27 S J^ew Algebraic Series. 



Culm about 20 inches high, leafy ; leaves sheathing, short-' 

 er below, reddish brown at the root, linear-lanceolate, 

 slightly scabrous on the margin, higher ones longer than 

 the culm ; sheaths white or yellowish white opposite the 

 leaf and like the leaves, pubescent : bracts scarcely 

 sheathing the peduncles except the lowest, leafy and lon- 

 ger than the culm, less pubescent than the leaves ; an- 

 drogynous spike from the same bract as the upper pistillate 

 spike : spikes filiform with a flexuous rachis, alternate, 

 somewhat pendulous and rather loose flowered, peduncles 

 longer than the sheaths ; staminate scale lanceolate, long, 

 acute, hyaline with a green keel ; pistillate scale oblong- 

 lanceolate, extending more than half the length of the fruit 

 and terminating in an awn beyond the fruit, hyaline with a 

 green keel. According to Muh. the upper spike is some- 

 times wholly staminate and the scale shorter than the cap- 

 sule. The latter variation I have rarely seen, and the for;- 

 mer has not occurred to me. 



MATHEMATICS AND MECHANICS. 



Art. X. — J^few Algebraic Series hy Prof. J. Wallace, Co- 

 lumbia, s. c. 



z .. z- 



Let 1 4-aY+o(a+^)YY+a(a+A-)(a-f2^) YTs"^^^' 



z 



and l+bj-\-b{b+k)-Y-^^b{b+k){b + 2k) J7^+^^-c. 



be the given series, the latter differing from the former on- 

 ly, in the substitution of b for a. If these two series be 

 multiplied into each other, the resulting series will be of 

 the same form; a-{-b being taken for a or 6 in the above; 

 or the form will be 



l4-(«+6)y +(a+6)(a+6-f A-)-^ +(«+&)(a+6+A;)fa-{' 



For the series being actually multiplied and the terms 

 placed in order, we shall have. 



