GRAYj ON A DOUBLY-REFLECTING PRISM. 277 



Now to apply tlie formulse. In Mr. Wenliam's prism p 

 will be equal to the semi-diameter of the back lens of the 

 largest object-glass with which the prism is to be used. (If 

 made to suit the largest, it will suit the smallest about equally- 

 well; but the converse of this does not hold.) The back 

 lens of the 3-in. and 2-in. being about -^ in. in diameter, 

 we may take for p '25. The value of d depends on the 

 length of the bodies (measured from the point where the axis 

 of the deflected pencil intersects the axis of the principal 

 body) and the distance of the eyes apart. If the length be 

 10 inches and the distance 2^ inches, then will d equal 

 twice the angle whose tangent is -125 = 2 (7° 80 = 14" 16'. 



Hence the angles of the prism are — 



A= 45° -f 7° 8'= 52° 8' 

 B = 135 -t- 7° 8=142° 8' 

 C = 45° 0' 



D = 135 -14°16'=120°44' 



Sum . . 360° 0'; 



And the sides — 



AB = -25 X sec. 52° 8= -25 x 1-6291 = '4073 

 BC =-2500 



If the other sides are wanted, they may be computed as 

 follows : 



In the A EBA (fig. 1) the angles are known, and also the 

 side BA. 



Hence EB=^5-^^g^= 1-3048, 



sm BEA 



and EA=^BiJ§#^= 1-0145. 



sm BEA 



Again, in the a ECD the angles are known, and the side 

 EC=EB-t-BC=l-3048+ -2500=1-5548. 



TT T7T-V EC sin ECD , ^^.m 



Hence ED = — . — .^^.^^-^ — =1-2791, 



sm EDu 



J ^T^ ED sin CED . , ,kc2 



and CD=— ^ — =^^7^=- — ■= 4458. 



sm ECD 



Therefore AD=ED-EA = l-2791-l-0145 = -2646. 



Or they may be otherwise computed thus : — If A and C 

 (fig. 1) be joined, the quadrilateral ABCD will be divided into 

 two triangles, ABC and ACD. In the former two sides and the 

 included angle are known, Avhence the angles at A and C, 



