the fixed Supports qfjtexible Szilstances. 143 



expression when x = of : hence — ya^— — a^ + J = 0, and the 



inclination in the middle is — ?/«' rr""^ + t:::^-^ ^ '/a* = 



-(vy— — a\ which must be the value of the inclination in the 

 second expression when x—.a; so that - ay — -:;- (r + ^za'x — 



a'z-{-c= }-a''y-^a*, and c= - -^ a'y + ~a* + -a-z: 

 when, therefore, x — a in the second expression for the ordinate, 



d=0 — —a=z +d, and d= ra^z: when also x=2a, -a'v—' 



6 o ^ 3 ^f 



— -a' + — a'« — 2a^x— ra-7/4- -~c.^ + a^z rc'a = = 



lis 

 °'y~ -a5+ --ft^;j, and s= -a'- — 6?/. Again, the pressure 



on the third point will be 2a^— y —z, the three hoops having to 



sustain the pressure 2a^ : this third pressure must also exceed 



the force y by -^^S i'^ order that it may hold in equilibrium the 



whole pressure 2a-, acting at the distance ~a from the middle 



point, considered as the fulcrum of a lever, so that y -\ a* = 



2a^—y-z, and z= —a^—2y; whence, subtracting the 



former value of z, we have iy— --fl- = 0, y= -^0% x = 

 5 j7 



— a*, and the third pressure kt^'' 



F. If a stave be supported at the ends, and by two intermo- 

 diate hoops at equal distances, the respective pressures will be 



1 9 V4 11 



iT' 45"' 46"' ^"^ 45- 



For the first and second portions of the staves, the values of 

 the inclinatfons and ordinates are determined from those of y 

 and z, as in, the last proposition: for the third, the inclination 



will be — yr* x* + — «x* — a«x + --■ nx'- •- 2aux + e, 



whichf at the origin of this portion, where x = 2fl, becomes 



2«*y — — a* + 2a^ — 2a'« + 2a^—\aHi + e, and this must be 



e^Ml to tbe fiaat inclination in the second part, or to 2a'y — 



