Art. 1. If x2 + y2 + z2 + 2 gx + 2 fy + 2 hz + c = 0, x2 + y2 + z2 + 2 g'x + 2 f'y + 2 h'z + c' = 0 be the equations of two spheres, these spheres will intersect orthogonally if the square of the distance between their centres be equal to the sum of the squares of their radii. Hence we infer that the two spheres whose equations are given above will intersect orthogonally if the condition holds, 2 ff' + 2 gg' + 2 hh' = c + c' . . . . . . . . . . (1) 2. From art. 1 we can easily find the equation of a sphere cutting orthogonally four given spheres, S', S″, S‴, S"". Thus, if the given spheres be x2 + y2 + z2 + 2 g'x + 2 f'y + 2 h'z + c = 0 & c .,