Length of a Wave, in the Undulatory Theory of Light. 11 
This equation must be true for all values of ¢, therefore 
@ U d* 
ept soe + gp ge =O (6.) 
a gl? ds 
Wg +878 + gage =O be) 
Now, as p is a function of z, it may be expressed by a 
series of such terms as a sin kz + 6 cos kz, where a, 6, and 
% are constant quantities. Substitutea sini z + bcoskz for 
Pp in equation (6.), and it will become 
n? —s? fe + % k4 = 0; hence 
Ae ae iter k 
a Aee aE i si? : 
=\/(*= 54") aA tome er art 3 ), nearly. 
As the equations (6.) and (7.) are similar, and as we have 
put asinkz + bcoskz for p, we must put a! sin £ + b' cos 
k z for q; a! and 0! being two more constant quantities. Hence 
£ may be expressed by a series of terms similar to 
(a sink z+ coskz) snnt+(a' sinkz+0' coskz) cosnt. 
With respect to any particular value of z, this term goes 
through all its values while ¢ increases by =, and with re- 
spect to any particular value of ¢, it goes through all its values 
; A Qa - 
while z increases by —3 consequently it represents a wave 
k 
bey pti : OE tn 
of light, moving in the direction of z, with the velocity a 
the value of which has just been found equal to 
Wd gl? ) 
(0+ s55-) 
Professor Powell’s expression for this quantity is 
n> 
 sin( #5") 
H , which is equal to H [or aay very 
2 BY 4 n 
l 
nearly. As the Professor considers only one term instead of 
the sums s° and s'*, and as rm and J in his notation are the 
Qa. : : 
same as Az and iy in ours, the two expressions are vir- 
/: 
tually the same. 
C2 
