the Axial Dioptric System. 341 



Now f=0 becomes 2- = 0, /. a;. = ^>3 = (13) 



Again ^rj = is equivalent to 



{a^.v 4- b,y + c^z) (^2.;' + % + Coz) = 0, and to 



ma!'^-\- (a — k).vy — If/^ = 0. 



Hence ala2 = "^ hh—~^y aih2-{-a.2b-^=—k + n . . (14) 

 Aho aiC2-{-a2^i = 0, hiCQ + l>-2Ci = 0y <:'i^2 = 0, . . (15) 



Similarly C2-ai = 0, c{% = 0, C2% = J" * ' ' ^ i 

 I£ Coil^O then c,=0, a^-O, hi = 0, which is impossible. 

 Hence Ci = C2 = 0. Thus 



A .^r=— fa2C. + r7«,Co, V, . . . (17) 



(10) becomes ^ = ^.^^^^.^y^^y^^^-^^^^ 



— Ci'2^^ -^(^W 



.(18) 



»i(/^20 — '''l^) +'^( — t'^20 + «l'7) • 



Using (14) this reduces to 



^ V (m + mn -I^'p^ -m^ ^^ \ ^[kl %^ t mn ^ + 2/m) 

 V l>i aiJ ^' hi cii 



This we may write 



^'■■v':t'=/?:im:K (19) 



which brings out the symmetrical character o£ the trans- 

 formation. 



It is easy to show that for this transformation, the locus o£ 

 the intersections o£ a set of old lines passing through a fixed 

 point fo 7;^ f,^, with their new lines is 



which in general is a conic circumscribing the triangle of 

 double-points, but reduces in the case when ^^ — to a 

 straight line through C. 



And the envelope of the lines joining the old points of a 

 fixed line l(j^ + iii^7j + 7i^^=0 to their new points, is the conic 



?o^(^-/0T + "V(A-/)V + -,/(/-5/)'?'-2^^o^o(A-/)(/--.^)^? 



-2nMf-fj){ff-h)i^-2l^mXo-h)(.^^~f)^V = 0, . (21) 

 which touches the three double-lines. 



