

OATEN AH!*. 



a section made by a plane perpendicular to the tanp-nt. the 

 mutual actin of the portions on opposite sides of this plane 

 must be in t: ion of the tangent, or else equilibrium 



would not hold, since the string is perfectly flexible 



184. A heavy string of uniform density and thickness is 

 suspended from two given points ; required to find the equa- 

 tion to the curve in which the strimj \ang$ ////< 

 librium. 



Let A, B, be the fixed points to which the ends 

 ied; the string will rest in a 

 -iiiir through A 



anl //. because there is no reason 

 why it should deviate to one side 

 rather than the other of this vcr- 

 tieal plane. Let ACB be the form 

 it assumes, C being the lowest 

 point; take this as the origin of 

 co-ordinates; let P be any point in 

 the curve; CM, which is vertical, 

 = ?/ ; MP, which is horizontal, = x ; 



The equilibrium of any portion CP will not be disturbed 

 if we suppose it to become rigid. Let c and / be the lengths 

 of portions of the string of which the weights r.jiial the 

 tensions at C and /'. Then CP is a rigid body acted on 

 by three forces which are proportional to c, $, and /. and act 

 respectively, horizontally, vertically, and along the tangent 

 at P. Draw PT the tangent at P meeting the axis of y 

 in '/'; then the forces holding CP in equilibrium have 

 direc: allel to the sides of the triauirl.- /'!//'. and 



therefore bear the same proportion one to another that 

 sides do (see Art. 19) ; therefore 





tension at lowest point 

 weight of the portion CP 1 



or 



therefore 

 therefore 



- and 



c 



tu 



.(i); 



