376 PROFESSOR CHALLIS, ON A THEORY OF THE POLARIZATION OF LIGHT 

 and T,, To must respectively satisfy the equations, 



-J-i + -ri + nkr, = (26), 



dx^ dy 



^ + -j^ + n?kT:, = (27). 



From the system of equations (24), (25), (26), and (27), it is required to determine the forms of the 

 functions expressing the values of ti and tj, that expressing the value of ctj being supposed to be 

 known. It does not appear that this can be done generally; but as before, approximate solutions 

 may be obtained applicable to parts of the rays contiguous to the axis. The process for this pur- 

 pose will be analogous to that applied to the ray of common light. 



Let Ti = m cos (gw + hy), and t.. = m'cos {g'x + h'y). 



Then equations (26) and (27) are satisfied if 



g^ + h' -n^k = 0, and g'^ + h'' - n'k = 0. 



Also equation (25) is satisfied if gg + hh' = 0. And these three equations of condition are satisfied 



if g = n v^k cos 0, h = - n \/k sin Q, g = n \/k sin Q, h' = n \/k cos Q. Hence since we have 



S — 



shewn that when the approximation is carried to the second powers of ,t and y, cr, = —co%n\/ky, we 



shall have by equation (24), 



\/ ky = m cos (nat \/k cos 6 - ny \/k sin 6) + m'cos {tix \/k sin 6 + ny \/k cos 6) .... (28). 



— cos n 



Hence, expanding to the second powers of x and y, 



S f n'ky''\ , mn^k , ^ . ^_, m'n'k , . ^ 



— 1 — 1 = 7K + m (.» cos y - 2/ sm ffy (x smO + y cos 6). 



S 

 Therefore - = m + m , 



2 



and -y' = {.in cos^6 + m'sin^O) x^ -2xy sin 6 cos (m - m') + (m sia'O + m' cos' 6) y', 



g 

 or, substituting m + m' for - , 



(m cos'0 + m'sin'''^) (y' - .r^) + 2 ^ry sin cos (m - tn') = (29). 



It appears, therefore, that equation (28) is not satisfied to second powers of x and y for gene- 

 ral values of these variables, and the functions assumed for ti and t^ are consequently true in 

 general only to first powers of x and y. It is however important to remark that the equation 

 (29), being put under the following form, 



y- 2y sin cos (m - m') , , 



x" X m CO!, 6 + m sin^ 



shews that for two directions at right angles to each other, the assumed values of t, and Tj are 

 true to the second powers of x and y. These two directions may be presumed to be the direc- 

 tions of the planes of polarization of the two rays. But because 



