The author investigates the solubility in rationals of equations of the form $$a_0X_0^4 + a_1X_1^4 + a_2X_2^4 + a_3X_3^4 = 0 $$ where $a_0a_1a_2a_3$ is a square, building on the ideas which Colliot-Thène, Skorobogatov and he have developed; see {\em Invent. Math.} 134 (1998) 579--650. He obtains sufficient conditions for solubility, which appear to be related to the absence of a Brauer--Manin obstruction. This represents the first large family of K3 surfaces which almost satisfy the Hasse principle, in the sense that the auxiliary condition which ensures that local solubility everywhere implies global solubility is nearly always satisfied. 1991 Mathematics Subject Classification: 10B10.