J. Trowbridge on Ohm’s Law. 115 
Art. XX.—Ohm’s Law considered from a geometrical point of 
vew; by JOHN TRowBRIDGE, Assistant Professor of Physics, 
Harvard College. 
Oxn’s law is briefly expressed thus: the strength of a cur- 
Tent passing in any conductor of a resistance R is equal to the 
aa force producing the current divided by the resist- 
ance or S=> Let us suppose that a certain quantity of elec- 
1. tricity pees at the point O, 
is transmitted by the conductor 
. BOC to the surface BC. The 
= c quantity passing through an 
unit ay of the conductor will 
vary inversely as the section bc, 
and inversely as the distance of 
. the section from 
Hence we shall have for an 
expression of this quantity g= 
x 2 gq. (1). In which Q repre 
0 
sents the entire quantity passing through any section S$; and 
«18 the distance of the section from O. If we suppose that 
the conductor B OC is formed by the revolution of any curve, 
Whose equation is y= F (a), about the axis of X, equation (1). 
becomes g= — 3; By substituting for y its value from the 
*quation of the curve which generates the conductor, we shall 
obtain equations which represent, when constructed as curves, 
the variations in the quantity of electricity passing through a 
Unit section of the conductor. 
When the generating curve is y? = e the equation becomes 
= —= constant; a straight line parallel to the axis of X. If 
the curve is an equilateral hyperbola, whose equation is =~ 
we shall have g= 
= ma where m is any constant. This 
aC2e 
° id 
1s the equation of a straight line passing through the origin. 
en the equation of the generating curve is y=’ the con- 
ductor BOQ becomes a cylinder ; and g= © where Cis any con- 
Stant. This is the equation of an equilateral hyperbola, and is iden- 
