involved in the Construction of Artillery. 195 



115. Poisson has shown, and Cagniard de la Tour has experimentally veri- 

 fied, a singular relation between linear and cubic extension or compression ; that 



if i = -^ be the proportional elongation of a bar whose length is L, and whose 



elongation for unit of length is I, and a the diminution of cross section, which 

 the original cross section A sustains as due to i, then 



so that the reduction of cross section is equal to half the elongation. From 

 which it follows that the total volume of the bar augments by a fraction = ^i, 

 although its cross section diminishes. The original volume of the bar = AL 

 becomes 



(A - a) X (L + l) = AL + Al - aL - al ; 



and, neglecting the product al, which is small with respect to the others, the 

 total volume becomes = Al — aL, and the increment due to i, 



But if the bar be exposed to compression in all three axes, L, B, D, simul- 

 taneously by forces perpendicular to its faces (assumed a square prism), and 

 the pressure on L, be that as before on A; L, B, and D being respectively the 

 length, breadth, and thickness of the bar, then the compression of the bar in L 

 shall be only half the former, and the volume of the whole bar becomes 



LBD (1 - yy = LBD - |-i LBD 

 the decrement due to i being 



^iLBD. (12) 



Neglecting the functions of the small firaction of i as before, the cubic contraction 

 or expansion of the bar in this case is measured by #i. 



116. Now, when a bar of a homogeneous metal is heated, and expands in all 

 directions alike, forces analogous to those above, but acting in opposite direc- 

 tions from within the mass, may be considered as applied perpendicular to the 

 faces of the prism ; but the increase or decrease of volume is altogether different, 

 being twice the former, f i = 3i (i being in this case the lineal expansion or 



