The exponent 2 in Coulomb's inverse square law of force between charges in empty space has been found experimentally to be correct to within 1 part in 109. The wellknown electrostatic experiment of Cavendish and Maxwell with concentric metal globes was replaced by a quasistatic method in which the difficulties due to spontaneous ionization and contact potentials were avoided. A "resonance electrometer" (undamped galvanometer with amplifier) was placed within the globes, the input resistor of the amplifier forming a permanent link connecting them, so as to measure any variable potential difference between them. It was shown theoretically that the presence of the resonance electrometer would have no effect on the result and that it could replace electrically a part of the inner globe. The galvanometer was observed through a "conducting window" at the top, made so by covering it with salt water. No effect was observed when a harmonically alternating high potential V (>3000 volts), from a specially designed "condenser generator" operating at the low resonance frequency of the galvanometer, was applied to the outer globe. The sensitivity was such that a voltage v=10-6 volt was easily observable above the small fluctuations due to Brownian motion. If the exponent in the law of force were not exactly 2 but rather 2+/-q then q<vVF(a,b) where F(a,b)=0.169, a and b being the radii of the globes. This gives q<2×10-9 in space remote from matter. The formula for F(a,b) was derived by Maxwell's theory in which the effect of gravity is assumed negligible. Reasons are given for believing that this assumption does not invalidate the result.