18 
and the formula (E), after some slight reduction, gives 
2 4 (D)(2D — l;(D — l)y=(4D — 3)(2D — 3)(D — 2) *- 6 y 
= c 2<? (4D + 5)(2D + l)(D)y; 
whence 
d?y 
dry 
Ay . 
2”(2’-*>^+2 (2 J .3 - 1 9*°) &-9X& = 0, 
dx'~ 
dx 
an equation which I find, on trial, is not satisfied by the roots 
of (a). In fact, the differential resolvent of (a), obtained by 
direct calculation, is 
2\2-.r)x ( ^+2\3{l-x) 
dry 
dx L 
■3.o|=0... W 
and the Boolian or symbolical form is 
2 4 [2D Jy— [4D — 3 Je°y = 0, 
which, expunging a common symbolical factor, may be 
reduced to the form 
2 4 D (2D — 1 ) y — (4D — 3)(4D — 5) Ay — 0, 
and returning to the ordinary form of a differential equation, 
the last result may be written thus : 
This equation I have also obtained by direct calculation. It 
is worth while noticing that (f 3 ) and (y) may be combined in 
one equation 
| 3.5 2 4 y(l — x) | ~ + '/u/=0...(2) 
in which /.i is a perfectly arbitrary function, either constant 
or otherwise. 
I content myself at present with simply recording these 
results, without attempting a discussion of them; but I 
believe they will be considered, by cultivators of the calculus, 
as not wanting in interest. 
