322 
On the Nest and Eggs of the Common Tern. 
Only two of these equations are independent and it is convenient to write 
them in the form : 
Vipi + (1 - v.^ (/, = 6," + |e.,") 
(v). 
v-ilh' + (1 - f.) qi = fi J 
These will give, if and are known, one equation for the determination of 
v„ and one equation of condition. 
Ghitches ofS eggs. 
If the distribution of clutches be N{ei" + e,/" + e.'" + e^'") we have : 
r,pi' +{l-v,)q.'^ =6/", 
Ty.Pi-qi + (1 - vs) q^ih = heJ", 
v-iP.q,- + ( 1 - I',) q-ip.^ = 1 63"', 
»'39i' + (I - V:^ pi =64"'. 
Only three of these equations are independent and these may be written : 
v.ip,^ + {l-v.^qi = e^" \ 
v,p^ + {\-v^qi = e;" + \ (vi). 
v,p, +{l-v,)q, =e;" + K" + K") 
These suffice to determine, v-,, pi and q^. 
Uniting the right-hand sides of (vi) f:'.",f-l",f\" respectively we find : 
v-i --- ifi" - q-^liPi - 7-2) (vii), 
which suffices to find v-^ when ^1 and q^, are found, and 
/..-(/."■-/."■'/.)/(/,■" -9.) I ■ 
p,' =(./;:"' -//WW" -s.) I 
which lead to the quadratic for q.. 
q^ i/r -A'"') - q. i/r -fr.fr) +A'yr -fr^ = o (ix). 
We could therefore solve (ix) and choose the appropriate root for g„, find the 
corresponding p^ from (viii), determine v.^ from (vii), v.^ from the first of (v) and 
from (iv). We might then use the second equation of (v) as an equation of 
condition. But clearly this would not be satisfactory as all our quantities are 
subject to considerable sampling eri'ors. The correct method would be to deter- 
mine i'j, Vo, V-,, pi and qo from the sia equations (iv), (v) and (vi) so as to get the 
best values of these variables. But this would be a very laborious process. We 
propose therefore to determine pi and q^ by the method of least squares from the 
three equations 
P. =(./;" -A"qdi(fr -?.)*) 
=(./;'"-/;'"'/.)/(/;'"-?.) I 
* Obtained by writing /i" = fi" + AE2 ", /■>" = ^\" and eliminating between two equations of (v). 
