I' -11 MISCELLANEOUS THEOREMS. 



t - for r; then 



therefore 



therefore ^ = 



Therefore, by integration, 



(n + 2) 6 + const. = sin" 1 (Xu"**) ; 



therefore -^ = 8m {( n + 2 ) + constant}. 



If we fix the position of the initial line so that r may 

 its least value where = 0, we shall determine <tant, 



and obtain 



r** = X sec (n + 2) 6 ; 



or denoting by a the value of r when = 0, 



111. Suppose a flexible string to be in equilibrium under 

 the action of a central force. Iniairiin- any ]><>rti<m of tin- 

 string to become ritrid : then it is kept in crjuilibrium by the 

 tensions at the ends and the resultant of tin- action of the 

 central force on the elements of the string ; this resultant will 

 be some single force acting through the centre of force. Tim* 



portion of the string may be considered to be in 

 librium under the action of three forces ; and these forces 

 will therefore meet at a point. Hence we obtain the follow- 

 ing theorem : The ro ' ntral force on any 

 portion of itie string is directed along the straight fine which, 

 joins the centre of force with the point of intersection of the 

 tangents at the ends of the port 



