Metrics of positive curvature with conic singularities on the sphere

A simple proof is given of the necessary and sufficient condition on a triple of positive numbers θ 1 , θ 2 , θ 3 θ _1,θ _2,θ _3 for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles 2 π θ 1 , 2 π θ 2 , 2 π θ 3 2π θ _1,2π θ _2, 2π θ _3 . The same condition is necessary and sufficient for the triple π θ 1 , π θ 2 , π θ 3 π θ _1,π θ _2,π θ _3 to be interior angles of a spherical triangular membrane.