GEODETICAL STANDARDS WITH THE ENGLISH STANDARD YARD. 
169 
of some particular lines, to the amount of one or sometimes two micrometer divisions. 
The only way of eliminating personal error is by the employment of as many expert 
observers as can be commanded. 
Flexure . — It is of the utmost importance that a bar be supported invariably at the 
same points ; if these points be altered, then (unless the divided surfaces be in the neutral 
axis) the length of the bar undergoes a change. The proper positions for the supporting 
rollers of a bar have been investigated by Mr. Airy in the Memoirs of the Royal Astro- 
nomical Society. In order to test the theory of flexure of bars (considered as elastic 
rods) by actual experiment,, a large number of observations as to changes of length cor- 
responding to change of supports were made on three iron bars specially prepared for 
the purpose. Each bar was 40 inches long by an inch square. It will suffice here to 
explain generally the process and give the results for one of the bars. If a bar be sup- 
ported by its extremities, it is clear that the whole of its upper surface will be compressed, 
while the lower surface will be correspondingly extended ; therefore the length of the 
bar as measured by the distance between two dots at the extremities of the upper surface 
will be less than if the neutral axis were straight ; in fact, if i be the inclination of the 
bar at either extremity to the horizon, and k the depth of the bar, its curvature will 
cause the dots to approach each other by the quantity i/c. This effect of curvature will 
be greatly exaggerated if the dots be engraved, not on the surface of the bar, but on the 
tops of bits of strong wire inserted (in a vertical position) into the bar at its extremities ; 
if h be the length of either wire, the approach of the dots to one another by the curva- 
ture of the bar will be Accordingly four such perpendiculars were erected on 
the upper surface of the bar, one at either extremity, and one 10 inches from either 
extremity. A box fitted with rollers was prepared to receive the bar; each roller, 
mounted in a frame, could be fixed in any required position, and could at the same time 
be moved in a vertical direction up or down by means of a slow-motion screw outside 
the box. Now suppose four rollers so fixed in the box that one is under each extremity 
and two more at 2 inches right and left of the centre of the bar. Suppose also four 
microscopes adjusted over the four dots, their axes vertical, and their outer foci ranging 
in a straight horizontal line ; by the working of the slow-motion screw (which it is to be 
particularly observed does not require the opening of the box or the touching of the bar 
with the hand) the centre rollers can be withdrawn, that is lowered until they cease to 
have contact with the bar, and then the bar resting on the extreme rollers can be 
adjusted to focus under the microscopes. The microscopes are then read. The extreme 
rollers are then lowered, and the bar comes in contact with the rollers near its centre, 
which are then raised until the dots are in focus. The microscopes are now read a 
second time ; and by these readings, compared with the former, are obtained the changes 
of length of the whole bar, and of its subdivisions resulting from the alteration in the 
positions of the supports. 
The supporting-rollers were placed at different times in the following positions : — 
(1) at 20 inches right and left of the centre ; when in this position the supports are 
mdccclxvii. 2 A 
