to the Longitudinal Expansion in Rods of Spring Steel. 33 



be submitted to experiment I first consider to be made straight, 

 which can be effected by bringing supports near its ends, and 

 placing them so that a level suspended upon either of the halves 

 of the rod shows it to be true. The axis of the rod is then 

 parallel to the f axis. Starting from a variable point of the 

 axis of the rod, I suppose three axes at right angles to each 

 other, which I call x axis, y axis, z axis, which are firmly con- 

 nected with the molecules of the rod, and in the above mentioned 

 position of the latter are parallel to the axes f, tj, f. If the bar 

 undergoes a change of position and form, the former axes make 

 with the latter angles whose cosine I will indicate by 



«o> A)> 7o> 



"v &i> 7i> 



" 2 > Ay 7s; 

 so that the indices 0, 1, 2 respectively refer to the x axis, the 

 y axis, and the z axis. Further, let f, 77, f be the coordinates in 

 reference to the £, rj, f axes of the points from which the x 3 y, z 

 axes proceed. The sign of these three coordinates and of the 

 new cosine I will designate by the sign °, or ', or" placed above, 

 when they are to refer to the points of the axis of the rod A , A', 

 A", fig. 1, Plate I. 



If the supports by which the bar is made straight are removed, 

 it becomes curved in consequence of its own weight, of the weight 

 of the mirrors, their supports and their cross-rods. In order 

 not to make the considerations needlessly complicated, I shall 

 assume that this curvature can be considered as being produced 

 by equal weights which act at A' and A"; let this weight be G. 

 The magnitudes of the equal weights which are to be suspended 

 in B' and B", or in D' and D", I designate by P as before. Half of 

 A' A" I denote by s } and a quarter of the sums of B'B" and J)'D" 

 by /. In order to simplify the calculations somewhat, I assume that 



A'A° = A"A°, 

 and 



B'A' = D'A'=B"A" = D"A"; 



but observe that the final result also holds good even if these 

 equations are not fulfilled. The radius of the section of the bar 

 which is assumed to be circular shall be p. The coefficients 

 of elasticity I set, in accordance with designations which I have 

 used in earlier papers on elasticity, 



the relation of the lateral contraction to the longitudinal expan- 

 sion in the case in which the bar is expanded by a longitudinal 

 Phil. Mag. S. 4. Vol. 23. No. 151. Jan. 1862. D 



