485 



Similar equations are to be formed for the remaining five sub- 

 divisions of P(x + k-}-l, k + l). 



Of the products under , the first factors are found by the pre- 

 ceding part of the process, and the second are given by the equa- 

 tions above written as solutions of the problem. The factors will 

 of course frequently be zeros. Finally, if z'=x 



Thus, to give an example, 



1(11, 2) = F(9, 2) . IP(9, 2, 0) + 1(9, 2) . 11(9, 2, 0) ; 



I(11,3) = I(8,2).II(8, 2,1) + R(8,2).IR(8,2,1) + 1(8, 3) .1(8,3,0) 



1(11, 4)=P(7, 2) . IP(7, 2, 2) +R(7, 2) . IR(7, 2, 2) 



+ R 3 (7,3).IR 3 (7,3, 1); 

 F(ll, 2) = I 2 (9, 2) . PP(9, 2, 0) ; 

 P(ll, 4)=P(7, 2) . PP(7, 2, 2) ; 

 R(ll, 2) = R(9, 2) . RR(9, 2, 0) ; 

 R(ll, 3) = R(8, 2) . RR(8, 2, 1) ; 



R(l 1, 4)=R(7, 2) . R(7, 2, 2)+R 3 (7, 3) . RR 3 (7, 3,1); 

 R(ll, 5)=R(6, 2) . RR(6, 2, 3). 



The result is 



ii=61 +7 + 12 = 80. 



XV. " Notes on British Foraminifera." By J. GWYN JEF- 

 FREYS, Esq., F.R.S. Received June 19, 1855. 



Having, during a great many years, directed my attention to the 

 recent Foraminifera which inhabit our own shores, I venture to offer 

 a few observations on this curious group, as Dr. Carpenter, who has 

 favoured the Society with an interesting and valuable memoir on 

 the subject, seems not to have had many opportunities of studying 

 the animals in the recent state. 



