22 Mr. Mac Cuutacu on the dynamical Theory of 
dy _ dy dx’ dy pet dy dz 
FE: ~ da’ dz * dy’ dz ' dz’ dz’ 
_ de dx’ wigs dg dy’ , & dz 
E= dui dy ' dy dy ' dz dy’ 
in the right-hand members of which we have to substitute the values of the dif- 
ferential coefficients obtained from ) and a Thus we get 
a1 (Fe mak; ee e+e © cos p”) cosy 
ce one 75 el ‘See cos 
Coy ) 
— = 7 bee — T cos B fe a cos p’) cos 7", 
=! ) 
) 
A 
Ce 
d. 
z ovr Shs +E cos 
cos B 
( ee dnt aos ? 
ae a y y cosy + Se, c08 7 cos B 
= e cosy + 5 cos ETE = =, COS YY’ ’) cos 63 
and when we subtract these equations, attending to the formule in Lemma I. 
we find 
dm dt_ df d¢ de ae ; dé dy 
de ay = (ae = gy) coset (ais — Gz) eos + (Fe — da’ ii) 8" 
or simply, 
x =x’ cosa+ y’ cosa’ + 2’ cos a’, 
which is the first of formule (p). And in like manner the others may be proved. 
The same things will obviously hold with respect to quantities derived from 
x, Y, z in the same way that these are derived from ¢, y, ¢. That is, if we put 
_dy dz nd dx _dx dy 
‘~dz dy 1 des. a” A = dy dx’ 
and then suppose the axes of coordinates to be changed, the formule for the 
transformation of the quantities x, y, z, will be similar to those for the trans- 
