44 



OF ^PRESSURE AND EQUILIBRIUM.' 



crensed, where x is greater than 78G thou- 

 sandths of the radiusj so as to he every where 

 inversely as x— x'. 



■The equilibrium requiring that xxy should be at least 

 equal t& l — . j/, where the thickness is equable, if the thick- 

 ness at any part be to the mean thickness of the supeiin- 

 cumbeiU pardon as 7 to r, the equilibrium may be pre- 



. _ r(l~y) 

 served while qxxy is equal to r.(i — y), or 9— . 



Now the whole weight being p, the mean thickness r is 



— ^ — . c being the circumference of the circle of which 



the radius is 1, hence q~ — ^— ani the increment ;/• is 



CXXTf 



expressed by the increment of the circular circumference r', 

 multiplied by cxq ; therefore pzzcqxs; but :=-, andf— 

 ^3±=IL, and i =-i-=(h.l.iy. since. (i)=^ 



y ^yy p ^yy \ y ' . }y' v 



■ ■ . . ,. '.MM 



■Til .T X V V ' — •TV/ , 



_^, which divided by - is -=■ - ■ „,; ' ■ i but xjr+i/y 



1/a y X y xyy 



=:i,thereforexiz:—jy)andtheexpres§ion becomes —^^^^^j — 



— _ — ; consequently h.l.p— h.l.- ±0, and p—- , or — ; 

 xyy ' ^ ■'• y y y 



then V^~ — ^^~- Therefore the thickness must be 



cxxy cxyy 



inversely as xyy, or as x—x' ; but if we estimate the 



b 



thickness in a vertical cUrection, it becomes 



cxy 



If we 

 wish to give a certain degree of stability to the domej we 



must make q 



dr(i — y) <ip , . . 

 zz — ■ ^=; — , a bemg some constant 



xxy 



cxxy 



d.i 



multiplier greater than unity ; then L— -; , and h. 1. ;>— 



p ^y 



dCh.l.-)±aandp=:4 (-) , therefore if d=:i+e, 9= 



_*£_.( 5 V. And the constant quantities may be so deter- 



cxyi/\y/ 



mined as to correspond to the weight at any particular part, 



whether the centre of the dome be closed nr open, b being 



I - J p, and the slability will be secure if all the lower parts 



bFmadeof the thickness 9 •, forthelower parts can never force 

 up the higher, however they may be loaded, since their 

 pressure will always be resisted by the collateral parts of the 

 course. Py making 9 a miniraum, we find that the thick- 

 ness is least where i^iv'f—--), or, if rfi=i,wheni= 

 .J78, if d=:i.5, when 1=. 408, if d=2, when 1=0, so 



that in this case the dome must become gradually thipkef, 

 from the vertex. In practice, considering the friction of 

 the materials, it will be amply s\4fficientto make dzizi.i, or 

 even J, and in this case 9 is least when r:::. 5, consequently 

 the thickness of the lower parts must begin to be augmented- 

 at the distance of 30° from the vertex, at 60° it must be- 

 come 3.28 times as great, and if the dome be continued 

 much lower, it will be proper to employ a chain to confine 

 it, since at 80° from the vertex a thickness 50 times as great 

 as at 30° would be required for the equiiibritjm. 



310. Theorem. When a weight is sup- 

 ported by a bar resting on two fulcrums, the- 

 pressure ou each is inversely as its distance 

 from the weight. 



>or, by the property of the lever, it is to the wholeweight,. 

 as the distaiice of the weight from the other fulcrum to the 

 whole length of the lever. 



."Jll. Theorem. The strain on a uiven 

 point of a bar, supported at the ends, from a 

 weight phiced on it, is proportional to the 

 rectangle of the segments into whicb the 

 point divides the bar. ~ 



For, considering A as the fulcrum 

 of the lever, the weight B produces \^ Bq 



at C a pressure^::- 



AB 

 AC" 



C 



and the 



strain at B is as the length of the lever by whieh it is ap- 



AB BC 



plied, or as — 7^:; — j it is therefore equal to the strain pro- 



duced by the weight applied at the end of.a, levei of whicb 



AB.BC 

 the length is — 77;— • 

 At/ 



312. Theorem. The strain produced by 

 the weight of an equable bar at any pointof 

 its length is equal to the strain produced by 

 half the weight of one segment acting at the 

 end of a lever equal to the other segment. 



_ o C 



The strain produced-at any point A. j^ 

 by a weight B on either side is equal ^ 

 to the strain of the same weight act- 

 BC.AD 



li 



ing at the distance - 



DC 



; therefore the strain produced 



by the portion AC of the bar, of which the weight may be 

 imagined to be collected into its centre of gravity, is as 



AC. — . — ; and for the same reason the weight of AD 

 2 DC 



AD AC 

 produces a strain AD. =5^ ;. therefore, both togethwr 



