470 



(1.) For up. 



CRYSTALLOGRAPHY. 



The triangles pay,psd give us ap : 

 But as-zzVa* ; ds=V& 



Matherfta- 



Theory, <?.?/ '■ ■ «#+«« ■ ds. But as=Va z ; d s=Vg--j r p-. 

 v— ~ v ^»- To obtain an expression for ay, we have (from the si- 

 milar triangles'*/ &^ day) dk ifk :: da : ay. Or, 



£ ; Vg 2 +i» ■'■ ~ 



P 



Ipn x y 

 x — y 

 Hence the proportion ap 



x—y 



becomes a p 



Plate 

 CCXXIV. 



tig. 23. 



ii x y 



ai)— X ~^Vg'-\-p l . 

 nxy * 



ay : : ap-\-a s : ds 

 V / g ri 4-p * : : «p+ -/a' 2 : Vg 1 -f.p*. 



Hence we obtain a p~(- — -)apA -A/a'jandfi 



1 \n x yj nx y 



nally, ap — - 



x—y ,/-? 



Va- 



-y 



it x y — x 

 (2.) For dp. We have dp— V* (pi 



jr^-\-(dr)h But 



(d r)' 1 — tgi. We have only therefore to obtain ;j r, in 

 order to have an expression. for dp. 



Now przzap + arz=( ■ — - !- t ) V 'a 2 = 



1 ' ' \ra jr y — x -\-y J 



2 n x y-lf-x — y 



3 n xy — 3x-\-3y 



Therefore dp= ///SS^EpS 

 ' v \3iixy—3x-\-3yJ 



+ t£ ? - 0f 



consequence, the proportion ap 

 come 



al : : dp : aV will be- 



nxy—x + y 

 : \^-i 



■ » i > 



Vc 



v \3hxy— 3x+3y) 



x ~y . \fJaTg*. ' 



" 2 +T< 



Thus we get a l—- 



nxy — x -f- y 



J Jnxy + x-y_y 



/ \3nxy—3x+3yJ ^ TS 



Resuming the original proportion om : on :: ad : al, 

 and substituting the values thus found, the proportion 

 becomes 



x — y 



yx+yy . 



x—y ■' 



on 



nxy — x -f- y 



V4 



/( Znxy + x—y \ , 4ff2 

 V \^xl,-3x + 3y) ' Bt,& 



This proportion gives us 



a: — ,?/ 



nxy—X-\-y 



VI 



^l 



] nX y—3x + 3yJ T 3 to 

 Therefore /jo : on : : g : the preceding fraction. 

 And getting rid of the denominator of that fraction, 

 and dividing by g, we finally obtain bo : on : : 



^5(n:ry — x-\-y)f nxy — x-\-y i 



V \( L 2nxy-{ x—y) 2 a' 4 -f (nxy— x-\-yi)4g* ^/a 2 :(x-\-y) t/ a 1 . 



Nothing more would be necessary than this ratio, 

 ■and the law of decrement, to determine the accurate fi- 

 gure of those kind of crystals which we are consider- 

 ing. But an example may be necessary to make the 

 method obvious to beginners. 



Let us suppose that HX (Fig. 23.) represents that 

 variety of carbonate of lime called paradoxal by Hauy. 

 If we attempt to divide this crystal mechanically, ' we 

 find that each section, such as s2£, commences from one 

 of the shortest edges, and rises in such a manner that 

 the angle S contiguous to the edge QX is about 45°. 



This being understood,, draw the rhomb ub%u (Fig. 



25.) similar to the primitive rhomb, and, from the point Mithema- 

 2, draw h, S£, each of which makes with the diagonal T1 " cal 

 a d an angle of 22^°. It is evident that these two lines s .„- eor ^./ 

 represent the position of the two decreasing edges of p LATE 

 the same plate, so that h i and b § are to each other as CCXXIV. 

 the number of lengths of molecules subtracted 'from the Fig- 25. 

 two sides of the angle on which the decrement takes 

 place. But, on comparing these two lines, we find that 

 b ? is apparently double of b i. Hence we conclude, 

 that in the preceding formula x—Z and y=l. As we 

 are ignorant of the value of n, we give it at first the 

 most simple value, making nz=\. 'We know already 

 that a z z=9 and g l —3. Substituting these values in the 

 preceding formula, we obtain ho: on: : ^29 : V / 27- 

 This gives 92° 3' 10" for the incidence of CXQ on NXQ- 

 But this measure agreeing with observation, we con- 

 clude, that the decrement really takes place by the sub- 

 traction of one range of double molecules. 



To determine the incidence of CXB or CXQ, we 

 must have an expression for h r. (Fig. 24.) 



But h r=zic x—a x-\-a uzzap-^-px-^-a u. Now ap— 



— — — J- */ a z : and awrz-rVV. We have only there- 



n x y — x -\-y 



fore to find p x. 



The similar triangles ba x, lap give us a I: hi: : ap: 



p x. But b l-=in zz=.o n — o z=zo n — \ a I. So that the 



proportion becomes alio n- 



found formerly, that 



\l: : ap :.px. But we 



.r — ■?/ 



alzz 



n x y- 



■x -j-;/ 3 b 



A 



2n xy + x-y , 

 3 n x y — 3 x -j- 3 y 





and 



T 3 to 



x+y 



11 x y—*-x 4- u 

 11=. -1 ' J 



Vi 



a*g> 



If 2nx 

 * \ 3 n x i 



2 11 x y -\- x — y 



ly—s.x+syj ^ 3to 



From these values we see that a I and o n are equal, 

 the first to (x — y) a/4> or to 2.r — 2y; the second to 

 x 4- y; multiplied each by the same fraction. We may 

 therefore state the proportion as follows : 



2x—2y:x-\-y—x-\-y:: 



-y 



''Xy—x+y 



11 



This gives us px-=. — - — \/ a 1 



° 2 nxy-^-x-\-y 



Therefore we have at last 



y 



hr= (-^ 



\n x y — x- 



+ 



-fc 



Va z ■• p x. 



*}*#« 



•y—x+y nxy— x+y 

 nxy + Zx+y ^— 



K»xy—x+y) 



Let us now conceive a plane, which, passing through 

 some point of the edge CX (Fig. 23.) is perpendicular Pl4 J e , 

 to the axis FIX. Let qxv, b xv ( Fig. 1 . ) be the two ccxxv> 

 portions of the faces BXC, QXC (Fig. 23.) intercepted Flg * lm 

 by this plane. For the greater simplicity, let us sup- 

 pose that the plane passes at such a distance from the 

 summit, that the part x r ( Fig. 1 . ) which it intercepts 

 on the axis, is equal to p r (Fig. 24.) In that case, 

 q r (Fig. 1.) or b r will be equal to d r (Fig. 24.) 



Having drawn;; t (Fig. 24.) parallel to qx, produce 

 d r till it meet pt. The line vr (Fig. 1.) will be 

 equal to r t (Fig. 24. ) 



We have then x r (Fig. 1.) : 



2nxy + x~y_ v - 

 3n«y—3x-^-3^/ 



