24 MEASUREMENT OF OSCILLATIONS. 



7T T 



If the angle a were null, this equation would give ^ = = ; we 

 may put / x = +_#,__ 6 being a very small quantity, and, neglecting 

 terms of the second order in a and in 0, we obtain 



and therefore 



It will be seen that the time of oscillation is no longer divided into 

 two equal parts by the period of passing through zero, the time of 

 the descending half oscillation being increased by as much as the 

 ascending half oscillation is diminished for the time of the entire 

 oscillation has not changed. 



These various conditions are not those which we most frequently 

 meet with. Experiment shows indeed that the resistance of the 

 medium acts as the square of the velocity only for greater velocities 

 than we shall most frequently have to consider. 



680. RESISTANCE PROPORTIONAL TO THE VELOCITY. The 

 hypothesis of m = i is that which best corresponds to the practical 

 case of the resistance of the air, and it is always realised for the 

 effects of induction; this hypothesis has been investigated by 

 Gauss.* 



Equation (n) becomes 



dx 



its general integral, as we have seen, is 



(14) * = A^ + A> 7 , 



A and A' being constants, and p and p' roots of the equation of 

 the second degree. 



(15) p 2 x 2e/> + 2 = 0. 



These roots are real or imaginary according as we are dealing 

 with one or the other of the two conditions, e 2 - n? > 0, or 



2 - 2 <0. 



* GAUSS. Restiltatc aus den Beob. des Magn, Vereins. 1837. (Euvres, v., p. 374. 



