Preston— Application of Parallelogram Law in Kinematics. 473 
Lemma.—Before going further it may be well to state the 
result made use of in the last investigation as a separate lemma 
for subsequent use. We have seen that 
Se gill RAE 
dd le wae a 
_ and, since MQ is the radius of curvature of MP’ regarded as an 
element of path described with the velocity r, it follows that the 
radius of curvature of the radial velocity is equal to the radial 
velocity divided by the angular velocity, or 
UQ = 
In the general case of polar coordinates in three dimensions the 
acceleration can be written down with almost equal ease. In this 
case the sides of the element of volume are dr, rd@, and r sin 0 
respectively, passing from the corner P (fig. 5) to the diametrically 
Bret 5: 
opposite corner P’ in the direction in which 7, 0, @ increase. In 
this case the velocity may be resolved into three components, 
r,70,rsin 0 along the radius vector, along the meridian, and 
