anti-symmetry required for the piezo-electricity of quartz investigated so admirably by the brothers Curie[18], who found that a thin plate of quartz crystal cut from any position perpendicular to a pair of faces of a symmetrical crystal, becomes positively electrified on one side and negatively on the other when pulled in a direction perpendicular to those faces. But this assemblage has not the chiral piezo-electric quality discovered theoretically by Voigt[19], and experimentally in quartz and in tourmaline by himself and Riecke[20], nor the well-known optic chirality of quartz. [Illustration: FIG. 16.] [Illustration: FIG. 17.] § 49. Change now the directions of the grooves and fillets to either of the oblique configurations shown in Fig. 16, which I call right-handed, because the directions of the projections are tangential to the threads of a three-thread right-handed screw, and Fig. 17 (left-handed). The prisms with their grooves and fillets will still all fit together if they are all right-handed, or all left-handed. [Illustration: FIG. 18.] Fig. 18 shows the upper side of a hexagonal layer of an assemblage thus composed of the right-handed molecule of Fig. 16. Fig. 15 unchanged, still represents a horizontal section through the centres of the molecules. A prism built up of such layers, and finished at each end with a pyramid according to the rule of § 48, has all the qualities of ternary chiral symmetry required for the piezo-electricity of quartz; for the orientational differences of the alternate pairs of prismatic faces; for the absolute difference between the alternate pairs of faces of each pyramid which are shown in the etching by hydrofluoric acid; for the merely orientational difference between the parallel faces of the two pyramids; and for the well-known chiro-optic[21] property of quartz. Look at two contiguous faces _A_, _B_ of our geometrical model quartz crystal now before you, with its axis vertical. You will see a difference between them: turn it upside down; _B_ will be undistinguishable from what _A_ was, and _A_ will be undistinguishable from what _B_ was. Look at the two terminal pyramids, and you will find that the face above _A_ and the face below _B_ are identical in quality, and that they differ from the face above _B_ and below _A_. This model is composed of the right-handed constituent molecules shown in Fig. 16. It is so placed before you that the edge of the prismatic part of the assemblage nearest to you shows you filleted faces of the prismatic molecules. You see two pyramidal faces; the one to your right hand, over _B_, presents complicated projections and hollows at the corners of the constituent molecules; and the pyramidal face next your left hand, over _A_, presents their unmodified corners. But it will be the face next your left hand which will present the complex bristling corners, and the face next your right hand that will present the simple corners, if, for the model before you, you