74 Royal Society : — 



Il(x,k,l)=ii(x:,k,l). 



IP(2^+ 1, 2^, l)=i{ii{2x+ 1, 2k,l)-ii{x+\,k, i/)}, 



llX3x+l.3k. l)=^{ii(3x+ 1, 3k, l)~u(x+l,k, i/)}. 



I-P(2a-+ 1, 2k, Z)=uGr + 1, k, il), 



PP(3cr +1,3^.0 = «('«' +l,k,^l); 



RR(2a:+ 1, 2k, l) = u{x+l,k, ^l), 

 RR(2.r+ 1. 2k+l, l)=ii{x, k, i(/— 2)), 

 RR(2a?, 2k, l) = ii(x, k, ^l} + ii{x. k, ^{1- 1)) ; 



IR(2a)+ 1, 2A, l) = ^{ii{2x+l, 2k, l)—ii(x+ l.k.^l)}, 

 lR{2x+\.2k + l,l)=^{iii2x+l, 2k+l, l)-ii{x, k, |(/-2))}. 

 IR(2a 2k, l)=\{ii{2x, 2k, l)—ii{x, k, \l)-ii{x, k, K^— 1)) } ; 



R=RX4a?+ 1, 4k, l)=ii(x+ 1, k, ^l), 

 l-Br{'ix+ 1, 4k, l)=i{ii(2x + l. 2k, il)-ii(x+ I, k, ^l)\, 

 RR'(4^ + 1, 4A, l) = u(2x+l, 2k, il)—u(x+ 1, k, ^l), 

 IR-(4^+ 1, 4k, l)=^[ii(4x+l,4k, l) + 2ii{x+l,k, {l) 

 -3iii2x+l,2k,^l)']; 



RW(6^+ 1, 6k, l)=ii(x+l,k, ^l), RW(7, 3, 3) = 1, 

 PRX6a?+ 1, 6k, l)=^ii(2x+\,2k, ^l)-u{x+l,k, ^l)}, 

 'RR\6x+\,6k,l) = iii3x+i,3k,il)-ii{x + l.k.^l), RR='(7,3,1) = 2, 

 IR'(6.r+l, 6k, l)=^{ii(6x+l, 6k. l) + 3it(x+l, k, ^l 



—ii(2x+l,2k,^l)—3ii(3x+l,3k,^l)}. 

 IR3(7,3.2) = IRX7,3, 1) = IR'(7,3,0) = 1; 



V^W'(x+l,k,x-k)=0. 



By the aid of the above, together with the following, equations, 

 the (.r + A+/)-edra having k + l triangular faces, an (x + k + l—l)- 

 gonal base and triedral summits, are successively found. 



I(x+k4-l. k + l) = -2{l(x + k') . n(x, k', l') + l%x, k') . U-(x, k', I') 

 + lHx, k<) . IP(^. k\ V) + R{x, k<) . m(x, k', V) 

 + W{x, k') . lW(x, k<, V) + R\x, k<) . IR^J^, k', I')} ; &C.&C. 



taken for all values of k' + l'=^k+L 



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

 divisions of F(x + k+l, k + l)^ 



Of the products under S, 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 x'=x + k-\-l, 



P,+,+,=P.,,=P(^',2) + P(y,3) + .... + P(^',i(-r'-l)). 



