150 On a Dynamical Theory of 



the very same relations which connect the co-ordinates x ', ?/', z 

 with z, y, z, or the displacements ', r/, ' with , r?, 2. 



That is to say, if the axes of #, ?/, s make with the axis of x 

 the angles a, /3, 7, with the axis of y the angles a', |3', 7', and 

 with the axis of s' the angles a", j3'', 7" respectively, we shall 



have 



X = X' COS a + T' COS a' + Z' cos a", 



Y= X' cos /3 + F' cos /3' + Z' cos /3", (D) 



Z' = J' cos 7 + F' cos 7' + Z' cos 7", 

 and 



X' = X cos a + Y cos )3 + Z cos 7, 



F' = X cos a' + F cos j3 r + Z cos 7', (D') 



just as we have, for example, 



% = % COS a + i/ COS a' + ?' COS a", 



n = ' cos )3 + / cos j3' + T cos )3", (d) 



= ' cos 7 + r! cos 7' + y cos 7", 



and 



#' = x cos a + y cos ]3 + 2 cos 7, 



?/ = # cos a' + ?/ cos j3' + s cos 7', (d') 



2' = # cos a" + y cos /3" + s cos 7". 



For, the change of the independent variables #, y, 2 into 

 of, y', z' gives us the equations 



di\ dr\ dx drt dy drj dz' 



= __ _ 



dy ~ fa' ~dy dy' dy dz' dy 9 



in the right-hand members of which we have to substitute the 



