THEORY OF THE REFLECTING EXTENSOMETER. 97 
3. Relation between extension and _ scale-reading.—Let one 
terminal of the apparatus be supposed fixed on the test specimen 
£ Fig. 1; and on the application of the stress S, the other terminal 
viz., the knife-edge /’, also touching the test piece, to move from 
F to F’, that is through the distance e, which is therefore the 
extension due to the stress. For simplicity also, let the knife- 
edges HG of the rotated prism, be supposed to move from the 
position of adjustment, that is from a line perpendicular to the 
test piece, and to take up the position F’G’ in consequence of the 
extension e. It is obvious that as the point H moves toward Lf", 
the point G will move in the are of a circle whose centre is Z. 
Then since ZG = EG'—the distance between the knife-edge @ and 
the point Z being, as indicated, invariable—we shall have, # denot- 
ing the length ZF of the test piece, the extension of which is to 
be determined, 7 the distance between the knife-edges, and w the 
angle of rotation of the line joining them, 
E+e=lsin w+ v(H? +1? —1? cos? w) 
which by expansion and transposition gives 
: sin w — a 
22 8k 
The final term can never be greater than one hundred-thousandth, 
hence it may be at once rejected, and the expression thus reduced 
accurately denotes the relation between the extension e and the 
_‘Totation of the prism. 
e=Usin w (1+ sin® w+ ete.)...... (1) 
If the knife-edge occupy the position /'”, Fig. 1, instead of F’, 
we may regard F'F’” as negative, as also the rotation angle w: in 
other words, if the point / move toward /’” it is a minus extension, 
°r a compression instead of an elongation. In such a case the 
formula still holds good, e and sin » are negative, and the term in 
Y/ 2E is numerically subtractive from unity, instead of additive as 
in the preceding case. The formula may therefore be regarded 
_88 quite general, 
: If the seale 7U be supposed parallel to the test piece,’ that is, 
ifit be parallel to FF’ Fig. 1, and if the distance between the scale 
1 The more general case is stated later in § 11. 
G—Aug. 4, 1897, = ee eee 
