[From Mature, Vol. x.] 



I. XVIII. OH the application of Kirchhojfs Rules for Electric Circuit* t<> the 



Solution of a Geometrical Problem. 



THE geometrical problem is as follows : Let it be required to arrange a 

 system of points so that the straight lines joining them into rows and columns 

 shall form a network such that the sum of the squares of all these joining 

 lines shall be a minimum, the first and last points of the first and last row 

 leing any four points given in space. The network may be regarded as a 

 kind of extensible surface, each thread of which has a tension in each segment 

 proportioned to the length of the segment. The problem is thus expressed as 

 a statical problem, but the direct solution would involve the consideration of a 

 large number of unknown quantities. 



This number may be greatly reduced by means of the analogy between 

 this problem and the electrical problem of determining the currents and po- 

 tentials in the case of a network of wire having square meshes, one corner of 

 which is kept at a unit potential, while that of the other three corners is zero. 

 This problem having been solved by KirchhofFs method, the position of any 

 point P in the geometrical problem with reference to the given points A, B, 

 C, D, is by finding, the values of the potentials p a , p t , p e , p d of the corre- 

 s]x>nding point in the electric problem when the corners o, b, c, d respectively 

 are those of unit potential. The position of P is then found by supposing 

 Pa> Pb> Pe> Pd placed at A, B, C, D respectively, and determining P as the 

 centre of gravity of the four masses. 



