Mr Sang's Analysis of the Vibration of Wires, 311 



its peculiar value in the case of minimum or maximum rigidity ; 

 and equating the differential to zero, we obtain 



/© z . dz . s (^ cos <i> + ^ sin *) ( — a: sin <fr + ^ cos <i>) = ; 



but this is just the value of tlie disturbing force belonging to 

 the direction <i», so that, when the wire is deflected either in the 

 direction of least, or in that of greatest, rigidity, it returns di- 

 rectly to the position of rest, and its extremity oscillates in a 

 straight line. This property aids us in detecting the directions 

 of greatest and least flexibility. 



Extracting, from the above equation, the value of % we ob- 

 tain 



9, 1 . Qzdz'L . xyds 

 / . Qzdz-L^x^ — y^)ds 



Now this expression never can become imaginary ; every elastic 



wire, therefore, has one direction of greatest, and another of 



least flexibility, these two being at right angles with each other, 



«r 3 '^ 



since <t, <i> 4- 5 , o -f ^'j <l> H-, -5-5 &c., all the roots of the above 



equation pertain only to two such lines. In that case, indeed, 



when both I . q z . dz .^ . xyds and / Qzdz s (a?* — y^)ds 



are zero, the value of would become indeterminate, and the 

 wire would become equally flexible in all directions. 



In order that X and Y may be the directions of greatest and 



least flexibility, tan2<p must be zero, whence /© zdz .^ccyds =0. 



Let us then suppose this to be the case, and, for abbreviation's 



sake, put f&z . dzi: a^ ds = A; / ®z . dzi:y*ds = B; A 



being supposed less than B; the redressing force pertaining to the 

 direction <p will then be e(Acos(p^-\- B sin <?>'), and the disturb- 

 ing force pertaining to the same direction e(B — A) sin <p cos (p. 

 These forces decomposed into others parallel to the axes X 

 and Y, and collected, become 



Force in the direction X, — A . e cos (p ; 

 Force in the direction Y, — B . e sin <p. 



