592 Proceedings of the Royal Society of Edinburgh. [Sess. 
\S dv \> hvH f ( u ) cos uv du = du\l e - kv J(u) cos uv dv 
= Lit [ f(u)x / ^r-e~Mdu 
t=0 J q V 4 
= L /^ / — e ~h [ Q f(u)du = 0, 
mV 
which proves the theorem. 
§ 6. For the next theorems we require the following lemma : — 
Lemma. — If f(u) is monotone decreasing with zero as limit at infinity , 
and 
F (u, v, t) is e~ kv2t f(u) cos u(y - V), or e~ kvn f(u) sin u(v - V), 
we may reverse the order of integration and write , 
Io'H7 F ( M ’ r, t)du = \*du\*1£(u, v, t) dv, 
both these repeated integrals existing, provided that when F(u, v, t) 
has its second value, we know that [ ^du exists. 
J q U 
To prove this, we affirm that 
J>P>, v, t)du — ¥{u, v, t)dv, 
and J^ r c??;| e F(w, v, t)du — J 3 duyf¥(u, v, t)dv, 
from which, by addition, the required result at once follows. 
The proofs of these two results are very similar; we shall therefore 
content ourselves with proving the latter. 
Replacing the lower limit V by Y + c, for the variation of v, so that 
V + c, Lv 'ZB, 
we have, for the range of values of v in question, using the Second 
Theorem of the Mean, 
F (u, v , t)du | = e~ kvn f(q + 0) 
'W(*;-Y)cfo | L2f(q)/c. 
q sm u(y — V) j - •' w// 
Hence, firstly, the left-hand side vanishes when q moves off to infinity, 
so that J“F(u, v, t)du certainly exists; secondly, J^F(u, V, t)du converges 
boundedly to its limit JjF(u, v, t)du, and the latter integral is a bounded 
function of ( v , t) for the range of values of v considered. Hence we may 
integrate and get 
Jv+c<H7 F (“> v ' t ) du = $+cdvpjy{u, V, t)du = pj1 +c dv\y(u, v, t)du 
using the Second Theorem of the Mean, B' lying between Y + c and B. 
