134 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and that if a, b, c, d, e, and h are the roots of the equation formed by 

 equating to zero the determinant, 



Lx+r Mx Mx Mx 



Mx Nxx + Ri i NiX + NiX -\-Ri\ f A^x + N^x + N^x 



'( +i?4 S \ +R,+R4+R, 



Mx^ iNiX^ + NiX^+Rixl Nix- + Rix Nix- + Ri (Q) 



\ +E2^X + 1/A'2J . -^ 



Mx^ Nix^ + Rix C Nix'~ + Nix"- + Nzx"- ) j Nxx"- + NiX"" + Rix \ 



• +R,x+RiX+Rzx\ \ +RiX ( 



( + 17/^3 ) 



then 



V 



I = Ae''' + Be^^ + Ce'^^ + De^' + Ee'^ + He^' + - 



H, 



h = a.oAe''' + ^-.Be^' + ^^Ce'' + We'^' + e.Ee'' + jyoFe''' (4) 



h = a^Ac'"' + ^M' + TsCc'^' + asDf'" + e-iEc'' + r^sfl'e''^ 

 /a = a,Ac^' + /355f'''* + l^Ce^' + SaDe'^^ + eaEc^' + -n^He^'' 



If these values be substituted in one of the Kii'chhoff equations above 

 and the coefficients of the different exponential expressions separately 

 equated to zero, it will appear that the a's, /3's, 7's, 5's, e's, and t^'s are 

 determinate functions of the constants of the circuit and in no way 

 dependent upon the manner in which the currents are managed. 



The other six constants {A, B, C, D, E, H) have to be computed 

 from a knowledge of the electrical conditions which determine any 

 problem concerning these two fixed circuits. 



If, for instance, there is no ciu'rent in any branch of the circuits 

 at the outset, and if the gap, O, be suddenly closed at the origin of 

 time, the values of the constants for all positive time satisfy the 

 equations 



A+ B+ C+ D+ E+ //--^ 



a^A + P2B + 72C + hiD + eoE + €0// = 

 azA + ^zB + 73C' + bzV + e;E + €3// = 

 aa + (3,B + 75C' + 8:,D + e,E + e-JI = 



and, after these have been solved, it is easy to comiDute the whole 

 flow of electricity through the galvanometer, for 



fl it = _(-! + &« + 2i^ + h^_ + ^A + "f ) (e) 



Jq \abcaehj 



