DETERMINATION OF OPTICAL AXES. 379 



If we have observed the extinction angles and determined the position of 



the section (S and therefore Cm and Sm), we can find a = <^ Cy and V = 



^ AB from (12) and (13). If we have determined the relative retardation 



/ — «" 

 / / , (14) maybe used as a check; or we can determine o. directly from 



(14) and use one of the others to determine V. We have then the remark- 

 able result that a random section of an augite twin may be enough to de- 

 termine both a and 2 V, unless it occurs in certain zones. Positions of the 

 section near these zones will be unfavorable for finding one or the other. 

 The case of figure 2 is unfavorable for finding 2 V, but a must be near 45°. 



Solution in the Case in which the Section must be in a given Zone : Application 



to Tourmaline. 



§ 9. The last case we take up is the common one in which we know at 

 least approximately in what zone the section lies. Such knowledge is given 

 us v/hen twins have symmetrical extinctions, when long, slender prisms lie 

 wholly in the thin section, when microliths are wholly in focus at the same 

 time, etc. Then (p and 6, the coordinates of position of the section, are con- 

 nected by a relation of the form cos ^ h cot 0. We can, moreover, take 

 the plane of projection in or perpendicular to the given zone, and thus make 

 ip ov 6 {)° or 90° ; so that our substitution in previous equations is quite 

 simple. If we know, for example, from the extinction of hornblende or 

 augite twins parallel to the twinning trace, that the section is in the zone of 

 the orthodiagonal (6), equation (3) of § 7 becomes — 



sin 6 = — .., , and B^O; i. e., n + m = 0. 



Hence — 



m 

 sm 6 = — T • 

 cot a 



For the case of long prisms, we may assume that they lie at right angles to 



a face perpendicular to the prism axis, and that = 0. Therefore, from 



p 

 equation (5) of § 6, tan <p = Y)' ^^d — 



A cos (p -\- C sin cr =^ — E sin cr cos (f -\- F. (1) 



Now, if tan <p= — Bjl), cos <p = — Dj^B' -[- D"), and sin c = B\{W -f B") ; 

 so that we have from (1) — 



_ _GB_ = EBP +J., .2) 



