1900. No. De ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. 9 
We shall enquire whether it is possible so to determine A as to resolve 
it into linear factors. The form of the given equation suggests what 
the forms of the linear factors must be, if the resolution be possible. For 
Ou 
as the squares of — = and 2" =. both appear, and these squares only, in the 
; Må Ou We. JE 
function to be resolved, it is clear that = and 5: will be the only diffe- 
rential coefficients of z, which will appear in both linear factors in common. 
We are then led to assume, as the proposed equivalent of our function, an 
expression of the form 
Ou ou ou 1 24 
atmet re] LE - tm | 
where m, x and #1 are indeterminate quantities. 
Multiplying the factors of this expression together, and then equating 
the coefficients with those of the first member of (10), we have 
MO (VD) 
C+ 16 = — Im! — Dm 
A=m.m!' 
1D)= 7 12) 
14 = n .m! 
From the last three equations we find that 
DV CD AA, m FR 
These values reduce the second equation of condition to 
433 + Bi? + Ch + D=o, (A) 
so that A is determined by a cubic. The resolved form of equation (10) 
now becomes 
I Ou Ou Ou Ou 
[257 + te 3 [PG +42] =>. 
Let now A, and A, be two values of A determined by (A), then our 
given system (9) is eqivalent to the following equations: 
I Ou , Ou ou 
Rn a a I [2s +uaG]|=- 
I ou 
Ou Ou Ou ou | 
nat te] [Pe +242]=° | om 
QUN Ou cu\ Ou Ou ou 
a ele a p(s) or er VE ae 
