Remarks on Several Subjects. 131 
This arrangement seems preferable to the glass rods used 
in the great English trigonometrical surveys. 
Brown University. March 20th 1824. 
Ant. XJ.—Remarks on several papers published in former 
volumes of this Journal. (Communicated *y the Author 
to the Editor.) 
. Remarks on the “New Algebraical Series,” giwen by 
Pes don Watuack, in page 278 of this Journal for the 
month of February, 1824. 
These series can hardly be called new, since they are noth- 
ing more than the usual expansion of the binominal quan- 
, a 6 
tity 1—kz to the negative fractional power — gs Obta aEy 
etc. and nearly the whole theory of the functions he has 
named fa, fo, etc is to be found in the Complement des El- 
emens d’ Algebre of La Croix, [Page 161, Edit. 4.] where 
it is stated that the method was first published by Euler in 
the XIX, Vol. of the * Novi Commentarit Academiae Screnti- 
arum faperales Peteopolitane’’ about fifty years since: 
Mr. Wallace supposes, 
2 3 
fa=lt+a—ta(a+h). pa ta(a+k) (a-t2k). ra 
&c. and as the series in the second number of this 
equation is equal to (1 a a expanded intoa series by the 
un 
binominal theorem, if we for brevity: put F=(1—Az) * 
it will become equal to F*, whence 
ee 
Therefore the function fa is equivalent to the power a of 
the quantity F5 and in like manner fO=F’; f(a+')= 
Fe, and as F?x F°=F™, it follows that fa xfo=f(a+6), 
which is the fundamental iheskehi of Mr. Wallace, found by 
the actual multiplication of the series corresponding to 
those functions. In like manner his formulas II, III, IV, 
become respectively, 
