462 keport— 1878. 



Again let CV, C,V, be two consecutive tangents to the focal conic F of the 

 bicircular quartic, and let OAB, OA'B', he two perpendiculars to CV, CV' from 

 the centre of J. Now, if CV, CV intersect the generating circle in RR', it is 

 evident from geometrical considerations that 



RR' = £( BB '-AA'), 



hut RR' = pd& (dd heing the angle hetween two consecutive tangents to F). 

 Hence we have 



ds'-ds = 2pdd, 



and from the three other focal conies and circles of inversion we get three other 

 equations, viz. : — 



ds'-ds"=2 P ' ay 



ds"-ds'" = 2p" d6" 

 ds'"-ds =2p'" dff". 



Hence we have four equations for determining any of the four quantities ds, ds', 

 ds", ds"', in terms of the four quantities pdd, p'dd,' p"d8," p'"d6,'" each of which 

 is separately expressible as the differential of an elliptic integral. 



4. On tlie Eighteen Co-ordinates of a Conic in Space. 

 By Wm. Spottiswoode, F.B.8., Sfc, fyc, President. 



The six co-ordinates of a right line may he derived from the equations of the 

 two planes of which it is the intersection, by eliminating each of the co-ordinates 

 in succession. We may proceed in like manner with the equations of a conic in 

 space. Let the equations be — • 



(a, b, c, d, f, g, h, I, m, n) (x, y, z, t) 2 = o. 



ax + /% + yz + bt = o. 



Then eliminating x, y, z, t, in turn, we should obtain four forms which may be 



written thus : — 



(CC, BB, FF, BF, OF, BC) (y,z,ty = 0, 

 (AA, CC,GG, CG, AG, OA)'(s, x,tf = Q, 

 (BB, AA, HH, AH, BH, AB) (x,y, 2 = 0, 

 (FF, GG, HH, GH, HF, FG) (x, y, z) 2 = 0. 



The 18 quantities A A, BB, •• , the values of which are easily calculated, are " the 

 18 co-ordinates of a conic in space." 



If we represent the three equations — 



AAa + AB/3 + ACy = 0, 

 BAa + BB/3 + BGy = 0, 

 OAa + CB/3 + CCy = 0, 

 by the formula 



(A,B,C)(a,/3,y)=0. 



A,B,C 



The 18 co-ordinates will satisfy the 12 equations — 



(A,H,G)(o,/3, r ) = 0, (H,B,F) (a,5, y ) = 0, (G,F,C) (a,^,8) = 0. 

 A,H,G H,B,F G,F,C 



Eliminating a, /3, y, 8, from these we obtain the following identical relations 

 between the 18 co-ordinates, viz. : — 



o, o, o, HH,FH,BH, GG,FG,CG, AA,BA,CA = 0. 

 HH,GH,AH, o, o, o, GF, FF, CF, BA, BB, CB. 

 HG,GG,AG, HF,FF,BF, o, o, o, CA,CB.CC. 

 HA,GA,AA, HB,FB,BB, GC,FC,CC, o, o, o. 



These conditions are nine in number ; and, as we are concerned only with the ratios 

 of the co-ordinates, the total number of independent co-ordinates will be 

 18 — 9 — 1 = 8, as it should be. 



