12 Mr. Tredgold o)i the Flexure 



large or small, provided the divisions be graduated with equal 

 accuracy in either the one or the other. 



The equation niight very easily have been made more general, 

 so as to have shown the degree of extension or compression at 

 any point in the circumference : but since my object is only 

 to show the utmost change that can have place from the 

 causes I have imdertaken to investigate, in order to prove 

 whether it be necessary to provide against them or not, I 

 have not gone further than suited my purpose. But the 

 equation ought to be general, if it were necessary to compute 

 the total quantity of extension in any quadrant of the circle. 



We may in the next instance proceed to determine the de- 

 pression at the points C and D. 



. Tlie extension of any point in the circumference being as 

 the stiess upon it when other circumstances do not vary, and 

 that stress being as the area of a segment multiplied into the 

 distance of its centre of gravity, it would involve us in a consi- 

 derably complicated equation to use the accurate expression 

 for this product. I shall therefore consider the area to be 

 proportional to yx, and the distance of the centre of gravity 

 from the strained point to vary as .r ; when .r is the abscissa 

 and y the ordinate. Now it may be proved that when the 

 depression is represented by a progression, the .rth term of 



that Jirojiression will be — ^ = -; r ; when t is the ex- 



tension at A. And the sum of the progression is, when 

 x=r, ...tr[ — 2-5 -\- 4>h. log. 2) = tr X -27258872. But t = 



^■^ ( Equation F) ; therefore, the depression at C or D is 



c, -017036795 /'g?-'^ ,r^s 



^= i^v 



The error from depression at D, or C, is directly as the 

 square of the radius; consequently small circles have, as I'e- 

 spects this error, the advantage over large ones. 



The expression will perhaps be better adapted for compari- 

 son, if the quantity of depression be exhibited in parts of the 

 circumference; while the accuracy of the comparison will not 

 be materially affected by the change. In this case, the de- 

 pression at D or C will be the ?<th part of the circumference 



1 1)7-4 wi ,,, 



when 71= — . (11) 



The qufiiitity ?«, which is necessary to be known before the 

 equations can be a])plied, I propose to determine by some 

 accurate expfrimeuls, as soon as I have collected suitable spe- 

 cimens, which, with examples, will Ibrm the concluding part of 

 this paper. 



If 



