Lines of Force due to given Static Charges. 177 



The equation in i\ is — 



2^ l 6 ~40V+382V + 358v 1 3 + 121V + 18r 1 + l = 0, 

 which may also be written 



2u 1 4 («i 2 -20i? 1 + 191) + 358i' x 3 + 121V + 18^ + 1 = 0; 



and as the expression in brackets is positive for all real 

 values of vi we may pronounce that there are no positive 

 roots of the equation in v { ; therefore no real roots of <£(j/) = 

 between 2 and 3 ; hence no negative roots of the original 

 equation. All the roots of that equation are therefore 

 imaginary. 



XIII. Lines of Force due to given Static Charges. 

 By Prof. D. N. Mallik, B.A., Sc.I)., F.R.S.E.* 



[Plate II.] 



1. TAIAGRAMS of lines of force due to given static 

 \J charges are given at the end of the first volume of 



Maxwell's Treatise, in which also a brief description is given 

 of a method of drawing them [Art. 123, vol. i.]. The 

 importance of the subject, however, justifies a more detailed 

 treatment, and I have, accordingly, in the present paper con- 

 sidered with considerable fulness, a simple geometrical method 

 of drawing these lines which is obviously suggested by the 

 construction given in Roget's ' Electricity ' for Magnetic 

 Curves. 



2. Consider two charges e u e % at A, B (PL II. fig. I.). 



If F 1? F 2 are the forces due to e u e 2 at any point P, of 

 which the coordinates are r 1? X , r 2 , 2 respectively referred 

 to A, B, we have, since the resultant force along the normal 

 to a line of force vanishes, 



i. e. 



But 



from the triangle APB. 



£i 



dOi e 2 d0 2 

 ds r 2 * ds 





7*, sin 2 





r 2 sin $i ' 



0. 



* Communicated by the Author. 

 Phil. Mag. S. 6. Yol. 22. No. 127. July .191 1 I'T 



