« 
34 7 REPORT—1858. 
of the lines, and of the shift determining the epochs of maximum and minimum. It 
is, indeed, obvious that the result is equivalent to a movement of these lines; but 
whether the inducing action is experienced by the earth’s surface, or its atmosphere, 
was not evident. J think I[ can prove satisfactorily that the diurnal variation is not 
due to the sun’s heating power, whether on the earth’s surface or its atmosphere ; 
but that it may be due to the sun’s magnetic action on the atmosphere, is, it appears 
to me, much more probable. I have said in the letter already referred to that the 
diurnal variation of the magnetic declination is least near the equator where the dip- 
ping-needle is parallel to the terrestrial axis,—so I believe it will be found that the 
diurnal variation of dip and of force is least in mean latitudes, where the dipping- 
needle is nearly perpendicular to the earth’s axis. 
“Tam, &c., Joan ALLAN Broun.” 
Sounp. 
On the Mathematical Theory of Sound. By the Rev. 8S. EARNSHAW. 
The author adverted to the circumstance, that the only impediment to the com- 
plete development of the mathematical theory of sound has hitherto been the dif- 
: : saan : ._ dy\? d?y _. d’y 
ficulty of integrating the partial differential equation ee phe! oleae As an ap- 
2 
proximative mode of surmounting this difficulty, it has been usual to assume (2) as 
But the author suggested that the legitimacy of that step is by no means evident ; 
and that the true test of the allowableness of it is a knowledge of the change which 
: saute : d? a 
must take place in the constitution of the atmosphere, in order that (3) =p (a) 
may be the ewact equation of motion. In this way it will be seen whether the plitysical 
2 
change, represented by assuming (=) = 1, be of such a minute character as to be 
allowable. From this it was inferred that the equation which represents the proper- 
ties of sound does not admit of the assumption *=1. The reason why this 
a 
assumption, though analytically allowable, is not allowable in the problem of sound, 
is that in that assumption quantities are neglected which (in the case of sound) 
2 
represent essential properties of wave-motion, so that the equation 2 = a4 is 
x 
not an approximation, and from it nothing can be inferred but that the assumption 
ae 
(2) =1 is not admissible. The result of this reasoning is, that the equation 
a 
y =F (a + at) + f (a@—at), which has hitherto been the basis of explanation in 
treatises on Sound, has nothing whatever to do with sound, but represents the 
motion of a wave in an imaginary elastic medium of a constitution very different 
from that of the atmosphere and of all known gaseous media. The mathematical 
theory of sound is consequently put back to its differential equations, beyond which 
not an inch of ground can be maintained. Till the differential equation is integrated 
3 . — /dy\2 ‘ ; 
accurately, without assuming 2) =1, no advance can be made, and science remains 
without a mathematical theory of sound. The author then announced that he has 
succeeded in integrating the differential equation of sound without approximative 
assumptions; that he has, in fact, obtained its exact integral; and in the result has 
possessed himself of the key to the various properties of sound. Among several 
other things, it was stated that the exact integral accounts for the great difficulty 
which experimenters have found in obtaining accordant velocities of sounds,—for 
the sweetness of musical sounds,—for the rapid decay of violent sounds as they pro- 
