The Intensity of Transmitted Light, 167 



^ ?i+2 U"-F2 _ Jw+2 . IAA\ 



F„^2 denoting ^U„+i_ function of 

 Integrating (44) we obtain 



r,„j^i being some arbitrary constant. Hence U„+2 has been 

 obtained in terms with smaller suffixes, and these in their 

 ttirn will have been obtained in terms with suffixes still 

 smaller ; so that finally U„+2 will be obtained as a complex 

 function of Ui. 



Now give to n the values of o, i, 2, 3, &c. in succession, 

 and substitute the expressions so obtained for U2, Us, &c. 

 in the expansion /=Uo + Ui;r+ U2;i^^ + &c. ; then we obtain 



Now by equation (37) Uj is a known function of U<, for 



<l>{t) standing for 



a I E dt; 



if we substitute for Ui its value in U^, the undetermined 

 quantities in (46) will be this letter, and the letters r^, 7% n 

 &c. ; the values to be assigned will depend upon the initial 

 condition of cylinder with respect to the distribution of the 

 body A, but it was supposed to be initially uniformly 

 distributed, so that at time /— we have 



_P 



also when x vanishes/ also vanishes ; making jir and / each 

 in (46), we obtain for the value of Uo, hence we have 



