In a hypothetical simple type of valve, easily derivable from the more primitive type of _Petromyzon_, it may be conceived that, while not exceeding in width the semi-diameter of the intestine, the valve becomes disposed in several complete and more or less closely approximated spiral turns, the free edge of the valve being on the same level as its attached margin, and leaving an open axial canal along the centre of the gut. The nearest approach to this hypothetical type, which has been compared, not inaptly, to _un escalier tournant sans noyau_, is perhaps to be found in the Thresher-Shark (_Alopecias vulpes_). The structure of the more complicated spiral valves of other Elasmobranchs are well illustrated within the limits of the single genus _Raia_. In one specimen of _Raia_ sp. (Fig. 159, A) the last four coils of the valve are similar to those of the hypothetical type, but the more anterior ones, owing to the greater width of the valve, which here exceeds the semi-diameter of the intestine, have their free margins deflected downwards, while that portion of the valve which forms the first half turn is coiled inwards upon itself, so as to form a hollow cone, open dorsally, and having its apex directed forwards. In other examples a further modification is introduced by the increasing width of the valve, which now, throughout its whole length, equals the semi-diameter of the intestine; and by the formation of an axial columella by the thickened free edge of the valve, which is traversed by a central band of unstriped muscle, as well as by the intra-intestinal artery and vein, and takes the place of the central canal of the preceding types. The valve is, however, still regular, and its free margin remains on the same level as the corresponding portion of the attached edge. In other specimens, again, additional complications are introduced by a still further increase in the width of the valve, which now exceeds, often considerably, the semi-diameter of the intestine, and the consequent deflection of the free edge of the valve either forwards or backwards (C and D). As shown in C the valve, in consequence of the backward deflection of its free margin, presents the appearance of a nest of {266}imperfect truncated cones with their apices directed backwards, the successive cones adhering so closely to one another that they combine to form a central conical chamber with a spirally disposed cavity winding round it. In D, on the contrary, the free edge of the valve is deflected forwards, so that, as in C, a nest of cones is formed, but the apices of the successive cones are directed forwards instead of backwards. Notwithstanding these variations in the structure of the valve as a whole, the first coil or half coil nearly always resembles that described in A.