78 



Mr DE morgan ON THE GENERAL EQUATION OF 



Make the following abbreviations, to which, for facility of reference, 

 are annexed those which will afterwards appear in treating the general 

 equation of the surface, 



aaf + bif-ir cz" + 2ai/z + 2bzx + 2cxy + 2aa? + 2% + 2cs! +/ = (3), 



the co-ordinates of the center of which call X, Y, and Z. Throughout 

 this paper, all subscript indices indicate the dimension of the quantity 

 signified, in terms of the coefficients of (3) : 



p =/3'7"-/3"7', 

 / = /3"7-/37", 

 p"-=^y' -d'y. 



y'a"-y"a'. 



^1 ff If 



q = y a—ya , 

 q"= 7«' -7'«' 



r =a'/3"-a"/3', 

 t" = a"(i-a(i".. 

 r"=«/3' -a' 13. 



(4), 



a^^= be — a', 

 b^, = ca- b% 

 c„ — ab — (?. 



tto = sin' I, 

 b^= sin'*;., 

 Co = sin^ ^. 



.(5), 



a^= 6 + c — 2acos^, 

 b,= c + a — 2b cos rj, 

 Cf = a + b — 2c cos ^, 



l^^ —bc-aa, I, = &cos^+ccos»7-« — acos^, \ = cos v cos ^— cos ^, 

 m, = ca — bb, mj=: ccos^ + acos^—b -bcosrj, 7»o= cos^cosf — cos ^...(6), 

 91^1 =■ ab— cc, n,=a cos j? + 6 cos ^—c~c cos ^, «„ = cos ? cos tj — cos ^. 



=»?„«o-ao 4-^ cos A 



= «o lo-boMo-T- COSri\...(7), 



.= /o««o-Co«o-T-cosg 



(8), 



(9), 



Fo=l+2cos^cos.jcos^-cos^^-cos'»j-cos'^' 





V, = aai + bb^-\-cCa + 2ala + 2bma + 2cnf, 



Vi = a„ + b^, + c„ + 2/,, cos '^■\-2m„ cos r\-\-<iLn„ cos ^ 



= */;C//-/,; ^«1 



.(10), 



Vz~abc^2(ibc — a<i — b¥ — c& \ =.c„a„—m,J-^b\ 



r; = a,,a' + J,,6^ + c,,c^ + 2/,,6c + 2»?,,ca + 2w,,«F (11), 



= m„n„-a„l„ -=ra 



■.nj„ -b„m„^b 



= l„m„-c„n„ ^c 



W=-^ +/= aX+ 6 F+cZ+/ 



.(12). 



