34 
MR. HARGREAVE ON THE SOLUTION OF 
The solution here given is finite, in those cases only in which m is an integer posi- 
tive or negative. When m is. fractional, the undeveloped expression involves frac- 
tional operations. 
If however 6=— 2c so that the roots are equal, the solution assumes the form 
M = ' H- A:') 
without any restriction upon the values of m. 
The form (7.) deserves particular attention, as it will be found to include the most 
remarkable of the equations of the second order, which have heretofore been inte- 
grated by artificial methods. 
Thus, if %px=e‘‘"' we have the solution of 
-1- (^6 -f 2a -b Dm 4- ^c2 -f a6 -l-a2 + (6 -b 2a)^) M = P, 
of which the well-known equation 
D^u-i-^Du±c^u=P (8.) 
is a particular case ; the solution being 
M= (D24:c2)”'~4'^''*(I^^i^^)’'’”('^P) }• 
If P=0, this is reduced, taking the negative sign only, to 
M = (D2— } , 
which will be found to be 
and the solution is expressible in finite terms when m is a positive or negative integer. 
This equation is merely the simplest form of (y.), and is soluble by (5.) without the 
aid of (1.) ; for in it 4^^ is taken to be unity. 
Now let %pa:=a :” ; then the solution of 
D%+(6+M)d«+ 
If in this 6—0 and m=— 2Mi-bl, we have another form of solution of the equation 
D2^_?(!^DM-bc2M = P ; 
a? 
namely, M=j;^"‘-‘(D2-bc2)-4a'-^(D2-bc2)-(x-“P)}. 
If n——m, we have for the solution of 
D2M-b&DM-b (c2-’^^^^)m = P 
M =x’”(D2 +bl>+ c2)— * {o--’ (D2-b &D -b c2) ; 
