2 SEVENTH REPORT—1837. 
On some New Properties of Geometric Series. 
By Cuarves Biacxeury, A.B. 
This paper consisted of a series of geometric theorems, of a some- 
what novel character ; but from its length and abstract nature did not 
admit of being read to the Section. ‘The author explained one of the 
theorems, and in a brief statement enumerated some of the purposes to 
which they are applicable. 
It appeared that the paper contains formule, ates 
1. For finding the products of a variety of factors of particular 
forms. 
2. For reducing expressions hitherto considered fractional to integers. 
3. For reducing fractional expressions to equivalent ones, of which 
the terms shall be, of m and x less dimensions ; or to other equivalent 
fractions of more convenient forms. 
4. For the resolution of geometric series into any proposed number 
of factors. 
Of the first kind of theorems, the 2 following are given as examples. 
(1) 
Let m and be whole numbers, # and y any quantities; then 
n n n 
a” TT gam Py gg By 0g tym 34 ym 24 ym ol 
= { am — Ly gm—2y 4 gm—3 yo 4 phe Bi wooo DIYM—3 4 gym —2 syn} 
* { m(m=1) 4 ym Kaneohe J NPS pees cone ey” (m—2) +y™ ini} 
Aig” ce Vel eit Sohal Pe this eee an 
x { m=) 4 amma men" a atm: (m—2) eee (m1) } 
y 
(2) 
Let mn p, &c.=N, then 
ANAT gN-2y 4 N-B yey 28 yN—3 4 gyN-2 4 yN=I 
> { ml eg dal RO ye es ou eos . 2m +ay™—? 2 aioe } 
Xfm OD MBs pamym (en) ym (nnd p 
4 { gm (>—1) 4. rm (p—2), ee. hae ae rmymnn(o-2 4 ymn(p—1) b 
&e. &e. &e. &e. 
Other theorems will probably appear in some other scientific publi- 
cation. 
