356 Mr Murphy on the Inverse Method of 



roots, one a constant, the other variable, commencing' from 

 nothing- and increasing indefinitely; it is evident that the above 

 formula, will represent the variable root, as long as it is less 

 than the constant; but the instant the variable exceeds the con- 

 stant root, the formula will cease to represent the former, and 

 from thenceforward it will express the constant. This principle 

 when fully developed, meets every case of discontinuity. 



In the third Section, the preceding theory is applied, to the 

 phenomena of developed electricity, from thence deducing the 

 law of accumulation on the surface. The function which ex- 

 presses the action of a closed conducting surface, charged in any 

 conceivable manner with electricity is discontinuous, but the 

 parts are not independent in consequence of the known law of 

 force to each particle at different distances. By the principles 

 of the first two Sections, we may, by observing the law of action 

 of the electrised body, deduce the law of distribution on its 

 surface. When for instance the action of a sphere, in any manner 

 electrised is observed at distances greater than the radius to vary 

 inversely as the (« + 2) power of the distance, the law of accu- 

 mulation, which is expressed by a differential coefficient of the 

 n a order, is of a remarkable kind. There will then be « nodal 

 or transition lines on the surface, in which the electricity remains 

 unresolved, and which divide the sphere into n + 1 portions con- 

 taining alternately the positive and negative electricities ; in each 

 portion there is a line of greatest accumulation (which becomes 

 a point in the two extreme portions) ; the transition lines, and 

 lines of greatest accumulation, divide the surface into belts, con- 

 taining alternately in pairs, exactly equal quantities of the op- 

 posite fluids. And by the superposition of several systems of this 

 kind, all the phenomena of developed electricity are produced. 



