Convolution of Gaussians

Gaussians have an interesting property with respect to convolution. The convolution of a Gaussian (μ1, σ1) and another Gaussian (μ2, σ2) is a new Gaussian (μ3, σ3) with the means and variances being additive; i.e., μ3=μ1+μ2 and σ3^2= σ1^2 + σ2^2. (Jaynes, Mathworld).

Consequently, in an image processing application, a Gaussian Blur of radius 3 followed by a Gaussian Blur of radius 4 should produce the same result as a single Gaussian Blur of radius 5 (give or take some rounding error).