1890.] and the Curvatures of their Trajectories. 37 



The result will be simplified if we take the central axis of 

 the motion for the axis of x, so that u = V, co x = Q, while the 

 other components vanish, though their fluxions remain finite. 

 We thus obtain the equations 



L M-n 2 y N-il'z 



- = = (4), 



-37 z — y y 



wherein 



L = u — yw z + zw>^\ 



M = v — zco x + xw\ (5), 



N = w - xw v + y&J 



and «r is written for V/fl, the pitch of the given screw-motion. 



The equations (4) thus obtained may readily be verified by 

 intuition ; for L, M, N represent the component accelerations 

 due to the motion of the origin and the change of values of 

 the angular velocities, while 0, — 12 2 ?/, — Q?z are the components 

 of the centrifugal force round the central axis, and it is clear 

 that these together make up the total acceleration. 



These equations (4) represent the curve of intersection of 

 two paraboloids. To reduce them to the simplest possible form, 

 first turn the axes of y, z round that of x, so as to make w. zero. 

 Then move the origin along the central axis a distance h, so 

 that the equation referred to this new origin is obtained by 

 writing x + h in place of x, and take h = wjm y . We thus have 

 finally 



ii.+ zw y _v — Z(b x — Q?y — xcb,^ yco^— fl^x . 



\")i 



■sr z —y 



where we notice by the way that the numerators give the simplest 

 form to which the rectangular component accelerations for a 

 moving solid can be reduced. 



The equations (6) represent the curve of intersection of a 

 parabolic cylinder having its generators parallel to the axis of x 

 with a rectangular-hyperbolic paraboloid having its axis in the 

 same direction. These surfaces intersect on the plane infinity 

 along the line where any plane z — constant meets it. The finite 

 part of their curve of intersection, which is the proper inflexional 

 curve, is therefore a tivisted cubic. Its equations (6) may be put 

 in the form 



y=A + Bz(a + /3z)\ 

 ■x = y(a + (3z) J ^ h 



which are unicursal in the parameter z. 



We remark that a wire of this form is the most general solid 

 that can be moved with rotation so that all its points are in- 

 stantaneously describing straight paths ; also that any wire whose 



4—2 



