115 
(A)... (r+ nyt dz) (a+ Ky HAZ) = 
(m,° — 1, 1,) xa! + (4, m, — mz, m3) (yz + zy’) 
+ (m,? — 1; 1,) yy! + (L, mz — m; m,) (za + x2’) 
+ (m2 —1, 1.) 22’ + (1, m3 — m, mz) (ay + y2’) 
+ (el, + Km; + Am) (yz - zy’) 
+ (il, + Am, + um,) (22° — x2’) 
+ (Ad, + um, + Km) (ay’ - yx’), 
then this expression, which he proposes to call a QuaDRINOME, 
has many properties (associative, modular, and others), analo- 
gous to the quaternions ; which latter are indeed only that case 
of such quadrinomes, for which, 
(SpSeS 0 
mM, =M,= m;=0, 
of — ia ee 
He has, however, found another distinct sort of associa- 
tive quadrinomial expression, which has also several analogous 
properties, and for which he suggests the name of TeTrRaps ; 
the product of two vectors being in it, 
(B) .. . (e+ my + nz) (l' + my + nz’) 
+(kn — Xm) (yz - zy’) + (Al =m) (22 - wz’) 
+ (um —«l) (ay -— yx’). 
Dr. Allman communicated the results of some observations 
he had just made on Aphanizomenon Flos-ague. This mi- 
nute alga had begun about three weeks previously to make its 
appearance in great abundance in the large pond of the Zoolo- 
gical Gardens. The best account we possess of the plant is in 
an excellent paper on the Nostochinee, by Mr. Ralfs ;* but as 
the specimens from which Mr. Ralfs’s description was drawn 
up were not in a recent state, some important points of struc- 
ture have necessarily escaped him. 
A. Flos-aque shows itself in the form of little fusiform fasci- 
* On the Nostochinee. By John Ralfs, M.R.C.S., Ann. and Mag. of Nat. Hist. 
May, 1850. 
