On Systems of Rays. 105 
we shall then have also, by (Z'°), (the extreme media being supposed ordinary, and 
their indices p, »’ positive, ) 
Se. Gg hBe) 9. 288: Bee 0 OE, qs 
W) hames gay = ae = ay te ND 
that is, in this case, the guiding axes for the initial ray-lines are also the guiding 
axes of the same kind for the final ray-lines measured backward ; which is already 
a remarkable relation, and may be enunciated by saying that the eye-axes and olyect- 
axes of distortion are interchangeable, when the extreme media are ordinary : that 
is, for such extreme media, the eye-axes of distortion become olject-axes, and the 
object-axes become eye-axes, when the object is removed from the final to the initial 
perpendicular plane, and is viewed from the final instead of the initial point. And 
while the equations of the fifteenth number, 
Jaf SOs ry OB u 
Uae ee ov, y=z sy oY; (A") 
represent the initial visual ray-line corresponding to a final visible point B’ which has 
for co-ordinates da, dy, dz, the following other equations, 
Rete eae, Crete ein a Is 
aie eS ov, y=—2z. Rope? @) 
will represent by (J/") the final visual ray-line corresponding to an initial visible 
point 4’ which has for co-ordinates da’, dy, dz’; the initial visual ray-cone corres- 
ponding to any small object 
oy =f (8x) (G") 
in the final perpendicular plane is therefore represented by the equation 
6@=-EE). 
and the final visual ray-cone corresponding to any smal] object 
ef =f (82) (P") 
in the initial perpendicular plane is represented by the following analogous equation 
CFATTOT 6) in fae May GRAB cal Raa Viral 8 15 
are al =ah (= we (= ) ) a (Q") 
if therefore these two small objects, (G") (P"), at the ends of a given luminous 
path, be equal and similar and similarly placed with respect to the conjugate axes of 
distortion, that is, if the final and initial functions ff’ be the same, and if we cut the 
two ray-cones (H") (@Q"°) respectively by perpendicular planes having for equations 
i Z=wh, 2=—pRh, Ci) 
in which # is any constant length, while », »’ are the same constant indices as before 
VOL. XVII. 2E 
