586 
PHILOSOPHICAL SOCIETY OF WASHINGTON. 
came from considering motion in a curve; and since the radius of curva¬ 
ture p for the point xyz has the expression 
where s is the independent variable, we transform the expressions for the 
forces into others which have a more general form. Thus 
d 2 x d (dx ds \ 
dtf dt \ds dt ) 
d 2 x ds 1 d 2 s dx 
~ ds 2 dt 2 ^ dtf ds 
d 2 x dx d 2 s 
’ ' ds 2 ds dtf ’ 
with similar values for the components Y and Z. Squaring these values 
and adding, we have for the resultant, 
Itf 
^•{(f )'+©)'+©•} 
2 d 2 s j dx d l x ( dy d 2 y 
v ‘ dtf \ ds 
(l iV 
\ dt 2 ) 
d 2 x 
ds 2 ds 
dz 
ds 2 ds 
dtfz \ 
ds 2 j 
+ 1 . 7^2 
f dtf ^ dytf + d* \. 
\ ds 
ds 2 
ds 2 j 
From analytical geometry we have 
dx 2 -f dy 2 -f dz 2 — ds 2 . 
From this equation the second term in the values of E 2 is zero, and the 
coefficient of ( J is unity; so that we have 
2 * _ j^y j. f£iy. 
E 2 = 
+ 
\ dtf ) 
If we differentiate the equation 
dx 2 + dy 2 -f- dz 2 = ds 2 , 
making t the independent variable, and divide by 2 ds dt 2 , we have 
