H. S. Uhler — Deviation Produced by Pi'isms. 413 



locus t the tangents to the former are perpendicular to the 77, 

 coordinate axis. For, at these points cos y 1 = cote tan |/3 so 

 that the radical symbolized by P 2 on page 403 vanishes. 



Consequently, the expression for — becomes infinite. The 



theoretical values of the most important quantities pertaining 

 to figure 6 are collected in Table II. 



Table II. 









D, lt 











ft 



7i 



£>i 



discrete 

 minima 



T)x for D,„ 



D 2 



Vi for D 2 



a; 



9° 42' 0" 



90° 



27° 25' 27" 



10° 48' 33" 



84° 31' 0" 



10° 50' 53" 



84° 33' 23" 



b; 



20 



< ( 



36 7 50 



22 2 42 |78 15 37 



22 23 22 



78 37 49 



c ; 



43 37 58 



< < 



49 6 5 



45 22 46 57 47 41 



49 6 5 



63 24 48 



d: 



60 47 39 



< i 



58 24 43 58 24 43 







68 54 13 



49 52 



e ; 



77 42 



(i 



73 51 46 l 





89 4 



25 46 20 



The discussion of the mathematical properties of the curves 

 m and t has been deferred to this place because the relevancy 

 of these loci might not have been evident before the presenta- 

 tion of figure 6. The parametric equations of the curve m are 

 given by formula? (27) and (28), namely, 



sin \D 

 cos -q l 



\ cot c sin /3, 

 I- cot c tan ft. 



Elimination of the parameter 13 leads to 



sin 4 D — 



cos rj 1 



Vl + 4 tan 2 c cos 2 rj 1 



(56) 



In general, the rationalized equation of the curve 111 may be 

 written 



tan 2 r jl —cot 2 \D + 4 tan 2 c = (57) 



AT dD — 2 sin w, 



Now, - — = n 



ci Vi (l + 4 tan 2 c cos 2 77J V sin 2 r) 1 + 4 tan 2 c cos 2 rj 1 

 hence, when rj 1 equals and \ir, — — becomes and —2 respec- 

 tively. Thus, the locus of discrete minima of fig. 6 intersects 

 the rj^ndD coordinate axes at angles of 116° 33 r 54" and 90°, 

 in the order named. Both angles are independent of c. 



The equations of the curve t are given by formula? (15) and 

 (40), namely, 



