Simpler Methods in Or y stall ograpliy. 335 



each of the two fractions by any number we please. We 

 might, for example, take a at full length (i.e., three-thirds), 

 and h at three-fifths, and still get the same result as before. 

 In practice this is what is done. Indeed, the older writers 

 on Crystallography actually expressed the direction of the 

 planes by a method much the same ; and it is only because 

 the Whewell-Miller system is so much neater, and more 

 useful in other ways, that the intercepts are expressed by 

 fractions in which the numerator is simple unity — to be 

 understood, and therefore not to be written. 



30. The first step is to join eel, and produce the line (in 

 this case towards the left hand) until it cuts the continuation 

 of the line upon which amh stand. If the work has been 

 done well, the point of intersection thus marked will be the 

 projection of the pole of ??i, which lies in the quadrant of the 

 sphere of projection to the left of that represented by the 

 gnomonogram. It is well to sign this pole M. It forms a 

 radiant point for all the poles in zone with corresponding 

 position on a and h, as, for example, 102, 012; 201, 021; 

 506, 056; 605, 065, and so on; so that lines drawn from M 

 through either of each pair gives the position of the other, as 

 well as of any other pole in zone with them. 



31. Next join Mc, and join also Id, and produce the line to 

 meet mc; the point so determined is the pole of the unit 

 octahedron (answering to o) in the same quadrant as M. 

 Join also am, and produce the line until it meets the pro- 

 longation of ca ; the point of intersection is d in the quadrant 

 below, and is again a useful radiant point for many zones of 

 importance. Another valuable radiant point is found by 

 joining dm, and prolonging the line below until that line, the 

 gnomonogram, cuts the continuation of ch ; the point so deter- 

 mined is the e of the quadrant below. We might usefully 

 refer to these last two points d and e by italic capitals, D and 

 E respectively. In practice, having laid down the exact posi- 

 tion of D and E, fine needles are fixed at either of them, say 

 at D first, and from this as a centre draw lines through any 

 pole determined on ah to cut cb, on which the corresponding 

 pole will thus be found. Thus a line from D through 310 

 marks on he the position of 013, and of all the poles in that 



