130 
Proceedings of the Royal Society 
Finally we have, as the modification of the original wave, 
Ce- W * | ^ --^ Bx + ^ -B x 2 ^j cos (2 kiyiH - mx) - ^ B% 2 sin (2 kmH - mx) j 
These terms, when combined with the harmonic part of the assumed 
value of u, may be put in the form 
Ce -m i* cos (2 kthH - m 2 x) , 
where 
i \ 4 m 
m 2 = m( 1 - ^ Bx) . 
Ba?), 
We thus see the effects of the introduction of the quantities a and /3 
upon the amplitude and phase of the wave ; and it is evident that 
they are of the greater consequence the greater is the difference of 
mean temperature at the sides of the slab. 
Hence the only legitimate mode of applying Angstrom’s method 
is to keep the mean temperature the same throughout the slab. 
This can easily be effected. 
It is obvious, moreover, from the values of m l and m 2 above, 
h 
that Angstrom’s method gives the value of - for the mean of the 
mean temperatures indicated by the two thermometers. Only, 
there is always the extraneous factor 
which is usually very nearly unity. 
I have worked out by the above method the case of two harmonic 
waves (in the value of u), one of half the period of the other. Hew 
terms are thus introduced into m 1 and m 2 . They are such as to 
seriously affect the values of these quantities when x is small, but 
they rapidly diminish by increase of x. 
If the new term in u be 
P) e -maV 2 cog ( 4 Km 2 £ _ mx j2 + E) , 
the additional terms in m 1 are 
D e -W2 S mX- o /o V -t-««V8cosX 
4 m 2 ^ 2-1 m * 
Those in m 2 are formed from these by making the first term posi- 
tive, and interchanging the sine and cosine of 
X = mx(J 2 + l)-E. 
