out from the bottom stage the above mentioned embodiment shows an example where two cooling apparatuses are connected for two stage cooling but in another embodiment three or more cooling apparatuses may be connected for multi stage cooling and in still another embodiment as shown in fig4 heating wires 9 may be wound around the internal peripheral surface of the chimney 4 and the pitch made gradually closer upstream as best seen in fig5 so as to heat the cooling air and give it a temperature gradient so that the temperature gradually falls downstream whereby the cooling can can be made continuously stronger downstream further in another embodiment the flow path of the cooling air can be formed to be gradually narrower upstream or formed so that the pressure loss gradually increases thereby gradually decreasing the amount of the cooling air upstream as mentioned above according to the method of this invention the cooling is made performed weaker at the upstream side and stronger at the downstream side thereby improving the elasticity tensile strength and other physical properties of the filaments without enlarging the apparatus and further making molten breakage difficult and preventing mutual adhesion of filaments according to the method of this invention wherein the degree of strength of the cooling is changed by the spinning speed even if the spinning speed is changed to make it faster the degree of the cooling at the weaker portion of the cooling zone can be increased so as to avoid rapid cooling or elongation of the cooling zone in the cooling apparatus of this invention a plurality of cooling apparatuses are connected so.