1910-11.] Sommerfeld’s Form of Fourier’s Repeated Integrals. 591 
where, as t approaches zero, B' approaches infinity, provided u is not zero. 
Hence, by the extended theorem of Riemann-Lebesgue,* the integral last 
written down has, as t approaches zero, the unique limit zero when u is 
not zero. Moreover, since, whether u is zero or not, it is numerically 
L \™ve~ v2 dv, it is a bounded function of the ensemble (u, t). Hence we may 
write, by the preceding equality, 
U f du f e~ kvH u sin uv dv = I dvJ^-Lt f e~ kvH u sin uv dv 
t=oJ v u J o J p u t—Oj o 
= \ q M du . 
J p u 
By (1) this proves the theorem, whether p is zero or not. 
§ 5. Theorem. — If J“ | f(u) | du exists, then 
tolo dv\ q e~ kvH cos uv du = 0. 
For 
cos uv du 
is numerically not greater than the result of changing f(u) cos uv into 
| f(u) | , that is, J® e~ kvH dv x \f(u)\du, which, in consequence of the hypo- 
thesis made, has the unique limit zero when A is indefinitely increased. 
Therefore 
J- dv\ x x e~ kv2t f(u) cos uv du — 0. 
But this suffices to justify us in reversing the order of integration,]- and 
writing 
Jo dv\™e~ kv2t f(u) cos uv du = \™du^ e~ kvH f(u) cos uv dv , (1 ) 
Now the integral with respect to v on the right-hand side of (1) is a 
bounded function of ( u , B), since it is numerically L \™e~ kvn dv, and con- 
verges, since \™e~ kvU f(u) cos uv du exists. Also f(u) is summable in the 
infinite interval ( q , oo ). Hence, proceeding to the limit infinity with B, 
and zero with t, by the theory of the integration of sequences, (1) gives us 
the following : — 
* This extension to an infinite interval of integration may be found in my recent paper 
on “Fourier’s Repeated Integrals,” § 10, presented to the Royal Society of Edinburgh. 
For the original theorem see B. Riemann, Ueber die Darstellbarheit einer Function durch 
eine trigonometrische Reihe , § 10 ; and H. Lebesgue, Lecons sur les series trigonometriques ; 
also Hobson’s Theory of Functions of a Real Variable, p. 674. 
t See my paper on “Change of Order of Integration in an Improper Repeated 
Integral,” already quoted, § 9(8). 
