188 JSonway—The Partial Differential Equations of Mathematical Physics. 
to Whittaker’s Modern Analysis, chap. xm1.; T. H. Havelock, Proceedings of the 
London Mathematical Society, ser. 2, vol. u1., pt. 1; FE. W. Hobson, Eneyclopedia 
Britannica, 10th ed., 1902, vol. xxxu., p. 791. 
II.—CULASSIFICATION OF EQuarions AND FUNDAMENTAL PRINCIPLES. 
Considering, first, the case of homogeneous equations, the general form may 
be written 
Ay = () 
= 
OO Die 
Chae 
=P ayo = 
02 02> 
the independent variables being 2, %, . . . %. 
Let Ay, A»... be the minors corresponding to a, @:... ot the 
determinant 
MK = QQ), Ayo . . ° Ain 
| 
Ay Dy) ° ° . Aon | 
* | ’ 
| 
Any Ane . . ° Ann 
and let S denote 
Ay Hi" + Anna he Aero co 4 o 5 
then the following well-known results hold :— 
Bll =P thy Agy Sp 0 90 0 =P Opn Asn = 0, 
ab, 
or A, according as r + or = s 
0 oS’ oS’ 
Q,) Arp = Apo ane SC ip. 
1 “72, wn 
MjAy + 2a.4 +... = nA. 
as\? aS\ (08 ; 
Ay) G) ar 209 (e) i ago = 4 SA. 
1 Ei 09 
Substituting S” for ¢ in the differential equation, we get 
4m(m — 1) AS”? + 2mnaS”™ 
which vanishes if 
m = — s(n — 2). 
Hence S“"2 is a solution of the differential equation. This is what we 
shall call the first fundamental singular solution. 
