part 8). 



