S6 Mr. H. Hilton on 



A, B, 0, P be (Z l5 m h raj-, (Z 2 , w 2 , w 8 ), (Z 3 , w? 3 , w 3 ), (Z, m, n) 

 respectively. The projections A 1? B l5 C a of A, B, C from 



on 2=1 are the points f — , — , 1 ) &c. Hence if X, Y, Z 



are the areal coordinates of the point (x, y, 1) when A 1 B x Oi 

 is the triangle of reference in the plane z=l, 

 AX = w 1 (a?Z/+ym 1 / + w 1 / ) &c. ; 

 where Z/ is the cof actor of l x in the determinant [Zim 2 « 3 ] = A, 

 and so on. 



Therefore, by solving, 



ZjX Z Y Z 3 Z -niiX ?» 9 Y m 3 Z 



#= h - — I — , u= 1 — — + . 



n x n 2 n 3 n } n 2 n z 



Now Ix + my -r nz = meets z= 1 in the line Ix 4 my 4 m = 0, 



?'. £. - (Z^ 4 mffi; 4- wwi) 4- . . . +■ . . . =0. 



% 



Denote the angles between OA, OB, OC and the normal 

 to the plane of projection (the axis of z) by a, /3, 7, and the 

 angles POA, POB, POC by \, /*, v. Then the gnomonic 

 projection of the great circle in which a plane through 

 perpendicular to OP meets the sphere is the line 



X sec a . cos X 4- Y sec /3 . cos /jl 4 Z sec y . cos v = 0, 

 i. e. wX oY wZ _^ 



« cos a b cos /3 c cos y 

 (where a :b :c are the axial ratios), or ?«f 4- w? + tof=0 ? taking 

 a cos a . f=X &c. 



2. The axis of the zone containing (u v to) and (m' 1/ to') 

 meets the plane of projection in the point 



11% + vr) + 10 £= w'f -f ?/?; 4- m/?= 0. 

 It follows that the zonal axis [U V W] meets the plane 



f V %> 



of projection in the point ^ = = = ==r. 



3. Similarly, if A', B', C are the poles of the faces (10 0), 

 (0 1 0), (0 1) and A/, B/, C/ their projections, the projection 

 of the zone [U V W] is Uf + Y17 4- W?= 0, where 



£ = Xa sec a . cos AOA' &a, 

 and the projection of the pole of the face (?/, v, w) is 



A 2 ' B/ C/ being the triangle of reference. 



4. These results have many important applications ; for 

 example, the anharmonic ratio of the four cozonal faces 

 (tt! Vi tOj), (w 3 v 2 to 2 ), (?/•- r 3 to 3 ), (w 4 v 4 to 4 ) is that of the four 

 concurrent lines Wjf + ^^4 w^—Q &c, !. e. that of the four 



