5G2 
MR. G. H. DARWIN ON PROBLEMS CONNECTED 
Since L, M, N, R are constants as regards the time, 
d-yjr d~\]r 
~di = K d^' 
rp=Y — SPe - *? 7 cos qx is obviously a solution of this equation. 
Now we wish to make d=V, when x—y a, for all values of t ; since \Jj=S when 
7T 
x=±a, this condition is clearly satisfied by making g=(2&+l)—. 
Hence the solution may be written, 
^=V_[L«V-McrV+N^-R]-|P 2;+1 e-^ i ^] s cos (2i+ 1) ~ . (29) 
and it satisfies all the conditions except that, initially, when t— 0, the temperature 
everywhere should be V. This last condition is satisfied if 
P 2;+1 cos (2f+l)^=P,-La% 3 +MaV-Na; 6 
la 
for all values between x— ±ci. 
The expression on the right must therefore be expanded by Fourier’s Theorem; 
but we need only consider the range from x=a to 0, because the rest, from x—0 to 
—a, will follow of its own accord. 
Let y— 77 , ; let m be written for \ ; let M'=—, N' = -^ and P/^It—; . 
A 2 a 2 w 2 nr 4 a b 
Then 
R-La^+M«V-Nx G =^[R / -L x 2 +M , x 4 -N' x 6 ], 
00 7 r 
and tins lias to be equal to £P 2 ;+iCOS (2/+ l) x from x =— to 0. 
0 * 
Since 
[2 cos (2i+ l)x cos (2/4- l)x f ^X— 0 unlessy=?, 
mcl . 
Therefore 
[2 cos 2 (2f+ l) X ^ X = T 7r = 2 OT , 
J n 
W - iy+My - Ny ■1 cos ( 2 i+1 k d x - 
Now 
