243 
EXAMPLES. 
By this method may be treated the equation first proposed for solu- 
tion by Euler, but, since his time, more completely identified with the 
name of another German mathematician. The equation alluded to is 
that known as Pfaff’s differential equation, recently discussed by Dr. 
Boole in his valuable work on Differential Equations, and which, as re- 
marked by this latter distinguished author, includes all examples of the 
second order, which are susceptible of reduction to the binomial type. 
It is of the form 
(a+a'x”) Dut (b+b/a") eDut(et+cx”)u=X. 
By the same method may be treated the still higher equation of the third 
order 
(a+ a’x”) 2 Du + (b+ b'x”) a? Du + (0 + x") cDut (d+ Ve”) u=X, 
or, more generally, the differential equation of the r” order, 
(a+a'x") a’ Du + (b+ 0/2") x Dut &e. + (k+ Wa") u=X, 
as is plain from observing that this last equation may be reduced to the 
form 
(ax Dr + bx D™ + &e. +h) ut 2” (a'a’ D" + Wa" D™ + &e. + K)u=X, 
or, transferring x” to the right-hand side of the second operator, 
{acD (ecD-1)..(e@D-r+1)+baD (eD-1).. (eD-r+2)+ &e.} uw } 
+ a 
{a (aD -n)(zD-n-1)..(eaD-n+r-1)+)/ @D-n) 
..(e@D-n+r-—2) + &e.} aru 
=X, 
which is obviously of the required form. 
It will suffice to exhibit the application of the method to Pfaff’s equa- 
tion. This plainly can be reduced to the form 
(xD - a,) (cD - a) u+k («aD - B,) («&D-B,) xu ==Ma”. 
Hence at once we get 
(zD zr: B,) (xD - B,) Max" 
k — = _——$— 
c= .)\GDoa)  .  @—a)(n—s) 
or, finally, in full, 
+ A, 2,4 A, 2%, 
