10 Proceedings of the Royal Irish Academy. 



5. The result of such transformations may be examined in a more general 

 form. Let the equations of relative motion in their original form be 



dV 



dx 



dr 



x - 2ny - n'x = 



y + 2nx - n 2 y = , 



J dy' 



or, as they may be written, 



d fdT\ 

 dt\dx)~ 



dT 

 dx 



dV 

 dx ' 



d fdT\ 

 dt\dy) 



dT 



dy ' 



dV 



dy' 



= %(£- ny y 



+ *(«/ 



+ nx) 2 . 



x + iy = 



/« + 



k), 



dx dy 



~dT ~ HH' 



dx 



dy 



dK 



where 

 Let 



so that 



and 



&r dx (dy\ (dy 



d -12\ + ±- (°±i\ _ 

 d% J dt[ ? i 



Hence 



dl ' dr, dn' dl " \dV \di,J " \djj Vdl, 

 Then T = T t + 2 1 , + T , 



where T 2 = IJX? 2 + f ), 



T„ = §7> 2 (Z* + ^-). 



The equations of motion may be written 



d fdT % \ d fdTA dT, dT, dT dV 



+ ~^z + 



dt\dlJ dt \ d lJ dl dl dg dk ' 

 d_ /dTA d_ fdTA _dT 1 = dT 1 dT, dV_ . 



dt\dv) + dt\dv) di) dr, + dt, + dr, ' 

 and the integral of energy is 



T 2 = T + V - h. 



dT. dT dV ldJ , h .. dT dV 



m + tt + -dT = 2dt (li+ ^ + tf + -w 



I 2 7" o 



= j l{J(T, 1 + V-h)\. 



