The head for 0 (zero), figure 53, _s-w_, is always to be distinguished by the hand clasping the lower part of the face (). In this sign for zero, the hand probably represents the idea "ending" or "closing," just as it seems to have done in the ending signs used with {102} Period-ending dates. According to the Maya conception of time, when a period had ended or closed it was at zero, or at least no new period had commenced. Indeed, the normal form for zero in figure 47, the head variant for zero in figure 53, _s-w_, and the form for zero shown in figure 54 are used interchangeably in the same inscription to express the same idea--namely, that no periods thus modified are involved in the calculations and that consequently the end of some higher period is recorded; that is, no fractional parts of it are present. That the hand in "ending signs" had exactly the same meaning as the hand in the head variants for zero (fig. 53, _s-w_) receives striking corroboration from the rather unusual sign for zero shown in figure 54, to which attention was called above. The essential elements of this sign are[70] (1) the clasped hand, identical with the hand in the head-variant forms for zero, and (2) the large element above it, containing a curling infix. This latter element also occurs though below the clasped hand, in the "ending signs" shown in figure 37, _l_, _m_, _n_, the first two of which accompany the closing date of Katun 14, and the last the closing date of Cycle 13. The resemblance of these three "ending signs" to the last three forms in figure 54 is so close that the conclusion is well-nigh inevitable that they represented one and the same idea. The writer is of the opinion that this meaning of the hand (ending or completion) will be found to explain its use throughout the inscriptions. [Illustration: FIG. 54. A sign for 0, used also to express the idea "ending" or "end of" in Period-ending dates. (See figs. 47 and 53 _s-w_, for forms used interchangeably in the inscriptions to express the idea of 0 or of completion.)] In order to familiarize the student with the head-variant numerals, their several essential characteristics have been gathered together in Table X, where they may be readily consulted. Examples covering their use with period, day, and month glyphs are given in figure 55 with the corresponding English translations below. Head-variant numerals do not occur as frequently as the bar and dot forms, and they seem to have been developed at a much later period. At least, the earliest Initial Series recorded with bar and dot numerals antedates by nearly two hundred years the earliest Initial Series the numbers of which are expressed by head variants. This long priority in the use of the former would doubtless be considerably diminished if it were possible to read the earliest Initial Series which {103} have head-variant numerals; but that the earliest of these latter antedate the earnest bar and dot Initial Series may well be doubted.