190 Conway—The Partial Differential Equations of Mathematical Physics. 
infinite system of infinities lying on the hyper-surface S = 0. Now the type of 
solution which we shall require is one which will be infinite for a series of values 
of the variables which satisfy the equations 
mh SIMO)% BB = ja(@) ooe H = Ip) 
which include, as special cases, the case of the functions 
Si@>y Fo(B) oo » 
being all constants, and the case of all but one being constant. Hence we shall 
classify the equations according to the different forms which the surface S = 0 may 
assume, or, what amounts to the same thing, the various forms assumed by the 
differential equation when expressed in canonical form. 
By linear substitution, the equation may be written in one of the following 
three forms :— 
op , od ed 
ary . . . ‘ nT GY = I 
oe v ae Ons? 1 : OL? 0, (1) 
> a mp 
pS LS eerie — —— 0 II 
ah” Bae Or,” 0x? j ey 
Hp Ob th Oh ip 
poe Bake —-—- —-...-~% =0 I 
On," v Ane © v ae" Oy Wx On 1 (1) 
according as the equation in / 
Ay ra k Ay e ° ° Gin 
A951 Ay, — k O 5 6 Usp, 
[= 
| 
Bos Dp Sy PATE CREE. | 
has (1) all the roots of the same sign, (ir) all the roots but one of the same sign, 
(111) not less than two roots of sign different from the others. 
The second and less obvious type of singular solution is obtained as 
follows :— 
If the number of variables be x + 1, the differential equation may be 
written 
@ 
0 oe 
bet dng + Btn en ee Ne Ok 
where there is no term containing the operator = to the first power. ‘Then, with 
