The Intensity of Transmitted Light, i6i 



we shall obtain by differentiation 



To find the value of 



dx 



Y 1 dN\ 

 "VUi + V dxP 



we may apply the theorem of Leibnitz for finding the n^ 

 differential coefficient of the product of two functions of 

 x\ if in this result we make xz=.o we obtain the equation 



r_^/ 1 



^\\ = 



+ Ain(7?. + l)U„4.i + A2(7i - 1 )7iU„ + «fe 



+ A^(7i - r + l)(7i - r + 2)U„_^+2 



+ A^+i(7i - r){n - r + l)U„_^+i + <fcc. + A^SUg]. (29) 



Hence the coefficient of ;i;^+^ in the expansion of log(Ui + V) 

 will be 



— — [Ao(?i + \){n + 2)U„+2 + k^n{n + l)U„+i 



+ A2?H'?i - 1)U„ + &c. + A,,2U2 'y 



giving to n the values 0, i, 2, 3, 4, we shall obtain the 

 following results — 

 Coefficient of 



X = 2A0U2 



„ ar2 = l(Ao3-2U3 + Ai2U2) 



^3 = |(Ao4-3U4 + Ai3-2U3 + A22U2) 



x^ = i(Ao5-4U6 + Ai4-3U, + A23-2U3 + A32Ua) 



„ x' = ^(Ao6-5U6 + Ai5-4U6 + A24-3U4 + A33-2U3 + A42U2) ; 



by a continuation of the process expressions might be 

 found for the remaining coefficients. 



To determine the values of the letters Ao, Ai, A2 &c. in 

 terms of Ui, U2, Us &c., we may proceed as follows ; sub- 

 stituting for V its value in terms of x from (27), then from 



