338 Proceedings of the Royal Physical Society. 



of 056, 065, 032, 023, we see that the zone upon which they 

 all occur is parallel to a and is not parallel to c, whence it 

 follows that it must coincide with ch. We proceed, as in 

 last case, producing oe, signing it E, and erect a perpen- 

 dicular to oz to touch E, figuring the line 6. Also a 

 perpendicular to ox is raised also to touch E, and the line 

 figured 5. All indices in which the second figure is greater 

 than the third are to be measured on 5, and those in which 

 it is less upon 6. Five-sixths is measured from c, or one- 

 sixth from E, and the distance projected on to cb by a line 

 radial to as before. For 065 we reverse the fraction as 

 before, and measure five-sixths on 5 from ox, or one-sixth on 

 5 from E, and proceed as before. In like manner 032 is 

 measured on 5 as two-thirds from ox or one-third from E, 

 and 023 is obtained on 6 on the same principle as already 

 described. 



38. Assuming that we wish now to determine 710, 170, 

 530, 6.10.0, the final figure shows that these all lie in a zone 

 parallel to c ; they are therefore on ah. om is to be produced, 

 signed M, a perpendicular, 1, to be drawn to M from ox, and 

 another, 2, perpendicular to OY. The cases in which the first 

 index figure is greater than the second are to be measured 

 on 2, and those on which it is less on 1. For 710 one-seventh 

 of 2, measured from OY, gives the position, which is projected 

 into am by a line radial to 0. 170 is obtained by measuring 

 one-seventh of 1 from ox. In the case of 530 the first and 

 second figures are transposed, and three-fifths of 1 are 

 measured from ox, or two-fifths of the same are to be 

 measured from m. 6.10.0 is the same as 350 or as 12.20.0, 

 just as 10.6.0 is the same as 530, and so on. 



39. Now we are in a position to consider the cases in 

 which the plane whose pole is to be determined cuts all three 

 axes at distances less than unity. Suppose it is required to 

 lay down 672, 953, 12.7.4, 6.10.3, which between them 

 illustrate the principal cases. 672 can be shown to lie at 

 the intersection of a line drawn from 670 to c with another 

 line drawn from 602 or 301 to h. In like manner 953 lies 

 at the intersection of a line from 950 to c with one from 903 

 or 301 to h. 12.7.4 in like manner is determined by drawing 



