PEIRCE. — OSCILLATIONS OF SWINGING BODIES. 87 



Here again the apparent moment of inertia is nearly constant but 

 the damping coefficient increases rapidly as the field about the magnet 

 becomes more intense. 



Many kinds of physical measurements concern themselves with the 

 behavior of oscillating systems, and it is often necessary to determine 

 what the apparent moment of inertia of a system is if the motion is in 

 air, and what the exact value of the damping coefficient is at any time. 

 If this is not constant throughout the whole motion, — as it should be 

 if it follows the Gaussian law, which assumes the existence of a fixed 

 logarithmic decrement, — it is necessary to find out how it varies with 

 period and amplitude. If one uses a d'Arsonval galvanometer to meas- 

 ure changes of magnetic flux in a large mass of iron, and for reasons of 

 sensitiveness at some point of a hysteresis diagram needs to introduce 

 extra resistance into the circuit or to remove some which is there already, 

 one cannot compute the effect of the change unless one knows, not the 

 real, but the apparent, resistance of the galvanometer coil, and this de- 

 pends upon the "constants " of the motion which must be determined 

 with some care ; it would not be difficult to show that such deviations 

 from the Gaussian law as one frequently encounters in practice need 

 to be carefully taken into account in accurate work. The fact that the 

 swinging system comes to rest in a comparatively short time suggests 

 that the law may not be exactly followed at any part of the motion. 



If, then, a swinging magnet or galvanometer coil is exposed to a 

 relatively strong air damping, we must expect that unless the amplitude 

 is very small there will be an appreciable departure from the Gaussian 

 law. If the system be turned out of the position of equilibrium through 

 a considerable angle and then released, it moves rapidly through this 

 position and out on the other side to a new elongation corresponding to 

 a displacement much smaller than the one from which it started ; and 

 this modifies profoundly the theories of some ballistic instruments, 

 but after this the subsequent decrease of the amplitude takes place 

 slowly and regularly, accompanied usually by a slowly decreasing 

 logarithmic decrement. For any small number of swings after the 

 first few, however, the constancy of the logarithmic decrement can 

 often be assumed with sufficient accuracy for ordinary purposes. 



The moment of inertia of the swinging system cannot as a rule 

 be computed with any fair approximation from a knowledge of the 

 masses and the geometrical dimensions of the bodies of which the 

 system seems to be made up, for a comparatively large mass of air 

 accompanies the visible system and materially increases the inertia. 



