432 Mr. J. H. Cotterill on the Further Application 



two equations which determine the law of variation required. 

 These results agree with those obtained by M. Lame and Pro- 

 fessor Rankine. 



2. A thin perfectly elastic spring is bent by the action of 

 forces acting on it ; it is required to find its form. 



Here the work done in the spring is very approximately 



J'M 2 

 n^j dsj where M is the bending moment at the point consi- 

 dered, E the modulus of elasticity, ds an element of arc -of the 

 spring-curve. Consequently fM 2 cfe must be a minimum; but 



it is shown in works on applied mechanics that Moo-, where p 



is the radius of curvature of the spring at the point considered ; 



Cds 

 and it follows therefore that 1 —^ must be a minimum. Adopt- 

 ing this principle, the problem is discussed in Jellet's f Calculus 

 of Variations/ where it is ascribed to D. Bernoulli. 



3. A solid smooth hard cylinder fits into a semicircular recess in 

 an indefinite solid of hard material, lined with a thin layer of soft 

 material; to find the law of variation of the pressure on the 

 cylinder. 



Here, if the pressure at an angular distance from the bot- 

 tom be p, the work done will be approximately proportional to 

 ^p^dOy which integral must consequently be a minimum, subject 



CL 

 to the condition that I 2 p . cos . r . dd shall be equal to half 



the load on the cylinder. Consequently \ ^-{-2kp cos 6}d0 

 must be a minimum, where k is an arbitrary constant. Therefore 

 p = — Jc cos 6 ; but if W be the load, 



W=2rC*vcos6dd 



' \ 2 p cos 6 

 Jo 



whence 

 and 



= -2r/cf 



cos*dd0=:-i7rrk', 



k= 2W^ 



7JT 



2W 



p = — . cos 0. 



an equation which gives the law of variation of the pressure, and 



