Hypersurface representation and the image of the double S^3-transfer

We study the image of a transfer homomorphism in the stable homotopy groups of spheres. Actually, we show that an element of order 8 in the 18 dimensional stable stem is in the image of a double transfer homomorphism, which reproves a result due to P J Eccles that the element is represented by a framed hypersurface. Also, we determine the image of the transfer homomorphism in the 16 dimensional stable stem.