Conway—TVhe Partial Differential Equations of Mathematical Physies. 189 
If, however, A= 0, then S becomes a perfect square, so that the 
fundamental solution becomes 
: (it + Nets M5 6 6 hnLn eo). 
where 
2 — An; Ni Az —— Aj», &e. SUNS : 
but in this case any arbitrary function of 
WR) eS Ny Sg oe yan 
will be a solution. 
Introduce, now, new variables €), €, ..., €,, such that 
Sh db diy dbo 5 6 So 
(Sy == Ni TS Now 45g 4 6 Se Ne 
Sr = Ma ap do) aP 00 05F rag 
where the only restriction on the accented \’s is that the modulus of the 
substitution should not vanish. The differential equation will now be free 
from &€,, and will contain only »—1 variables. Proceeding as before, we can 
again form a singular solution. If its discriminant vanishes, we reduce the number 
of variables by one; and so, finally, we arrive at an equation of the form 
FOO = O, 
or else a singular solution whose discriminant does not vanish. ‘This latter 
solution we shall for the most part study in what follows; and, unless otherwise 
stated, it will be supposed that the discriminant of the differential equation does 
not vanish. 
If aiz,... 2, are simultaneous values of the variables (not all zero) 
satisfying 
Alba -- Ane Lo + ee 0, 
and if %,%... are any other values such that 
| = | SG Pa eee 
where ¢€ is a small positive quantity, then 
Ay 2," + QA @2o +. 
is of the order e, but if 2), 7, . .. %,, are all zero, it will be of the order &, 2.e. 
the solution S has an infinity of order » — 2 atthe origin, and an (7 — 1)-ply 
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