472 Mr. L. Schwendler on the General Theory 



To make /#, therefore, as small as possible, a sensitive con- 

 struction of the differential instrument becomes requisite ; fur- 

 ther, cells of high electromotive force and low constant resistance 

 are best adapted for forming the signalling-battery. In order 

 to get the widest limits in the variation of w, it is clear that that 

 /3 should be selected which is calculated from the maximum 

 number of cells required to produce the signals with sufficient 

 force. The greatest number of cells is obviously required when 

 the line is at its lowest insulation in India, during the monsoon. 



T 



The value vz=- is what has been termed the mechanical ar- 



rangement of the differential instrument*. 



If b = w + /3 has been determined by fixing /3, then v has its 

 smallest value for L largest, which is the case when the line is 

 perfect in insulation — when the coil a must be closest to the 

 magnetic pole acted upon, and the coil b furthest away from it. 



The highest value of v we obtain by substituting the lowest 

 L, i. e, when the line is at its lowest insulation — when the coil 

 b must be nearest to the magnetic point acted upon, and the 

 coil a furthest away from it. 



Hence, the two limits of v being fixed by the known limits 

 between which L varies, the extent of movement of the two coils 

 is also fixed ; and consequently, if q is chosen arbitrarily, the 

 construction of the differential instrument is determined. But 

 even q is not quite arbitrary, since we know the form, dimen- 

 sions, and resistance of the coils which, for instance in Siemens's 

 polarized relays on any given line, have to produce the magnetism 

 in single circuit to get the signals with engineering safety. 



The solution of the first problem of the differential method is 

 therefore : — 



1. Balance in each station must be obtained by a movement of 

 the two acting coils or their armatures, either singly or {better) 

 simultaneously in the same direction, and not by an alteration of 

 the resistances in the branches. 



2. If this mode of adjusting balance be adopted, then the 

 solution is 



d=h=0, 



f=b=w+£, 



2L + b* 



* Journal of the Asiatic Society of Bengal, vol. xli. part 2, p. 148 ; 

 Phil. Mag. vol. xliv. p. 166. 



