240 STRING ON SMOOTH SURi 



From (5) we obtain ultim: 



dT 



-y- as pR mg sin 0, 



T 



and from (6) R = + mg cos 6 ; 



fore nr- = -- + g (/* cos sin 0), 



P 

 jn 



therefore p , p, T= mg (ji cos sin 0) p, 



dT 

 that is -JQ p T= mg (/t cos - sin 0) p. 



Thus we have a differential equation for finding T. and we 

 may proceed in the ordinary way to obtain the soli it inn. 

 .Multiply both sides of the last equation by e~ fi - t thus 



-77; ( Te~p) = mqe~* 9 (u, cos sin 0) p ; 

 au 



therefore Ter** \ mge~^ (ji cos sin 0) pdO. 



nee when p is known in terms of we shall only have 

 to integrate a known function of in order to obtain the value 

 of Tin terms of 0. 



To form the equations of /i/t'/>rium of a ,s//7//y 

 stretched over a smooth surface and acted on by any forces. 



Let * be thedength of the string measured from some i 

 point D to the point P; a;, y, z the co-ordinates of P; '< 

 length of the element of the string between Paivl an adj 

 point <2; m&s the mass of the element; J&s the resistan 

 the surface on this clement, the direction of which will Le 

 ultimately the normal to the surface at P; let a, , 7 1" the, 



s which the normal at P makes with the axes; A 

 YmSs, ZmSs the forces parallel to s acting on tin; 



element, exclusive of the resistance EBs. Hence, in the 



