402 PROCEEDINGS OF THE AMERICAN ACADEMY. 



rian logarithm of the decrement during the interval, or simply as the 

 log- decrement in the interval. If the log- decrement be denoted by 8, 

 then 



8 = J-^i. numeric (57) 



If the rotating vector moves through one radian at the actual angular 

 velocity w, the time ti occupied in the passage will be l/a> seconds ; so 

 that 



Si = - = cot <^. numeric (58) 



to 



If the rotating vector moves through a half-cycle, semi-revolution, 

 or TT radians, the time occupied in the passage will be ti = tt/w seconds ; 

 so that 



o„ = TT- = -— = TT cot d), 3 numeric (59) 



which is the log-decrement of any two successive elongations of the 

 vector's projection in opposite directions on the axis of reference JT^. 

 If the vector moves through a whole cycle, revolution, or 2 tt radians, 

 the time occupied will be ^i = 2 tt/w seconds, and 



± ± T 



82,7 =27r- = - = — = 277 cot (h, numeric (60) 



which is the log-decrement of any two successive elongations of the 

 vector's projection in one and the same direction on the reference axis. 

 Consequently, from any pair of successive maxima of the oscillating 

 quantity y, the log-decrement is obtainable, and from this the angle ^ 

 of the equiangular spiral and of the stationary vectors for the quantity 

 is obtainable. In the case considered, 81 = 0.8165, K = 2.5651, and 

 h2n = 5.1302. 



Root of Mean Square of Oscillating-Current Quantities. 



If V be any vector oscillating quantity of the type V^e^^^ sin <ot, such 

 as an oscillating- voltage, current, or force ; then if the damping coef- 

 ficient J- is taken as zero, the square root of the mean square of the 

 quantity during any integral number of cycles is F= Vq/V2, as in 

 ordinary alternating-current theory. If, for instance, the initial volt- 

 age is 1000, and there were no damping, the r. m. s. voltage of the 0. c. 



' The condition expressed in (59) was first pointed out by_Prof. Clerk- 

 Maxwell, in a somewhat different manner. See Bibliography. 



