merwin: dispersion of carborundum 445 



7. Balance wheel of a watch. The well-known expression 



1 IS t 

 n = k~ \~j for the case of a weightless spring acting on a rigid 



T • 1 IS 1 



mass of moment of inertia I, may also be written n = — \— — 



r " m 2 ir 



in which r is the radius of gyration. This agrees with (8). 



Differentiating, n = | St — r. For the hair spring (pure bend- 



» 



ing), (9) becomes St = a +3 7, hence for the balance wheel 

 h = f a + 2 7 — t. The condition for compensation is therefore 

 that the radius of gyration shall be constrained to vary with 

 temperature at the rate r = \ (a 4- 3 7). 



8. Two springs coupled together. Let spring 1, strained purely 

 by twisting, be coupled to spring 2, which is strained purely by 

 bending; and let both springs be made of the same material. 

 Then \pi and 2 are constants, so that Ci = — 1, C 2 = 0, Ci 2 = 0; 

 .*. K = — 77, and (12) becomes n = | [ (1 - 77) a 4- r? (3 + 7]. When 

 the twisted spring is stiffer than the bent spring, the /3 term 

 predominates, and vice-versa. For springs equally stiff, tj = %, 



ol 8 *y 

 .". n = - + - + -. For two helical springs, Ci = C 2 = — 1, so 

 4 4 2 



that n = f (j8 -f- 7) ; for two leaf springs or elliptical springs, 



Ci = C 2 = 0, so that n ------ § (a 4- 7). These results, including the 



above formula for n in ';erms of a, /?, 7, and rj, and subject to the 



limitations stated, are true independently of the shapes and 



mass distributions of the two bodies, as well as of their absolute 



sizes, masses, and elastic moduli. 



CRYSTAL OPTICS. — Dispersion and other optical properties 

 of carborundum. H. E. Merwin, Geophysical Laboratory. 



Along the edge of a very pale green carborundum crystal 

 two areas, each about 0.5 mm. square, were found which were 

 suitable for measuring the ordinary refractive index. The 

 angles, measured from the base, were 32° 13' and 34° 15'. The 

 refractive index for the extraordinary ray was found by measure- 

 ments on the interference figure 1 of a colorless plate 0.675 mm. 

 thick and about 3 mm. square. 



1 See Bauer. M. Neues Jahrb., 2: -49. 1882-83; also Journ. Wash. Acad. Sci., 

 4: 533 and 538. 1914. 



