PROFESSOR TAIT ON MIRAGE. 5538 
A complete integral is 
Hence the equation of a ray is 
dy : 
C=2—-—a gia : st @): 
[This result might, of course, have been at once obtained from the corpus- 
cular theory. For its principles give 
eee Y= Je +y— a J 
Equation (2) has two distinct forms according as a is greater or less than 
a, These are separated by the limiting form when a=a, viz :— 
zr 
y=Ce* , 
a logarithmic curve asymptotic to the axis of 2. When a is less than a, the ray 
passes through the plane y=0, and we need not consider it further. 
We may therefore assume 
=a +1, 
and it is obvious that y cannot be less than 7. With this expression for a, the 
mere form of the equation (2) shows that the curve has a vertex at the point 
y=n, and that it is symmetrical about the ordinate through that point. 
We must remark, in passing, that this property of symmetry about an axis, 
at the extremity of which is a vertex, is common to groups of rays in all media 
in which the refractive index depends only on the distance from a particular 
plane :—the groups which possess it being those which either do not reach that 
plane, or pass through it more than once. 
3. Let us now consider only rays which have vertices, and which pass 
through a particular point z=0, y=. Thenif € be the x-cdordinate of the 
vertex, equation (2) becomes 
ta ldttf ae ete 
Vy—7 
This is the equation of the Locus of Vertices of all rays (having vertices) which 
pass through the point 0, 6. We may write it in the form 
2 — 
£= Ja? +7? log SS . (3). 
