85 



X r = TT z 1 -{- 2 TT/ y -f- TT t/ which is seen to be identical with the expression 



tn Eq. 4, for the area of the corrective circle. Hence, calling the area of the correc- 

 tive circle C, we have for Case II 



X 



andX=C-f-r 



(8) 



It will be seen from the above equation that when z cos a =/, the forward rota- 

 tion of the wheel just balances the backward rotation, and the result is for a read- 

 ing. This occurs when the tracing point moves on the arc of the corrective circle 

 and the area passed around is X = C. When zcos0 is less than/, then the back- 

 ward rotation preponderates over the forward rotation, and the result is a minus 

 reading ; this is Case III. 



Case III. When the anchor point is inside the figure to be measured, and the 

 rotation of the wheel is backward with reference to the figures on its circumfer- 

 ence. In this case r is negative, and instead of X = C + r, we haveX = C r. 

 Hence, we have for Case I, X = r, Case II, X = C -f- r, Case III, X = C r. 



Suppose it to be required to make the instrument record 100 for every square 

 inch passed around by the tracing point. As the wheel is divided on its circumfer- 

 ence into 100 divisions, and by the vernier can read tenths of these, then ^ of a rev- 

 olution will give a reading of 100, make X = l sq. in., and n = then by Eq. 3,t/r^ 



= 1, and y = : the length of the arm varies inversely as the reading, so for any 



c 



other reading we may obtain the length of y by simple proportion. Suppose it is 

 required to read v for every sq. in. passed around, we have 



159-i. from which , = **. 

 v 10_ cv ' 



c 



The range of the arm renders it impossible to set it so that for one sq. in. of area 

 it shall read more than 250, or less than 50, so the value of v must lie somewhere 

 between these limits. Having determined the value of y for any particular scale, 

 the value of C may be found by substituting in Eq. 4 ; the values of z and /being 

 measured on the instrument. 



A New Prismatic Stadia. 



Devised by Prof. Robert H. Richards, 



Massachusetts Institute of Technology. 



In this prismatic stadia there iy placed in front of the objective a prism or wedge 

 of glass which half covers it. 



If we hold up such a prism with a narrow angle, say 1 to 2, and compare the 

 transmitted image with the image seen above or below the prism, the former will be 

 found to be thrown to one side by an amount varying with the angle of the prism. 

 Speaking of the two rays as the direct ray and the bent ray, we may say that when the 

 bisecting plane of the prism is at right angles to the line of sight, the angle between the 

 direct ray and the bent ray will be constant for any given prism. 



.300. FT. 



400 FT. 



Fig. 1. 



If now we place a prism in such a position that it half covers the objective of a 

 telescope, as seen in Fig. 1, we shall obtain, upon looking through it, two images of 

 every object seen one image by the direct ray, which comes through the uncovered 

 half of the objective, the other by the bent ray, which com^s through the prism. The 

 angle of divergence of these two rays will be constant and unalterable, whether the 

 telescope is directed to a near or a distant object. 



