146 
Again, operating on the equation 
pr=m7p+l, 
w( )a, 
with 
we get 
pw =r p-7?*; 
from which we deduce 
pr =7"p-nr". 
So that the equation 
pr = "9 +27", 
holds good for any integer value of n. 
From this again we infer that 
p Yr= dr ptr, (5) 
where Ym represents any function of integral powers of 7. 
And from (5), finally, we can ascend to the more general 
theorem, 
, ‘it 1 "4 4 
op da = Ur op tyr ppt Tam op tke. (6) 
Changing 7 into p, and p into —7, in the last two equations, 
we obtain the correlative ones, 
7 $p= 9p ™- Pp; (7) 
U 1 " a 
vr gp=9p Yr-9'p a+ > op Yr Ke. (8) 
In the theorems here given the reader will recognise an ex- 
tension to the symbols 7 and p of the theorems respecting x 
and D, stated by Dr. Hargreave at the commencement of his 
remarkable paper on the Solution of Differential Equations, 
printed in the Transactions of the Royal Society for 1848. 
Having obtained the theorems, 
oD pr= a oD+Pa ¢D+ a vx 9’ D+ &e., (9) 
