Conway—The Partial Differential Equations of Mathematical Physics. 195 
Hence 
~ SEHR) _ “6 PPG) PFO) 
af LEGO ower [ns 24 LD. 9 
where £ is a power-series. 
Hence, in the function ;,,., the term of most importance at the origin 
will be f(¢)r“’* multiplied by a numerical constant. We can get the asymptotic 
expression as follow:—Putting w=t¢-—r—w, we get 
K _ (flt-— 7 — w)dw | 
2(n-2) — | [ Qar ra we) ’ 
the path of integration goes from + « around the origin and back again. 
We can divide it into two parts—(1) from + « to 27 along the axis (this is 
traversed twice), (2) a closed path surrounding the origin and passing through 
the pomt w=2r. The first will become of less importance, as 27 becomes 
great, and its limit is zero when multiplied by 7%", and r made infinite. 
In the second we can expand 
[er + w? |-2e-)) 
in a series of inverse powers of w. The principal term is 
1 (f(t —7r —w) dw 
pal [2a 2) y 
which, when 7 becomes very great, is 
1 (“f[t-—r —w]|dw 1 
aa) aay) — = aC, say 
This, then, is the first term of the asymptotic expansion, for it satisfies the 
definition of such an expansion, 7.e., 
Lt. 72) E 3(n-2) — wo py 
t 
y=? prr-l) 
Further terms of the expansion can be found by means of the differential 
equation. It may be remarked that the phase of any disturbance in a space 
of odd dimensions depends on ¢—7, but that, in a space of even dimensions, 
the phase of a periodic disturbance will differ by an odd multiple of 
| 
TRANS. ROY. DUB. SOC., N.S., VOL, VIII., PART XVI. 2M 
