OF ARTS AND SCIENCES. 437 



Other terminals of the muchine and galvanometer were permanently 

 connected with the other ends of the three shunts. 



Another and better method, both as requiring no very small resist- 

 ances and as employing but a single switch connection, is the following. 

 Call G the resistance of the galvanometer, connect a resistance r' to 

 one of its terminals, and shunt bv a second resistance s'. Attach to 

 one end of this a coil r", and shunt again by the coil s". If necessai'y, 

 shunt again until a sufficient reduction is attained. Now connect 

 one terminal of the machine with one end of G, s', and s", and bring 

 the other in contact with the other end of either of them by a 

 simple switch, and we shall have tlie elFect of three shunts of three 

 different sensibilities. The total resistance and the relative con- 

 stants may be computed in each case, or they may be measured 

 directly. Calling the total resistances Ji^, 7?^, and i?g, and the shunts to 

 which they are equivalent S^, S.^, S^, we may deduce proper values by 

 the usual formulas for divided currents. As, however, the case is a 

 little complex, it is best to reduce it to the following symmetrical form : 

 Let / (x, y,z) =z xy -^ xz -\- t/z ; tlien we have : — 



p _r f{r\s',s"-\-r") ^ _ f{r',s',s"-\-r") 



^ ~ / ( ^' + '■', s', ," + r") '^1 ~ /( G + '■', s', s" + r") 



-/( G + r', s', s" + r") ^2 — /( (; + r', s', s" + r") 



p _ y .f{G + r',s',r") c _ s' s" 



■^3 '^f(G-\-r', s', s" + r") ^ f{G-]- r', s', s" -}-r")' 



In these equations, G would generally be given ; and we may, theo- 

 retically at least, assume any five other quantities, and then deduce 

 the remainder. As, however, these equations are too comjilex to be 

 used with any convenience, let us see how they may be simplified. 

 Suppose that s' = s" = G, and that r' = r" = n G, then our six 

 equations become: — 



P_^l + 3n + n2 „_ lJ-^3nJ-_«2 



1 ~~ 3 + i/i + >fl 



Oo 



3 + 4n + «2 

 1 



8 1 + 3« + «2' 



If now ?i= 1, or all the resistances equal C, the three values of R 

 become .625, .5, and .025 ; while those of AS'are .625, .25, and .2. If 

 n = 2,R becomes .733 G, .6 G, and, .733 G, S^ .733, .2, and .091 ; ?i = 5 



