172 
inflection of curves. I have not pursued these inquiries, 
though I hope soon to be in a condition to call attention to 
a class of functions for which I shall propose the name of 
“ differential covariants,” the existence of which is suggested 
by that of “ differential critical functions.” My memoranda 
however have not yet arrived here. 
With respect to the theory of transcendental solution, I 
would venture to suggest the publication of Professor Boole’s 
solution of a certain quartic resolvent, in a form involving a 
single definite integral. Mr. Rawson’s researches, as yet 
unpublished, on the algebraical equations on which given 
differential equations depend are, too, of high interest. 
In reference to my communication printed at pages 16-17 
of the “ Proceedings of the Literary and Philosophical 
Society of Manchester,” (meeting of December 2, 1862), and 
to Mr. Harley’s remarks thereon, let me observe that we may 
write the Boolian of (a) in the form 
4 3 [2D] 3 y - 2 2 [4D - 3fe°y=0, 
and that the portion 
[4D-3] 3 e 0 
of the sinister is deducible from the portion 
t d ,-in-r-ir n 
" r ar r - 1 J ‘ [r 
a 
x — — 
n + r~\r 
dx 
]V 
of the sinister of (F) by changing r into r — 1 in the exponent 
of the first factorial and throughout the whole of the second 
expression and substituting. For this change and substitu- 
tion gives 
r d i n d n + r— In 
I D-T*3rW3T-] 
l-W-l 
x' 
,r-l 
or, making w=4, r—2, 
[ 
5 ] 
i 
x 
or 
