600 Proceedings of the Royal Society of Edinburgh. [Sess. 
under the summation sign in (1) the following inequality, for all integers 
n A m. 
[ G (u, v, i)(a n cos nu + b n sin nu)du L g(q)e~ kvH (—*— + — i ( | a J + \b n | ) 
J a \n + v n-vj 
Lg{q)f(\%\ +l*.|) • ■ • • ( 2 ) 
7c 
Again, J® e~ kvH cos uv dv is a function of bounded variation of the 
variable u, since it has a bounded differential coefficient with respect to u, 
viz., — \*e~ kvH v sin uv dv. Therefore, the product of two functions of 
bounded variation being a function of bounded variation, g(u)\* e~ kv2t gosuv dv 
is a function of bounded variation of the variable u in the interval ( q , Q), 
and h(u) — s m , where s m denotes the sum of the first 2m — 2 terms of the 
Fourier series of h(u), is summable; therefore we may integrate term-by- 
term, as before, and write 
e~ kv2t g{u) (Ji{u) - s m ) cos uvdv = % j® duj* e~ kv2t g(u)(a n cos nu + b n sin nu) cos uvdv. 
n—m 
Similarly g(u)^ e~ kvH sin uv dv is a function of bounded variation, so 
that in the preceding equation we may change cos uv into sin uv. Thus 
we may write, with either signification of G (u, v, t), 
\^du^G(u, v , t){h(u) - s m }dv = % J '^du\*G(u, v, t)(a n cos nu + b n sin nu)dv (4) 
Now we may write the equation (1) in the following form : — 
f Q G(w, v, t) { h(u) - s m }du = 5 f Q G(w, v, t)(a n cos nu + b n sin nu)du. 
Bearing in mind that the individual integrals in this last summation 
continue to exist when we write infinity for Q, the inequality (2) shows 
that the equation last written down continues to hold when we put 
Q = oo .* When this is done, the right-hand side becomes a series of 
functions of (v, t) which converges boundedly for all values of v in the 
* See § 15 of the paper already quoted on Fourier’s integrals. The lemma here used 
is as follows : — If the series of proper or improper Lebesgue, or Harnack-Lebesgue , integrals 
n^ilq ^ n ( x )d x converge to the proper or improper Lebesgue, or Harnack-Lebesgue, integral 
J^f(x)dx for all values of q and z in the completely open interval (p<q<z<B), in such a 
manner that a convergent series of positive quantities^ c n can be found, each term of which is 
not less than the absolute value of the corresponding term of the series of integrals, whatever be 
the values of q and z in the interval, then, provided only in addition, J^f n (x)dx exists for each 
integer n, we can assert that J®f(x)dx exists and =^Jp f n (x)dx. 
