218 
PROFESSOR POWELL ON A NEW CASE OF 
that for the part retarded by r will be 
27r 
or. 
sin — {vt—x — r ) ; 
Sin — [vt—x) cos Y cos — {vt—x) sm — r. 
Consequently the intensity I for any point of the whole resulting wave will be the 
sum of the squares of the coefficients of (vt—x), or we shall find on squaring and re- 
ducing, 
27rr\ 
1=2 
cos 
A y ’ 
4r 
or if we assume p= it becomes 
I=2(^l-{- cos^p). 
(21.) And if we suppose p to be originally an even number and to increase by 
unity for successive rays of the spectrum, we shall have the corresponding values, 
TT 
p cos 2 j?= — 1, and therefore 1=0 
p-fl .... 
. . . . — 0 . . . . 
.... 1-2 
p-{-2 .... 
. . . . -4-1 . . . . 
.... 1-4 
p-\-^. . . . 
. . . . — 0 . . . . 
.... 1-2 
p+4 .... 
.... 1 = 0 
&c. 
&c. 
&c. 
Thus for any two values Pi, P 2 , if ^2=4, they correspond to a change from one 
dark band to another, and consequently, if for two rays j02=4w, n will be the 
number of bands in the space comprised between those two rays : and if they be the 
extremes of the spectrum, n will be the whole number of bands. Its value may be 
assigned by considering the nature of the retardation, or obtaining an expression for 
r, as follows : — 
(22.) On inserting the plate P of thickness r whose index is [Jtjp, as above, into the 
medium whose index is the retardation of the light which passes through the 
plate being the difference of the retardations of the plate and of an equal thickness 
of the liquid, will be expressed by 
P T 
But since ^ = - (20.), we shall have 
P / l^p [^m \ ^ 
And for any two rays whose indices are and wave-lengths we have 
P\ Vi . r n 
4 wrv ^ A-J 
This formula may apply to the whole length of the spectrum, taking the two ex- 
