1891.] Application to Periodic Electric Currents. 213 



We shall show presently that (30) can be reduced to zero; but for 

 the moment we will so far limit the generality of a 1} 0% as to suppose 

 that ! = xbi, 02 = rb 2 , x being real and positive. 



(30) then reduces to \ &i 2 (l + aj) ; and by (29) 



Mod (d 6) 



- 



\t 3 i /T 



Mod * = : - 



Accordingly, the maximum sensitiveness cannot be attained until 

 is reduced to zero, so that a l5 03 vanish. (31) may be regarded as 

 a generalised form of (24), free from the limitation that 6 2 = 0, pro- 

 vided a? be so taken that a^/b^ = a\fb\. 



We will now suppose in (30) that a\ and a? are both small, and in 

 the first instance that b\ is finite. We have 



..(32); 



and this reduces ultimately to its first term, depending upon the ratio 

 only of ai and a z . The expression vanishes if cti : a% be small enough, 

 so that (30) can certainly be thus reduced to zero. It is remarkable 

 that the expression for the sensitiveness should be capable of becom- 

 ing infinite by suitable choice of 2 . If we first suppose that a? is 



i absolutely zero, and afterwards that a-i diminishes without limit, the 

 ultimate value of (32) is | & 1 v / 0i 2 +&2 2 ) place of zero. 



From the practical point of view, these conclusions from our 

 equations are not particularly satisfactory. We began with certain 



^proposals which, in ordinary cases, could be carried out ; but in the 

 end we are directed to apply them to an extreme and impossible state 

 of things. We have found, however, in what direction we must tend 

 in the search for sensitiveness; and useful information may be 

 gathered from (32). In practice a x could not be reduced below a 

 certain point. The question may then be asked, what is the best 

 value of a 2 , when a! is given ? From (32) we find at once that 



............. (33), 



0i 

 then becoming 



61 </(<*!&,) .................... (34). 



In this case from (29) 



nd 



ependent of & 2 . 

 If we suppose in (32) that a 2 = 0, we have 



(36). 



