374 Prof. G. C. Foster on Graphical Methods of 



struction indicated above, BL represents the resistance equivalent 

 to A B and B C in multiple arc. Draw L M' parallel to A B C D, 

 make CM = BL ; draw M M' at right angles to B C, join M' D, 

 and draw C N perpendicular to CD. CN then represents the 

 joint resistance of the three conductors. 



Now consider a conducting-system such as that indicated in 

 fig. 9, where two points, P and Q, are connected, through three 

 conductors whose resistances are respectively r, r v and r 2 ; and 

 let a battery of electromotive force e make part of the first con- 

 ductor. The strength of the currents and the distribution of 

 potential in the various parts of the system can be represented 

 as follows : — Take A (fig. 10) to represent r, from draw 

 O B and C in opposite directions perpendicular to A, make 

 C C equal to C and perpendicular to it, draw B C' cutting 

 A produced in D ; then D represents the joint resistance of 

 7\ and r 2 . Draw A E perpendicular to A to represent the 

 electromotive force e, and join ED. Then tan Z ADE repre- 

 sents the strength of the current in the battery and the sum of 

 the currents in the two parallel branches. These may be ob- 

 tained separately thus : let the point of intersection of E D and 

 O B be denoted by F ; then F represents the difference of 

 potential between the points P and Q or the electromotive force 

 which is effective in the two conductors of resistance r } and r 2 . 

 In G A make OF=OF and draw F' B and F' C ; we have then, 

 for the strength of the currents in the conductors whose resis- 

 tances are represented by B and C, tanZOBF' and tan 

 Z C F' respectively. 



Next let two of the conductors connected together at P and 

 Q contain galvanic batteries, and, as before, let the resistance of 

 the branch containing one battery be r, while that of the branch 

 containing the other is r v and let the electromotive forces be e 

 and e Y respectively. If the batteries are so connected that both 

 tend to make the potential at P differ from that at Q in the same 

 sense, we have an arrangement of which a special case is pre- 

 sented by PoggendorfPs "compensation method" for the com- 

 parison of electromotive forces. To obtain a geometrical ex- 

 pression for the strengths of the currents in the various parts 

 of the circuit in the general case (that is, without assuming that 

 there is "compensation" in any branch), we may proceed in the 

 following manner : — Take A and OB in the same straight 

 line (fig. 11) to represent the resistances r and r l of the two 

 branches including the batteries, and C at right angles to 

 A B to represent the resistance r 2 of the third branch. In C 

 produced make OAj = OA, and OB^OB, also draw AiA 2 

 equal and parallel to A and Bj B 2 equal and 'parallel to B, 

 and join A 2 C and B 2 C. Let A 2 C cut A in N, and let B 2 C 



