74 Professor Hamitton’s Third Supplement 
Sv (ae ov ) ov Ge Sv ) 
Sa? \axdy sBdx/  Sad3\de* Sadr 
_sv (eV sv ov (eV cv). é 
Se ey eee ae Sh NS ce") 
and it is not satisfied by extraordinary rays, except in particular cases. We may how- 
ever always consider the paraboloid (Z°) as an auxiliary surface, with which the final 
ray-lines of the proposed system (V°) are connected by a remarkable and simple 
relation. For if we take the rectangular planes of curvature of this paraboloid for 
the co-ordinate planes of az, yz, and denote the two curvatures corresponding by 
7, t, so as to have the following form for the equation of the paraboloid 
z=thre’+tty’, (G") 
we shall satisfy the condition 
éa 6B = 10 
ay +32 = (H") 
and may employ the following expressions for the four coefficients of our problem, 
oa oa 6B _ op _ aris 10 
Sac esky ae ae sy be (1 ) 
the ray-lines of our system (V*) may therefore be thus represented 
L=8E—2 (rex + ney), 7 
y = by—2(ty—nda), J 
while the normals to the paraboloid are represented by these equations 
(K") 
T=or—zrex, y=sy—z2tdy ; (L”) 
from which it follows that the angle év between a ray-line (A) and the correspond- 
ing normal (L"°) may be thus expressed 
év=nel, in which V= / 82" + dy’, (M”) 
él being the same small line BB, as before; and that the plane of this angle 8, or 
in other words, the plane containing the ray-line and the normal, has for equa- 
tion 
wou + ydy = dl? — 2(rda* + bey”): (N") 
this plane therefore contains also the right line having for equations 
von +ysy =0, z= (O”) 
rbae+ toy? 2 
that is, the axis of the osculating circle of curvature of the normal or diametral section 
