150 On a Dynamical Theory of 



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

 with x, i/, z, or the displacements ', r/, % with , tj, . 



That is to say, if the axes of a-, y, z 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 z' the angles a", j3'', 7" respectively, we shall 



have 



X = X' COS a + P" COS a' + Z' COS a", 



F= X' cos /3 + F' cos /3' + Z' cos 0", (D) 



Z = X' cos 7 + Y' cos 7' + Z' cos 7", 



and 



.X' = X cos a + Y cos j3 + ^ cos 7, 



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



Z' = X cos a" + Y cos 3" + Z cos y" ; 



just as we have, for example, 



= ' cos a + i/ cos a" + ' cos a", 



TJ = ' cos j3 + ?' cos j3' + ' cos /3", (d) 



= ' cos 7 + j' cos 7' + ' cos y", 



and 



x = x cos a + y cos j3 + 2 cos y, 



i/ = x cosa + y cos /3' + s cos 7', (d') 



z' = x cos a" + y cos /3" + 2 cos y". 



For, the change of the independent variables x, y, s into 

 x', y', 2' gives us the equations 



dt\ di\ dx' dr\ dy' dt\ dz' 

 dz dx' dz dy' dz dz' dz ' 



= __ 



dy dx' dy dy dy dz dy' 



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



