468 



M.uhema- And simplifying 



f2?i+ly 



CRYSTALLOGRAPHY. 



ig z 



/2 b 

 V n- 



(Sf-g*)::g' 



3p'*-g* 



Plate 

 CCXXIV. 



Fig. 21. 



Taking the products of the extremes and means, get- 

 ting rid of the denominator (b — 1)'-, and transposing, 

 we obtain tliis equation. 



— (n—1) 2 12 gtp". 



Developing the quantities (« — l) 2 , (2 »-|-l)% and re- 

 ducing, we get (2n+iy p 2 g' 2 + (1 — 4»)g 4 g' 2 = 

 (« — l) s 4g* p' J , which gives us this proportion, 



g' : p' : : \/(«— 1) 4g 2 : -/( 2 n -(- 1 ) J p* +( 1— 4 n)g r 

 Let g=*/3, £>=a/2, as in calcareous spar ; and let us 

 suppose n—Jr. We will get 



g':p'::^3:\fvT. 

 Such is the ratio of the semidiameters in the variety 

 of calcareous spar called contrasting by Hauy. 



If in the formula a pz= \^Q p 2 — 3 g z we make 



h=1, we get a p—^9p z — 3g 2 , as we did for the de- 

 crements on the inferior edges, with this difference, 

 that the vertical faces result from a decrement by two 

 ranges. This case holds in the regular six-sided prism 

 of calcareous spar. 



Let us now proceed to the decrements which take 

 place in height on the same angle. Let o u (Fig. 21.) 

 be one of the oblique diagonals of the secondary rhom- 

 boid, and o p the adjacent edge, from which we see 

 that the first of these lines corresponds with the edge 

 ds of the nucleus, and the second with the oblique dia- 

 gonal a d. Let d h e be the measuring triangle, in which 

 dh: e h::p: nV 'g^+p 2 - Let us, in the first place, find 

 an expression for a p. 



Produce a d to I. The triangles pal, psg, being 

 similar, we havegs: as-{-a p: : al: ap. But 

 gs=2p_ 



as=^9p 2 — 3g z . 

 We must find the value of a I. The similar trian- 

 gles gal, d he, give nseh: dh :: ga:al, or n Vg* -+-^ 2 



^g l +P 2 



:p 



al=^ 



n 



The above proportion gs : as-\-a p 

 therefore 2 p -. Vdp 2 — Sg l + ap : : - 

 1 



:al: ap becomes 

 : a p. Hence we 



have ap=- t*Ah 



' 2 b — 1 * 



conclude, that dr : ur : 



— 3 g*=u s. From this we may 

 . /2n+2\ . 



■■■■^^•■{wn^r^p'- 3 ^ 



Let us ascertain the ratio between the diagonals g' 

 and p' of the secondary rhomboid. 



The semiperpendicular on the axis of this rhomboid 

 is to the third of that axis as d r : u r. 



Therefore 



^¥¥- 6^=3 ^9^-3g':: V^: f</9p"-g' 2 . 



Simplifying and getting rid of the radical signs, this 



proportion becomes ( 2 n— 1 ) * 4 g* : ( 2 b -f- 2 ) 2 ( 3p* g 2 ) 



: : g' 2 : 3 p' 2 — g'K 



Taking the products of the extremes and means, 

 transposing and dividing by two, we get this equation ; 

 (« + 1) 1 3pV 2 + (2« — l)Vg' 2 -(«-f.l) 2 g'g"= 

 (2«~l)'3g7 ,J . ' 6 6 



-1)*, (2 « + !)*, and 



Developing the quantities (2w 

 reducing 

 (re+1 y 3 p 1 g' 2 + (3 »*— 6 b) g* g' 2 =(2 b— 1)2 3 g z p\ 



Hence we obtain this proportion, 

 g':p':: */(2n—iy3g 1 : \/(b+1) x 3p z + (3/? 2 — 6n)g*. 



There is a variety of calcareous spar so nearly cubic, 

 that it was distinguished by the epithet. It is not, 

 however, an exact cube, the faces differing about two 

 degrees from being rectangular. Let us see how we 

 may determine the law of decrement which takes place 

 in this variety, knowing the angles of the faces. 



The solid being a little more acute than a cube, it 

 follows that the ratio dr: ur, which results from the law 

 that produces it, must be a little greater than that of 

 1 : */2 which exists in the cube. It must at the same 

 time be commensurable. But if we substitute succes- 

 sively for the ratio 1 : */2, the equal ratios a/2 : ^/4, 

 \/3 : a/6, a/4 : a/8, we perceive that it is sufficient, in 

 this last expression, to increase the number 8 by unity, 

 changing it into a/4 : a/9, to have the commensurable 

 ratio 2 : 3, which will be a little greater than the for- 

 mer. Let us therefore try this ratio, and suppose d r 

 : nr : : 2 : 3, or 



^¥i ■■ |SJ ^9p*-3g* : : 2 : 3. 

 And because g= a/3, and p— a/2, we have 



Mathema- 

 tical 

 Theory. 



/2_b + 2\ 

 Wn~^3) 



^6 b— 3) 



From this we obtain 6n — 3 = 2n + 2 and « = {. 

 Therefore, since n expresses the number of ranges sub- 

 tracted in height, the decrement takes place by 4 ran- 

 ges in breadth and 5 in height. 



Let us ascertain, according to the same hypothesis, 

 the ratio between the semidiameters g' and p'. We 



have had alr eady 



g' : p' : : V(2n— l) 2 3g 2 : V(n + 1 ) 2 3p* + (3 b 2 — 6 - B)g 2 . 

 And makin g n= {, g =*/3, p=*/2 , we have _ 

 g' :p' : : ^.3.3 : V^.3.2— ^.3 _: : /l2 : Vl3. 



This gives us the smallest inclination of the faces 

 87° 47' 45", which is conformable to observation. 



Let us now enquire whether, among all the possible 

 secondary rhomboids, there be one which is similar to 

 the nucleus. In such a case g' : p' : : g : p. Substitu- 

 ting the second ratio for the first in the proportion give n 

 ab ove, we will have g : p : ; -/(2 b— l) 2 3g 2 : 

 </(« -l. i) 2 3 f 4. (3 n 2 — 6 n)g 2 . From this, getting 

 quit of the radical signs and developing the quantities 

 (2b — l) 2 , (n+iy, we obtain 



S H y + Qnp* + 3p s + 3B 2 g 2 — 6«g 2 = 1 2b* P 2 — 1 2np 2 + 3p\ 

 And reducing, n 2 (3p 2 — g*)=n(6p 2 — 2g 2 ). This gives 

 us b=2. Heace we learn that the result in question 

 may take place in consequence of a decrement by two 

 ranges in height. 



If we compare the ratio dr : pr 



iJLT— */9 p 2 — 3 g*, with the ratio dr : ur (Fig. 21.), Piatb 



2m+2 



- */9p2 — 3g l , ratios the first of which 



3 b — 3 

 or Vf| . 



is for decrements in breadth, and the second for decre- 

 ments in height ; we find that they differ only by the 

 quantity, which in the second term multiplies the ex- 



. 2 b -j- 1 

 pression for the axis, and which in the first is -■> 



2 b-4-2 

 and in the second ^ — ^~. Let us change n in this last 

 6 b — 3 



expression into »', and let us equate the two, making 



CCXXIV. 



Fig. 21. 



