115 
(A)... (r+ yt dz) (a+ Ky + AZ) = 
(m,? — 1, 1s) xa’ + (1, m, — mz ms) (yz + zy’) 
+ (m;? - 1; 1,) yy’ + (hi, m, — msm) (22" + 22’) 
+ (m;? — 1, 1;) 22 + (1; mz — m, m2) (xy + yx’) 
+ (1h + xa, + Ams) (y2 = 2y/) 
+ (xh, + Am, + m,) (Z2" — xz) 
+ (Ad; + um, + Km) (ay’ — y2’), 
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, 
= oy Ae 
Mm, =M,= m,=0, 
Pe fc fee 
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 Terraps ; 
the product of two vectors being in it, 
(B) . . - (le+my + nz) (la + my + nz’) 
+(«n — dm) (yz - zy) + (Al-m) (22 - 22’) 
+ (um=«l) (ay' - yx’). 
Dr. Allman communicated the results of some observations 
he had just made on Aphanizomenon Flos-aque. 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- 
* Onthe Nostochinee. By John Ralfs, M.R.C.S., Ann. and Mag. of Nat. Hist. 
May, 1850. 
