Ford and Tillotson — Orthoclase Twins. 153 



purposes of drawing it is quite accurate enough to assume that 

 this angle is exactly 90° and that accordingly the c face of the 

 twin will occupy a position parallel to that of the b face of 

 the normal individual. 



Figure 6 shows the forms observed of the crystals both in 

 normal and in twin positions, the faces in twin position being 

 indicated by open circles and a prime mark (') after their 

 respective letters, while the zones in twin position are drawn in 

 dashed lines. Starting out with the forms in normal position, 

 the first face to transpose is the base c. This form, from the 

 law of the twinning, will be transposed to c' where it occupies 

 the same position as b of the normal individual, and it 

 necessarily follows that b itself in being transposed will come 

 to V at the point where the normal c is located. 



In turning therefore the crystal to the left from normal to 

 twin position, the faces c and b travel along the great circle 

 I through an arc of 90° until they reach their respective twin 

 positions. We have, in other words, revolved the crystal 90° 

 to the left about an axis which is parallel to the faces of the 

 zone I. The pole of this axis is located on the stereographic 

 projection at 90° from the great circle I and falls on the 

 straight line II, another great circle which intersects zone I 

 at right angles. This pole P is readily located by the stereo- 

 graphic protractor on the great circle II at 90° from c. The 

 problem then is to revolve the poles of the faces from their 

 normal positions about the point P to the left and through an 

 arc of 90° in each case. 



During the revolution the poles of the n faces remain on 

 the great circle I and as the angle n^n = 90°, the location of 

 their poles when in twin position is identical with that of the 

 normal position and n' falls on top of n. We can now trans- 

 pose the great circle II from its normal to its twin position, 

 since P remains stationary during the revolution and we have 

 determined the twin position of c. The dashed arc IT gives 

 the twin position of the great circle II. The twin position 

 of y must lie on arc IT and can be readily located at y\ the 

 intersection of arc IF with a small circle about P having the 

 radius P ^ y. It is now possible to construct the arc of the 

 zone III in its transposed position III 7 , for we have two of the 

 points, y f and n' of the latter, already located. By the aid of 

 the Penfield transparent great circle protractor the position 

 of the arc of the great circle on which these two points lie can 

 be determined. On this arc, III', o' and m! must also lie. 

 Their positions are most easily determined by drawing arcs 

 of small circles about V with the required radii, b^o — 63°8 / , 

 b/\m = 59° 22 1/2' and the points at which they intersect arc 

 III 7 locate the position of the poles o' and in' . At the same 



