to the Longitudinal Expansion in Rods of Spring Steel. 41 



in which the magnitudes C and h, or the magnitudes C, C", 

 which are connected with them by the equations (8), were de- 

 termined. 



On the bracket which supported the rod submitted to experi- 

 ment, a point was fixed, the depth of which below the scale was 

 once for all determined on a large scale. In front of the elastic 

 rod a cathetometer was adjusted ; after the rod had been fixed 

 and made straight in the manner previously described, the depth 

 of a certain point of each reflecting surface below the above 

 point was measured with the cathetometer ; for after the height 

 of the point had been read off on the scale of the cathetometer, 

 its telescope was so arranged by turning around its vertical axis 

 that its vertical thread covered one of the plummets i, and then 

 it was lowered until the intersection of its threads appeared to 

 fall on the front edge of the corresponding reflecting surface. 

 The point upon which the telescope was then set, is the intersec- 

 tion of three planes, the equations of which have to be formed. 

 One of these planes is the reflecting surface ; it has the equa- 

 tion (if the mirror is the first) 



(?-aV + {V-V]P'+ (?-?)7 7 = 0. 

 A second plane is the vertical laid through the anterior edge of 

 the mirror ; let its equation be 



v -r ! = 0. 



The third plane is that which passes through the plummet i, 

 and the axis of rotation of the cathetometer ; if a lu and b 1 " are 

 the f and r) ordinates of this axis of rotation, the equation of 

 this plane is 



(f_flf) (b'" - V) - {<*) - V) {a"' -«') = 0. 



If Z' be the £ ordinate of the point upon which the telescope of 

 the cathetometer was set, we get from these three equations, 



c' = Z' + 



. J-V/-K 



(h-pSf*). 



7' 

 or approximately, 



r f— # / _ a'" — a' \ 



C ' =Z '- L 2C ((Y'-4')+;^rrf (X'-«')j. 



By a similar notation we may obtain in the same manner, 



c " =z "- yr Y " - J ") + w^v' v- a " ] ) ' 



c' and c" are calculated from these equations by taking an ap- 

 proximate value for C. 



