44 
MR. HARGREAVE ON THE SOLUTION OF 
du 
Again, equating the factors of ^ and u to zero, we have and ?a“=a 
therefore 
dx^'~ dxdy'^ ^ dy^' x dx ^ 
j,— Sttj , 
is soluble ; and if the variables be changed by the equations 
p = 
1 
q=.kx- 
'Vi (2771 1 ) * . 
2m- 1 
the form 
d!^u , li /'du , du 
' ' — ' 
dpdq ' 2 
fp-Vq\ fdu du\ 
(, 2yb j \dp'^Tq)-^^'P^i) 
becomes soluble. 
By a process of a similar nature applied to (10.), it will be found that the form 
has for its solution 
— D 
V. Connection with Dejinite Integrals. 
It is well known that many of the differential equations integrated by the above 
processes, and whose integrals are in some cases capable of an expression merely 
symbolical by reason of the number of operations to be performed being fractional, 
may be integrated generally, when there is no second term, by means of definite 
integrals. 
Now with reference to most of the equations of this description here integrated, 
I have observed that the symbolical form above given is capable of being instantly 
(and, as it were, mechanically) converted into a definite integral of the form 
the function <pz being typified in the symbolical solution by the form of the opera- 
tions preceding the factor x"*. 
To explain this, let us take the equation 
its symbolical solution is 
m = A:(D2-c 2)— {^’-’(r)2-c2)-”‘0} ; 
