exvili Proceedings of the Asiatic Soc. of Bengal. [N.S., XVII, 
oe Babies the rei unit of grouping, different values of y? 
ually obtaine 
(il) Withd diffrent waite of grouping, the different values of P may 
differ quite pepeerin ably. 
in ™m | ape aapectaliy in the case of small sam les. umerical 
examples show that samples of 100 or 200 (in the ease of anthropological 
) are not rk ae enough to avoid appreciable fluctuations due to 
am ~ 
observed sae by m, ’ the orresponding pang frequencies. 
It was found _ this led to a considerable improvement in practically 
eve case considered so far (i.e. in about 20 different numerical 
exam 
his pro sige also affords certain mathematical advantages. In 
many cases it is possible t vince a suitable analytical expression for 
mand m’. Inan Bispaleny. m a m’ are usually given (at least to a 
first iP eae anatige by the cs nor sal” ( eee ) curve of errors. 
in detail. The subject of integration may become infinite ponivs certain 
cks: with exception of such Special cases, it seems that the new 
formulae may be of some use in obtaining quickly the appropriate 
value of $2. 
Binary Commutative Algebras.—By R. VatwyYaNATHA- 
Swamy. 
1. General Theo A binary commutative linear algebra in two 
units — €2, is Goutpletaly defined by a multiplication table of the form :— 
Aye] H Age, § C)e,=u eee ey2= ye) + rp. 
The _— is of rank 3, the identical equation satisfied b 
umber eC) +X 2ey eing py )+xD(x)=o where L(x) As 4 Diz) 
are stiaiidion ovariant linear and quadratic forms i Z,% Writing x? 
Site ler: oa + Talley, the form /(x)= X\fg—2of, is alsoa covariant of the algebra. 
very covariant of the algebra is shewn to be a covariant of the binary 
forms ‘y Hee cL. In gephes tes the Soto invariants of the algebra 
are ogy as th > ages e A is the sina of f, 
Ag=(f, £4), os ae rag pee tes the. teeing of ¢. Two algebras whee 
equal, finite absolute case are equivalent under ace transfor 
mation of units, 
The e following special algebras ure per peenarg ae: Singular algeb- 
= “, D, L =O); (6) Se are algebras of ran (D, L=O); (c) 
tary algebras (f=0) ; (da) N ull algebras (D= , as ) Normal — 
(Lassy: ip iE copies ike a tages oe ee ; (g) Primitive algebra 
) The necessary a pa: — conditions for a principal unit 
are ig ADs: (2) Disa factor of f; (3) the remaining factor of / is not a 
fact 
2 . lied only algebras of rank 2 are (a), (c) and (f). 
(3) TF e only associative ee are (a), (6) and (/). 
3. For all the above speci algebras a certain invariant A‘ 
vanishes, resenting every aigrbra-type by a point in a plane with 
omogeneous co-ordinates,=th Ao, the algebras 
for which _ are represen ‘d by a radial cubic venom lle es i 
rere at the cusp and inflexional tangents ing a 
ng 
R be the oink on this cubic corresponding to the aioeees wih. a 
