ELISHA MITCHELL SCIENTIFIC SOCIETY. 7 



For a suddenly applied load P, causing a change of length 

 k^ and maximum stress jr, the work of the load is P k\ and 

 since the ?Ar^s?> gradually increases from o to s, the work 

 of deformation, by (2), is \- s k\ and since these are equal 

 s = 2 P, or the maximum stress is double the load; hence 

 by (I) the change of length is double that fo:' a gradually 

 applied load. After a series of oscillations this change 

 ultimately becomes that due to a gradually applied load 

 and s reduces to P. As, in what follows, we are only con- 

 cerned with the ultimate or statical stress, we shall always 

 compute the zvork of deformation of a bar, as for a gradu- 

 ally applied load, by the formula (2), 



1 I a 



— c s- = s- . . . . (3). 



2 2 e w 



3. Superfluous bars. When the figure of a truss has 

 more lines than are strictly necessary to define its form; i. 

 c, to fix its apices when the length of sides are given in 

 order, the extra sides are said to be ' ^ supetflitous.^^ 



The relation between the least number of sides w, or 

 the number of '' necessary " bars in a truss and the number 

 of joints or apices ?/, for strictly defining the form of a fig- 

 ure of invariable form, is easily arrived at. 



Thus for plane figures (which we shall alone consider in 

 this article) assume the position of one side, thus fixing 

 the two apices at its ends. From these apices we can fix 

 another with two new sides, then another with two new 

 sides from two apices previously fixed, and so on; there- 

 fore to each of the (n — 2) joints other than the first two 

 corresponds two sides, so that the total number of neces- 

 sary sides ;;/ = 2 (n — 2) ^1=2 n — 3. If the number 

 of sides exceeds (2 n — 3), the extra number are "super- 

 fluous " to strictly define the form. A less number will 

 give a figure that can change its shape without changing 

 the lengrths of its sides. 



