OF ARTS AND SCIENCES : FEBRUARY 14, 1865. 405 



Wc 



^' = W+|c' " (^) 



in which 



W = the breaking weight by formula (1). 



W' ^ the brealving weight for the combined effects of flexure and 

 crushing. 



c = the weight which would crush the pillar without flexure, which 

 Hodgkinson finds for Low Moor iron, No. 3, is 109,801 

 pounds per square inch of section ; consequently, 



c = 109,801 X 1 71 (D-— d-). 

 Substituting the value of c in (A), and reducing, we have 



_ 86238 (D^ - d"-) W ^ 



W+ 64678 (D^ — (/•-)• ^"^ 



* In computing the breaking weights of a series of pillars, in order to preserve 

 continuity in the results obtained by formulas (1) and (2), it is necessary at the 

 point where one formula is substituted for the other that we should have 



W = W ; 

 consequently, at this point 



86238 (D2 — of-) = "W + 64678 {I)- — d~). 

 Substituting the value of W in (1), and reducing, we have 



/D3.55 (P-io\^ 



I = 2.456 ( j,._,. yr 



For simplicity, confining ourselves to solid pillars, in which d = 0, we have, by 

 substituting this value in the last equation, 



I = 2.456 D°-^i 



/, in this formula, is the length, in feet, of a solid pillar of the diameter D, in 

 inches, for which formulas (1) and (2) give the same breaking weight. 



"When D = 0.1 .-.1= 0.3022 feet = 3.6264 inches = 36.264 diameters. 

 " D= 0.821 .-.;= 2.0525 " = 24.6300 " =30.000 " 



; " I)= 5.0 .-.;= 10.6240 " =127.4880 " =25.498 " 



" D = 10.0 .;. i = 19.9630 " =239.5560 " =23.956 " 



The substitution of formula (2) for formula (l),for pillars of lengths less than 30 

 diameters, is for the purpose of diminishing the computed breaking weights to con- 

 form to the diminished strength of such pillars, on account of the crushing effects of 

 the weights ; but in all solid pillars of more than 0.821 inch in diameter there are 

 certain lengths in which the substitution of formula (2) for (1) lias the contrary 

 effect; thus, in a pillar 5 inches in diameter, for all lengths between 25.498 diame- 

 ters and 30 diameters, the breaking weights computed by formula (2) exceed those 

 VOL. VI. 44 



