iM An hivestigalion ofjtexible Sulstances. 



2 111 



€ = —a'^i/ + t;t^<^"* -r — '2'^> ^° ^^^^ ^^^ ordinate will be 



— yx^ — —x^ + —2a;' — —azx'^+ —ux^ — ez^x' + 2a-2ix — 

 -J a'yx+~«-»x + -- a^zx + /; and this must vanish when 



x = 2a andx = 3q, or 0= -a'V ^0^+ - a^z — 2a'x; + 



i Q'u — 4a^u + Aa^u ~ahj + -^a^ + a'- + y = -^«'^— 



—(25+ ~a^z— -a^z-^ -a^u — 'iiahi-\-Qahi— 2"0'y+:^a'+ 



ii 7 7 1 



— a'z +/, and by subtraction Sj/— - a* + ■^2 + 7"=^, or 



36y — 2lrt« + 14s; + 2zi = 0. We have also, as before, from a 

 comparison of tlie evanescent ordinates of the second portion, 



~= -^(^*-^y, or Qz — 9a' - 36y, and by addition, — 12a« + 



8z + 2m = 0. On the other hand, considering the stave as a 



lever with its fulcrum at its lower end, we have -^ «' = 3cy 4- 



2az + an, and 2u = da'-Qy — Az = 12a''- 8x, and 6y = 42— 



Sa* = —a*— 2, whence 5z=— a* z= -^a% and the second 



-i 2 ^ 10 



hoop sustains one-fifth of the pressure; consequently y= 'l^^~~ 



1-1 'iX 



-7"^'= ^q'-^^} "^ ", "% ^i^d there remains for the force at the 

 bottom, Y^Q'. In a similar manner the calculation may be ex- 

 tended step by step to a greater number of points ; but as the 

 number increases, the ine(|uality of the distribution between the 

 neighbouring points must of course diminish, and if it became 

 infinite, the pressure on each would be simply as the depth. 



Augusts, 1315. A. B.C.D. 



P. S. — The " Radiatrix" mentioned as a " new curve," in the 

 last number of Dr. Thomson's Annals, p. 154, is the conchoidal 

 epitrochoid described, by Lahire and Maclaurin, and more par- 

 ticularly in Young's Natural Philosophy, vol. ii. p. 561. 



Your correspondent, who has endeavoured in your last num- 

 ber to remove some difliculties respecting the stiffness of timber, 

 seems to be mistaken where he asserts, that the stiffness is in- 

 versely as the square of the length only; as he will immediately 

 perceive, if he reflects that when the curvature is uniform, the 

 depression is as the square of the length, supposing the strain to 

 remain unaltered : but that from the properties of the lever the 



strain^ 



