of Edinburgh, Session 1869-70. 191 
been included, but is kept separate for a special purpose.) 
Hence 
du — — S (dpV)u 
= — ?7iS . 6<3~ dp 
= - 8 .Odr, 
if we put 
dr = mV. cr dp 
so that m is an integrating factor of Y. <r- dp. If a value of m can 
be found, it is obvious, from the form of the above equation, that $ 
must be a function of r alone ; and the integral is therefore 
u = F(r) = const. 
where F is an arbitrary scalar function. 
Thus the differential equation of Cylinders is 
S(a V)u = 0 , 
where a is a constant vector. Here m = 1, and 
u = F(Vap). 
That of Cones referred to the vertex is 
S (pV)u = 0. 
Here the expression to be made integrable is 
Y. pdp. 
But Hamilton long ago showed that 
dJJp dp Y. pdp 
Of = v 7 = (TpF 5 
which indicates the value of m, and gives 
u — F(Up) = const. 
It is obvious that the above is only one of a great number 
of different processes which may be applied to integrate the 
differential equation. It is quite easy, for instance, to pass from 
it to the assumption of a vector integrating factor instead of the 
scalar m, and to derive the usual criterion of integrability. There 
is no difficulty in modifying the process to suit the case when the 
right hand member is a multiple of u. In fact it seems to throw 
a very clear light upon the whole subject of the integration of 
partial differential equations. But I have not at present leisure to 
pursue the subject farther than to notice that if, instead of 8(<r^ V), 
