354 
Proceedings of the Koyal Society 
we get, because of (c), or <f>k = 0, 
4>P = ~ fr'Sip ~ fcj&jp • 
We assimilate this with our former expression (4) of cf>p, and we 
put in consequence : 
a =-<f>i, /?=-<&/, a 1 = i, Pi=j - 
Hence we have, first : 
m 2 = - (S i<J)i 4- S j<f>j) = - t . 
Secondly, we have, as 
S^ = S^ = 0, 
(fifk — iSi&h —jSjfik) 
= - <f>iS7c<f>i - cf)jS7c(f)j 
= Mb — Z?S 
(by its definition, page 678.) 
Finally, if 
T<f>i = e 1 , T<f>j = e 2 , 
we have 
Y. cfiicfyj = e 1 e 2 /3', when fi JL <£/ , /3' representing U . 
Thus all the terms of (h) correspond to the terms in the primitive 
equation (8) which we have thus established by a method different 
from that of the quoted memoir. 
2. On Mirage. By Professor Tail. 
(Abstract.) 
While seeking for a good elementary illustration of Hamilton’s 
general methods in optics, the author was led to consider, from a 
somewhat novel point of view, the path of light in a medium whose 
refractive index is a function of the distance from a plane. This 
is, at least approximately, the case of the peculiar atmospheric 
arrangements to which are due the phenomena of Mirage , so long 
as the curvature of the earth can be neglected. 
A considerable improvement in the usual theoretical treatment of 
this subject was introduced by Professor J. Thomson in 1872, and 
afterwards developed by Professor Everett in Phil. Mag n 1873. 
