17 
tribution to tlie subject of linear differential equations. I am 
not in a condition to make a substantial addition to Mr. 
Harley’s formulae, but the following, derived by a mere change 
of notation from his, seem to me to give a new interest to his 
valuable results. Write 
y n — ny" ’'4-r.r=0 . . . (Ill) 
n 
11 — i 
[(m — r)D] n 1 y 
(E) 
or, which is the same thing, 
(F) 
and, further, suppose that X is equal to ox 1 ', where 
orp=(l— 2]"“®, or /)=.lml[n— 2]”- 2 
n — r ax r — 1 
or any other discontinuous function of the like nature. Then 
(E) or (F) is, in the cases r=n — 1 or r— 1, the differential 
resolvent of (III). 
Mr. Harley added the following remarks : — 
Mr. Cockle’s formula, (E) or (F), is interesting and useful, 
as comprehending the differential resolvents of both the 
trinomial algebraic equations 
y" — ny-\- (n — l)x= 0 
and 
y n — }iy n ~ l x=z 0 
under one and the same general expression. For these 
cases, (r=l and r — n — 1,) to which the author confines it 
by express statement, the formula holds good ; but I do not 
find that it holds for other cases. If {ex. gr.) we assume 
n — 4: and r— 2, 
the equation which in Mr. Cockle’s communication is 
marked (III), becomes 
V K — fy 2 + 2.r=°. . . («) 
