Frc' 1' Gnomo"t' "'"'"'r?31."j t}i:4";\':#*ttli'.k/t' and' i/t ror the iaces TWO-CIRCLE CALCULATION IN THE HEXAGONAL SYSTEM

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1 TWO-CIRCLE CALCULATION IN THE HEXAGONAL SYSTEM A. L. PnsoNS, Toronto [Iniersity, Toronto, Cnd. Reently the writer hd osion to demonstrte to lss the oordintes tht must be used to lote the indies ZtSt in gnomoni projetion of hexgonl rystl, showing tht this ment mesuring two fundmentl units on the positive end of the o1 xis, one unit to the right prllel with the z xis, nd three units in negtive diretion prllel to the 3 o.xis, with the height of the plne of projetion bove Fr' 1' Gnomo"t' "'"'"'r?31."j t}i:4";\':#*ttli'.k/t' nd' i/t ror the ies the entre of the sphere of projetion s the lst index of unity. When this ws done he notied tht he hd loted point whih in the terms of Goldshmidt's ppo, qpo ws to be indited s 41 whih is supposed to hve lhe Brvis indies Further observtion showed tht in every se there is n intimte rel-

2 582 TH E A MEMCAN MINERALOGIST tion between Goldshmidt's o-ordintes ppo nd q?o nd the Brvis indies hkil, whih is quite difierent from wht he nd ll his pupils hve thought, for when these indies re given their proper signs nd when h)kti., f o: h-i, Po where h nd I hve difierent signs nd Afo: k-i, Po nd t -2^ y0-3l, In the ompnying gnomoni projetion (Figure 1) of some of the ommon forms found on beryl grphil determintion oi po hs been mde nd the indies for the IeslT2l,TZTz, t101, nd 2131 re indited s o-ordintes, with the proper signs, in hevy lines, giving in hhi eh se; Po,, p6,nd; po. IIT The following tble shows the Brvis symbol for eh of these fes together with the proper rrngement of the indies to give the oordintes on the three xes. Brvis Indies Ar 02 Os Co-ordintes At 02 oz Tr2r +++ lpotpo2lo ltt 1212khi121 t, t t 2'Po TPo,Po 1101Lhi I I t llo L?o opo 2rSrLih i 1 t 2Po r[o 1Po

3 JOURNAL MINERALOGICAL SOCIETY OF AMERICA 583 Insmuh s we must onsider positive nd negtive indies, nd positive nd negtive xes, omplete lultion for hexgonl rystl should determine the sign nd position of ll three of the oordintes for eh nd every fe. To omplish this, fter the unit first or seond order pyrmid hs been seleted by mens of the informtion gined from gnomoni projetion, ttention must be pid not only to 9 nd p but to the setnt in whih the fe lies nd the positive or negtive hrter of sin 60o whih ppers s the divisor in seuring?po nd. QPo. In oider to obtin the proper signs nd positions for the o-ordintes, ngle p should be mesured lokwise nd 60o-p ounterlokwise, in whih tseg my be greter thn 30o but not greter thn 600. In the lultions under these onditions sin,p, sin 60o-9 nd tn p re lwys positive but the divisor in the fundmentl equtions, sin 60o, is positive when the hypothenuse is prllel with the positive end of n o xis, nl negtive when the hypothenuse is prllel with the negtive end of n o xis. with the introdution of positive nd negtive signs for the indies we find tht. k-i E-t' + h- lt'-i' qfo:iy'o,ndqpo:. po,ndfpo:,?o,ndffo:, fo' llit Our generl formule then beome for pyrmidl fes sin (60"-f) p * sin 60" 'tn p (tl1-(td t o bo:- * sin 60o I Do nd prisms sin (60'-,p) t:'" j0" : tp _Gh)-(+1.). sin<p Tq (+e)-(+i) T sin 60o The prism 1120 is pprently n nomlous se for the vlues of I nd g re 1 nd 0, but where either p or q is zero the other Goldshmidt o-ordinte for pyrmidl fe is (3/2) h. po/l, nd inthe se of this

4 584 THE AMERICAN MINEKALOGIST prism the o-ordinte 1/0 must be looked upon s (3/2)h/0 from whih the indies 1120 n be redily dedued. To illustrte the hnge involved, the mthemtir rultions for single fe of eh of the following forms on beryl (1011), (llzd, (llrd QI3l), (1010) nd (2130) ur" glir"r, in the foilowing tble, using the vlues of 9 nd p tht orrespond to the positions of these forms s determined by the Kokshrov xil rtios. In the olumn devoted to the ngle rp the ngle Z, is given. Z, is the ngle mesured from the zero position of the projetion while,p is the ngle obtined by subtrting n integrl multiple of 60o from V,. with this hnge in the mgnitude of ngle 9,nd with ppo rwysgreter thn qpo s given in the formule bove, the position of p?o ni qps in tn".ut."tu,to" O. nu"*"ubt". Brvis Symbol V ' e log sin 60o - 9 p log tn p log!!o log sin p logqpo h-i!po:, po k-i APo:, Po Pu Po,:!o^ 2' , , 29"57' s I2I2 0.00, 0"00, 26"31' 94s s "00, TT2t 0.00, 44"56' s "53.4' "53.4' 56.44' o5219s s "53.4' "53.4' 90"00, p -p h-i. rogq -rq, -: _ k-, "00, 1r , 90"00/ ,p q p _h-i -q h-i

5 JOURNAL MINEKALOGICAL SOCIETY OF AMEMCA 585 The lultion gives no new determintions of r:o but by using the ngles tht hve been determined by others onfirms the vlue,obtined by Kokshrov. For omprison the vlues re given below: Kokshrov : Prsons (Arithmeti) Goldshmidt : \/ 3 There hs been muh onfusion in the tretment of hexgonl rystls by the two irle method beuse in either the Gr or G2 positions multiplition or division by {3 or multiple thereof is omitted. This is very evident when the Goldshmidt symbols re the sme s those of other workers for in tht se the vlue of hs been multiplied by V3 to obtin the Goldshmidt. Where Goldshmidt's is the sme s tht of other workers this is not so evident but it shows in his trnsformtion formul b-2q b- q pq(gz):--(g,) whih when multiplied bv t/s beomes b-20 b-o P1/ 3' q\/ 3(G) :-'- (G')' '\/3 \/3 While the mthemtil lultion of the rystllogrphi onstnts in the hexgonl system is quite simple, the theory underlying the eqution for plne in this system in terms of its interepts, in terms of the o-ordintes for the line whih is perpendiulr to the plne, nd in terms of its diretionl osines, is deidedly omplex. It is extremely esy to beome onfused s is illustrted by tbultion of the vlues of f s given by Goldshmidt in his Winkeltbellen nd by Dn in the sixth edition of his System of Minerlogy, s shown in the tble below where single exmple is given for eh type of differene. It thus beomes evident tht omplete revision of the hexgonl system is neessry for redy interprettion of two-irle mesurements, for there is lk of greement in the xil rtio or the symbols or both in every hexgonl minerl. In spite of the onfusion tht hs risen from the two orienttions Gr nd Gz the ft remins tht in his Gr position Goldshmidt disovered the simplest method, lthough on n inorret ssumption, of determin-

6 TH E A M EKICA N M I N ERA LOGI ST ing grphilly the Brvis indies of ny fe in the rystl fter the unit pyrmid 1011 hd been seleted. In his Gr position he hd the simplest wy of determining grphilly?o nd. Unfortuntely fter finding tht in every other system his o-ordintes ould redily be onverted into :b: ndhklhe ould not believe tht when he hd the Dn Goldshmidt Bery _ -X\/3 3, 1010 Dn: 1010 Goldshmidt Mirosommite _ -X2t/s e -Gr 10T0 Dn: 10T0 Goldshmidt Breithuptite - -xz\/3 -Gr 10T0 Dn: 10f0 Goldshmidt Clite L -Gz 10T0 Dn: 1120 Goldshmidt Dioptse -x2 -Gz 1010 Dn: 1120 Goldshmidt Millerite -x3 -Gz 1010 Dn: 1120 Goldshmidt Covellite o -x1.5 - G,* & 1010 Dn: 1120 Goldshmidt * Should be Gz. indies determined by others his rtio of,: o ws inorret nd tht when he epted the stndrd rtio of : o his indies were wrong. Lrgely on ount of the onfusion used by the Gr nd G2 positions there hs been onsiderble opposition on the prt of some ble rystllogrphers to the two-irle method. It is hoped tht this pper my help to lessen this opposition, nd serve s preliminry to omplete revision of the ngle tbles for hexgonl minerls. After this pper hd been sent in for publition muh simpler method of lulting indies nd xil rtios ws found. After seleting the first order unit pyrmid, mking I not greter thn 30' with h> k> i, then

7 TOURNAL MINER-ALOGICAL SOCIETY OF AMERICA Jd/ COS g'tn,:+ + os (60"-9)*" r::'+ os (60'f9) O",:+ + nd for prisms os e: os (60'-p) :os (60of9) : h: k:i. This will be tken up in greter detil in lter pper.