310 Mr Sang's Analysis of the Vibration of Wires. 



wire round OG as an axis of torsion. These two forces, esti- 

 mated as acting on the extremity of the wire, may thus be re- 

 presented by e@z . dz . IK^ ds and e®z . dz , IK . IL, . ds. 

 Hence the aggregates of all such forces occasioned by elements 

 of the section EF, are 



Force parallel to GO, e . ®z . dz . 'Z . IK^ . ds, 



to HO, e ®z . dz . i: . IK . IL , ds ; 



and thus the whole force urging the extremity of the wire to 

 return in the direction GO is e . / . Gz . dz . •£ . IK^ . ds ; 

 while the entire disturbing force acting in the direction H O is 

 e I Qz . dz . iL .IK .ILi . ds. 



Or, since I K = a: cos (p 4- «/ sin (p ; I L = — a? sin <p + ?/ cos (p, 

 these forces are, 



In the direction GO, 



e j Qz . dz . z {pc cos <p + 2/ sin <p)* d s. 



In the direction HO, 



e / &z . dz . 2 (^ cos <p 4- 2/ sin (p) ( — xs\n(p -\- y cos <p) d s. 



If for (p we substitute <p +-5-, we obtain for the value of the 



rectifying force pertaining to a direction at right angles to the 

 former, 



e l&z . dz . "E ( — .r sin (p -I- 2/ cos (p)* d s, 



which, added to the former, gives the constant amount 



e / Qzdz . 'Z {x'^ + y^) d s ., 



and thus we arrive at this very remarkable conclusion, that the 

 sum of the rectifying forces pertaining to two directions at right 

 angles to each other is constant for the same wire. Whenever, 

 then, the stiffness in one direction is a minimum, that in the 

 perpendicular direction is a maximum. 



Let us inquire, indeed, for the directions of greatest and least 

 rigidity. Differentiating the expression for the redressing force, 

 41 being regarded as the primary variable ; supposing that * is 



