Prof. Thomson on Electrified Spherical Conductors, 287 



culars from o on the sides adjacent to an acute angle we draw 

 parallels to the sides (remembering that ea is perpendicular to 

 ag, and eb to bg), then drawing pohm, qokn perpendicular to 

 these, and meeting the other two sides respectively in h and k, 

 we find the harmonic mean between om^ op = %oh ; between on, 

 oq=2ok ('.' ee' = ob). Aho in tYmngle bo. oe =ao.og = bo {oe^^ bo). 

 Hence producing on, om so that oq' = oq, and op' = op, also 

 oh'=2oh; ok' = 2ok. The points mn, p'q' are in a circle in 

 which h'k' is polar of o and oh' + 2op' = ok' + 2oq^. Construct 

 another figure where p'z is parallel to oq' and h'w, and q'y to op' 

 and k'a:, oc meeting M in e ; then we can show without difficulty 

 (by considering co^—cq'^ and ck^—cq'^, &c.) that 



{oh—ok){x2 + yw + 2oe) = 0j .-. oh=ok^ .-. om^oUj .*. og^oe, 



oa=ob, &c. 



More easily as follows : — suppose that the angle b is not less 

 than aj and describe a circle through adb cutting af in m; then 

 the angles abm, dab are acute, and am is not greater than af. 

 But also by hypothesis abm is not less than cab, and both are 

 acute; .'. «m is not less than bd or af; .*. dis equal to af and 

 abc=bac. Q. E. D. 



XLV. On the Mutual Attraction or Repulsion between two Elec- 

 trified Spherical Conductors, By Prof. W. Thomson*. 



IN a communication made to the British Association at Cam- 

 bridge in 1845, 1 indicated a solution adapted for numerical 

 calculation, of the problem of determining the mutual attraction 

 between two electrified spherical conductors. A paper published 

 in November of the same year in the first Number of the Cam- 

 bridge and Dublin Mathematical Journal, contains a formula 

 actually expressing the complete solution for the case of an insu- 

 lated sphere and a non-insulated sphere of equal radius, and 

 numerical results calculated for four different distances for the 

 sake of comparison with experimental results which had been 

 published by Mr. Snow Harris. The investigation by which I 

 had arrived at this solution, which was equally applicable to the 

 general problem of finding the attraction between any two 

 electrified spherical conductors, has not hitherto been published ; 

 but it was communicated in July 1849 to M. Liouville, along with 

 another very different method by which I had just succeeded in 

 arriving at the same result, in a letter the substance of which 

 constitutes the present communication. Formulae marked (8) 

 .... (18) in that letter expressed the details of the solution 



* Communicated by the Author, having been read at the Meeting of the 

 British Association in Belfast, Sept. 1852. 



