332 Dr. T. H. Havelock on 
moved gradually nearer to the source and arrived at the 
source when the ear was 16 inches from the wall. 
A similar experiment with a note of! wave-length 19*4 
inches gave the image : — 
on line parallel to the wall, with ear distant 17, 39, 61 inches 
at the source, with ear distant 24, 48, 58 inches 
40° from the wall, with ear distant 32, 52 inches. 
From these experiments it is clear that in noting the 
direction of a fog-horn at sea the observer should be well 
away from any reflecting surfaces of any kind. (In one 
experiment an umbrella held to one side of the head at a 
distance of 2 feet displaced the sound-image 20°.) I find 
that it adds to correctness in fixing the direction to have a 
flat board slung on the shoulders vertical and parallel to the 
axis of the ears. This increases the intensity in front and 
shuts off sound from the rear. I think it would also be 
better to have two short blasts of 3 seconds each, every half 
minute, at sea, rather than a long blast every minute. 
Also fog-horns should be placed well above any reflecting 
surfaces, but it might add to their carrying pow r er if a large 
disk or sounding-board was placed horizontally directly over 
them. 
XXXII. On certain Bessel Integrals and the Coefficients of 
Mutual Induction of Coaxial Coils. By T. H. Havelock, 
M.A., D.Sc. ; Felloic of St. John's College, Camhridge ; 
Lecturer in Applied Mathematics, Armstrong College, 
Newcastle-on- Tijne *. 
THE calculation of coefficients of mutual induction has 
been discussed by several writers from the time of 
Maxwell to the present, more accurate expressions being- 
required as experimental methods have become more refined. 
The expressions are generally in one of tw r o forms : they are 
either given in elliptic integrals, in which case numerical 
calculations are tedious, or given by a certain number of 
terms of a series. 
The present paper brings forw r ard another method of 
expressing the coefficients, namely in terms of integrals 
involving Bessel functions ; series are obtained from these 
integrals, their general terms found, and their convergence 
tested. In certain cases series are obtained which seem to be 
simpler and better adapted for numerical calculation than 
* Communicated by the Author. 
