250 Mr Sang on the Construction of Circular Towers. 



CDEF, it is requisite that the joints 

 of the stones be perpendicular to that 

 curve : let GHIK represent one of 

 these stones. The upper surface GH 

 is subjected to a pressure in the di- 

 rection of the tangent to the curve at 

 the point D ; the under surface KI to 

 a pressure upwards in the direction 

 of the curve at the point E. Put p 

 for the pressure 2A,D\ p-\ dp for that 

 at E ; put also i for the inclination 

 of GH, i+di for that of KI to the 

 horizon. 



The pressure on GH may be de- 

 composed into two, one vertically 

 downwards, represented by p cos i, 

 and one horizontally towards the concavity of the curve p 

 sin i. Again, the pressure on KI is decomposed into 



(p + dp) cos {i-\-di)=p cos i-\-dp cos i—pdi sin i 

 vertically upwards, and 



(jt? + dp) sin (i + di) ~p sin / + dp sin i-\-p di cos i 

 horizontally from the concavity. Let dw be the weight of the 

 stone ; then we have, as the sums of all the actions hitherto 

 considered, 



dw — dp cos i+pdi sin i upwards, 

 dp sin i+pdi cos i horizontally 



from the concavity. 



It thus appears that there cannot be an equilibrium from 

 these three sources, unless i and di be each zero ; that is, un- 

 less the wall be vertical throughout. In all cases, either of 

 inclined or of curved walls, there must be some means of sup- 

 plying a pressure from the convex side. 



There are three ways in which this pressure may be ob- 

 tained : two of them relating to a cylindrical wall, such as we 

 have been considering ; and one to a wall curved in two di- 

 rections. 



In the first place, we may conceive the end^GK of the stone 

 to be dressed vertically, and to abut against some firm ob- 



