158 MR. R. A. SAMPSON: A NEW TREATiMENT OF OPTICAL ABERRATIONS. 
of origin from one surface to the next, we have for the relation between any original 
ray 
y = ^x + b, z — yx + c, 
and the corresponding emergent ray 
y' = ^'x' + b', z' = y'x' + c\ 
fG+^G, 
H + ffl 
(b, 
Lk+<^k, 
L + ,^L 
r G+(5G, 
H + ^H 
(g y)-< 
[k+sk, 
L + 
= {h' + Sb', + 
— {c' + Sc', y' + Sy'), 
(13) 
where //, /3\ c', y are the values that would result if (5G, ... were all zero; and 
turning hack to the expressions (6) we see that oi,., are expressed in terms of the 
incidence—-(5, /3, c, y)—upon the plane Oijz hy the equations 
— i" (l + 
where 
b^. = gj.) + // c, = g,e + /qy ; 
.(i-i) 
where 
— kj.b + lj.i8, y^ = J\ J.C1j.y, 
= kr+ib + lj.^-1^, y\ — Zv+iC + /,.^iy. 
We notice that the determinants— {gl—hh )—of the scheme multiplied by each 
will always be zero, and that of the scheme multiplied by w,. also, for the case of the 
sphere. This supplies a useful check. 
The numerical management of these formulae for actual systems is dealt with later. 
I shall now consider their analytical and geometrical properties. 
If 
fG + ^G, H + ml 
Lk+^k, l+^lJ 
and ... are quadratic functions of h, c, ji, y, with the symmetries implied in the 
forms above, we .may put 
,!G = i{,S.G(h“+c*) +24G(i/3 + cy) +i,,G(,8" + 7‘')}, 
,!H = J{^,H(i*+c")+24H(i/3 + cy)+4HO'+y“)}, 
JK = i{S,K(V+c^) +2S,K(bl3 + cy)+S,K{l3‘+y‘)}, 
il, = J {^iL (i* + c*) + 24L (i/3 -H!y) + (li‘ + y'*)}, 
(15) 
