306 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



Let m be the mass per unit of length, and T the tension at any 

 point, 8 and N the tangential and normal components of the 

 bodily force per unit mass in the senses of u and v. 



For the motion of the element between s and s-}- ds resolve 

 parallel to the line from which < is measured. We have 



mds r- (u cos <f> v sin </>) = (S cos $ N sin <) mds 



vt 



-Tco&4>+(T+dT)cos 



Performing the differentiations, and replacing d<j)/ds by 1/p, we 

 may equate coefficients of sin < and cos </> on the two sides of the 

 equation, since the line from which < is measured is any line in 

 the plane of motion. We thus have 



-ieM*+t>l 



W 



These are the equations of motion of the element by resolution 

 parallel to the lines which are the tangent and normal at one end 

 of it at time t. 



It is to be noticed that the left-hand members of these equa- 

 tions might have been written down in accordance with the result 

 of Article 263. 



*270. Invariable form. Interesting cases of the motion of a chain 

 arise in which the shape of the curve formed by the chain is invariable, but 

 the chain moves along the curve. In discussing such cases it conduces to 

 clearness to imagine the chain to be enclosed in a rigid tube, of the shape in 

 question, and to move along the tube while the tube moves in its plane. The 

 velocity of any point of the tube is then determined as the velocity of a 

 point of a rigid body moving in two dimensions, and the velocity of any 

 element of the chain will be found by compounding a certain velocity w 

 relative to the tube with the velocity of any point of the tube. The direction 

 of w is that of the tangent to the tube at the point, and its magnitude is 

 variable from point to point in accordance with the kinematic conditions of 

 inextensibility. 



Taking now the special case of a uniform chain moving under gravity, 

 we show that the chain can move steadily in the form of a common catenary, 

 the curve retaining its position as well as its form. The velocity w is in this 

 case the velocity of an element of the chain, and, with the notation of 

 Article 269, we have 



