The simultaneous zero-sum search game on the sphere C is considered. The point chosen by the hider is called 'discovered' if it lies in the r-neighbourhood of the points chosen by the searcher. The payoff of the searcher is equal to the number of discovered points. The case with a moving searcher is also considered. The optimal strategy of the searcher consists in the random throwing of the 'discovery region' on the sphere according to the probability distribution generated by the kinematic measure. The optimal mixed strategy of the hider and the value of the game are found.