195-197] ROTATING TUBE. 185 



and . . i|{ 



These may be written 



Of these the terms independent of o&amp;gt; are equivalent to v-j- 



along the tangent to the tube at P and v z /p inwards along the 

 normal to the tube. 



The terms containing 2o&amp;gt; as a factor are equivalent to 2a&amp;gt;v 

 inwards along the normal to the tube. This can be seen by con 

 sidering that r along OP and rO transverse to OP are equivalent 

 to v along the tangent in the direction in which s increases, and 

 that we have, as multipliers of 2&&amp;gt;, the components of this result 

 ant turned through a right angle. 



Now we can resolve a vector in the direction OP into com 

 ponents along the tangent at P to the tube and inwards along 



the normal by multiplying by -v- and ^ , where p is the perpen- 



CLS ?* 



dicular from on the tangent ; similarly for a vector transverse 

 to OP. 



Hence finally the accelerations resolved along the tangent and 

 normal to the tube are 



dv dr 

 v-j &) 2 r -y- + cop, 

 ds ds 



-f wv 4- ft&amp;gt; 2 + cor -7- 

 p ds, 



Now let the particle move in the tube under the action of 

 forces in the plane of the tube whose resolved parts along the 

 tangent and normal to the tube are S and N, and let R be the 

 pressure of the tube on the particle. Then the equations of motion 

 are 



m 



[dv dr 



s *s 



^2 /y y* 



m\- + 2a)V + co 2 p + cor y- I = N + R 



I Vo I 



