Page:Electricity (1912) Kapp.djvu/68

64 Let us now subdivide the straight line, along which the movement takes place, into a very large number of little bits, each in itself so small that we may neglect any change of the repelling force within the two ends of this little bit. The force varies by a small amount from bit to bit, but within the limits of one bit or small step on the journey we consider it constant. Such a conception is quite permissible if we take the steps or elements of the path small enough. The energy corresponding to each elemental part of the journey is the product of the length of the element divided by the square of the distance to the centre of the sphere, and multiplied by the charge on it. To each step thus corresponds a little bit of the total energy, and by adding up all these little bits of energy we get the potential. It would be very laborious to actually map out the whole of the journey in this way and make the innumerable calculations here indicated. Fortunately there is no necessity for all this arithmetical work. By the application of a mathematical method known as the infinitesimal calculus we are able to arrive at the result in a very simple way by one operation. The result is