456 Lord Kelvin on the Front and Rear of a 
Fourier harmonic analysis of P(«, 0), when subject to (22) 
and (23), gives 
21x 212 
P(a, 0) =A, cos + A3 cosa 
r r 
§ 15. Digression on periodic functions generated by addition 
of values of any function for equdifferent arguments. Let 
f(@) be any function whatever, periodic or non-periodic ; 
and let 
+A; 00857 +. .. (24). 
ee : 
Pia)= yp fata). . - . - (29)5 
which makes aad 
Pi2)=P(e-+a) 2. 
Let the Fourier harmonic expansion of P(a) be expressed as 
follows :— 
P(#) =A)+ A, cosa+ A, cos 2a+ A; cos dat... 27x 
; where a= —— 
+B, sinae+ B, sin 2a+B,sin da+ ... Xr 
(27). 
Denoting by j any integer, we have by Fourier’s analysis 
tole 
As (" cos .272@ 
j=) dxrP(#)". 9-3. 
2) % (7) sin’ be 
which gives 
i=+o 2 a8 Be we 5) = : 
aMTg he \ daf (a + ir) cosj=™" =| day (x) 0 | 
“le sat (29). 
1 arts) | Se ee si Lee 
2AB;= > \ daf(#+2r) sing - =| dif (2) nj 
I= — DY 0 a —n i 
§ 16. Take now in (29), as by (AS) (20), 
: r 
He\—=ol@, 0)—A(@7+5,; 0). iin ee ee (30). 
This reduces all the B’s to zero ; reduces the A’s to zero for 
even values of 7; and for odd values of j gives, in virtue 
of (22), 
P+ x . 
JrAA;= 2| dxo(z, 0) cos jan .. 2 ae 
Go back now to §§ 3, 4, (6), (12), above; and, according to 
the last lines of § 4, take 
sae A/ 2 | eae a 
b (x. 0)= VBS ¢ 7) ane bea 
