430 DR W. PEDDIE ON 



unit in terms of which y was measured. And, if we denote the quantity on the right 

 hand side of the equation by B, we get 



log (nli) = log B + n log k, 



which, when k is constant, is the linear relation above referred to. 



But the value of n is such, throughout each series of experiments, that it is 

 impossible to determine whether that relation, or a linear relation between log b 

 and n, is the more accurate. If one were strictly accurate in a given series, the 

 other cannot be so simultaneously. Yet the possible variations in the determined 

 values of n and b, for any experiment in a given series, are such that either 

 relation may be regarded as practically correct. The results for the latter are 

 exhibited graphically in figs. 9, 10, 11, and 12. 



Just as the maintenance of a linear relation between log nb and n, in a given 

 series, implies the existence, throughout that series, of a Critical Angle at which the 

 loss of energy per oscillation is independent of n ; so the maintenance of a linear 

 relation between log b and n, in a given series, implies the existence, throughout that 

 series, of an angle at which the loss of energy per oscillation varies inversely as n. 

 For the equation 



f(x + a) = b 



may be put into the form 



y' n (x+a) = b(^y 



by taking as the ?/-unit a quantity k! times greater than the unit in terms of which 

 y was measured. And k' can always be chosen so that the right hand side of the equa- 

 tion has a given constant value, /3 say. We then have 



log b = log (3 + n log k', 



which, when k' is constant, is the second linear relation. Also 



*t = - — ' " +1 



dx nfi ' 



Hence, when y' is unity, i.e., when y = k', dy'jdx and y'dy'/dx vary inversely as n, 

 the latter quantity is practically proportional to the loss of energy per oscillation. 

 For convenience of reference we may call k' the Inverse Angle. 



Existence of an Oscillation Constant. 



As we have just seen, we can always choose a unit k", which will make the relation 

 between y and x take the form 



y n (x + a) = A, 



where A is an absolute constant. We may call this quantity, k", the Ujiifying Angle, 



