Mathema- 

 tical 

 Theory. 



Plate 

 CCXXIV. 

 Fi?. 9, 10. 



CRYSTALLOGRAPHY. 



461 



and join the points;?, r. The angle f p r will be equal 

 to one half of the angle, which gives the incidence of 

 b' ay upony a y. Draw y e, y h, perpendicular the one 

 to b' o, the other to a b', then join the points e, h. The 

 angle y he will be the half of that which measures the 

 incidence b' o y on the adjacent face a b. We will 

 therefore obtain the two incidences wanted, if we find 

 the ratio between the sine f r and the cosine p r 

 of the angle f p r, and the ratio between the sine y e 

 and the cosine e h of the angle y h e. 



Having produced g n (Fig. 9.) till it meets a m, we 

 may suppose, for the greater simplicity, that the plane 

 b' y f o (Fig. 10.) is of the same height as gar (Fig. 9.), 

 so that a o (Fig. 10.) z= an (Fig. 9.) In that case, we 

 shall have likewiseyo or b'o (Fig. 10.) = gn (Fig. 9.), 

 andy o (Fig. 10.) = n x (Fig. 9-) 



Let us obtain separately /' r and p r. 



1. For f r. It is evident thsXf r is the half of the 

 line which joins the points b, f ; and since these points 



re conceived to be of the same height as g x, they co- 

 *cide with the points b,f (Fig. 7. ) Hence it follows 

 at /'r (Fig. 10.)=6 C (Fig.7.)=£. 



2. For pr. The triangles a o y, rpy (Fig. 10.) are 

 similar, from their position, and the equality of the an- 

 gles a oy, rpy, both right angles. Hence a y : ao: : 

 y r : pr. Let us obtain values of ay, a o, and y r. 



ay — </(yo) l + (aoy ; yo — nx (Fig. 9.) 



a u : m u : : a n : n x, or 



2ra — 1 , M-f-1 . .-_ 



__— - x/gp^ — 3^ : -^—Vfg*:: ^9 p>— 3 g* 



n+1 



nx = - — - — - 

 2n — 1 



n s ° 



ao = -f V9 p* — 3g". Let us, for greater simplicity, 

 denote the value of the axis Vdp" 1 — 3 g 2 by a, we will 



have ay= J(jL±Ly* g+ ^ a >. 



We still want y r. The angle f' or being an angle 

 of 60°, and the angle/r s of 90 °, or = If o = i v / fg' 



y r zz y o — or =^t- 1 \/Yg^—W^ = 



3 , 



2 n — 1 v 3 s 



Therefore the proportion ay : ao:: yr: pr becomes 



1 



2n 



~^T¥-- 



pr = 



2n — 1 



\Ti 



v V2ra— 1/ 



f §* 



Hence f r:pr:: g: — 



1 

 2n -TJ 



V- 



A^~^ 



"+i" 2 



^(^^V+f*.-^ 



n — l ^t ' 



>/(«+ 1 >' 4 g r + (SS » — 1). j. a i. fc 



cosine XY£^tf-~ the *» V * «* the 

 >r * e. We have y e= l/^)*__ (oe)3 . y , = 

 > as before determined. Further, on ac- 



\2»-lJ T < 



count of e j/= 60°, and _y c 0= 90°, we hav e oe=iyo. M a them»- 



tical 

 Theory. 



Hence 3/ e = </|^)*= V (^Tl) 2 



2. For c A. The similar triangles b' a a, b' h e, give 

 ab' : ao::b' e : eh. Now, 



ao = V'ia 2 



b' e~ b' o — c e = V'-Hr?" (—Hj~- — ] a/T~ZT _ 



(■-<. 



M - 2 >Kv= 3 



4n — 2 7 / r ! s 4k — 2 

 Therefore the proportion ab' : ao :: V e: eh, be- 



comes 



7 - . — 3n — 3 , . 



*$? + $ « s : */$ *> '• : 4]7^r 2 V* g 2 : e &. 



Therefore 



3n — 3 , 



e h = 



?« — 1 . 



V|g 2 +^a^ * / ?g 2 -K« 2 



«— J_ /— 



Therefore 



!«2g'2 



?!— 1 / 



_3a2g2 

 12g?^s 



::»+!: 



(»+ 1 )^12g^ + 9p 2 ~3^: (»— 1 )V / 27ie=9^i.. (b+ 1} 

 V^-I-jd 2 : (?i — 1) V3pi — g\ 



There is a variety of calcareous spar, whose summits 

 have each six faces resulting from a decrement by three 

 ranges upon the superior edges of the nucleus, and 

 which combine with other intermediate faces, of which 

 we do not take any notice. To apply the above for- 

 mula? to this variety of crystal, we must make k = 3, 

 g=*/3, p-^% 



When these values are substituted, we get 



1. b'r:pr;: */S9: a/3, which gives 79° 35' 47" for 

 the angle f'p r, and 159° 11' 34" for the incidence of 

 b'am on f' 'am. 



2. y e : ek : : ^/20 : a/3, which gives 68° 49' 43" for 

 the angle yhe,_ and 137° 39' 26" for the incidence of 

 b'am on the adjacent face a b'. 



Let us examine whether there be a possible law of p 1ATE 

 decrement for the dodecahedron with isosceles triangu- CCXXIV. 

 lar faces, or composed of two right pyramids applied F'g- 9. 



-- to base. In that case, y o=b'o. Hence likewise 



*/fgK This 



n x (Fig. 9.) =gn, or ^~J ^fg' 



gives n—2. Hence the form is possible by means of 

 a decrement by two ranges. 



In proportion as the edge amis elevated by its lower 

 extremity, by making the angles larger with the axis 

 ao (Fig. 10.), the angle which b'am makes with the Fi £' la 

 face adjacent to a b' increases in size, and there is a 

 term when these two faces are in the same plane. The 

 secondary crystal becomes then a rhomboid, the oblique 

 diagonals of which coincide with the edges a b\ a f 

 &c. J ' 



To find the law which produces this rhomboid, let 

 it be observed, in the first place, that when it takes 



