264 REPORT — 187b. 



where ft,=f^s 



y=y + 4/3(/3+l), 



and so on ; or, ■what is the same thing, calling the first generating function <j.a; 

 then the second generating function is =J{(f^,v)" + ^;(.r-)}. 



It will be noticed that the bicentre-trecs are not (as were the centre-trees) 

 divided according to the number of their main branches ; they might be thus 

 divided according to the sum of the number of the main branches issuing from 

 the two points of the bicentro respectively ; a more complete division would 

 be according to the number of main branches issuing from the two points 

 respectively ; thus wc might consider the bicentrc-trees (2, 3) with 2 main 

 branches from one point, and 3 main branches from the other point of the 

 bicentro ; but the whole theory of the bicentre-trees is comparatively easy, 

 and I do not go into it further. 



We have yet to consider the case of the limited trees where the number of 

 branches from a knot is equal to a given number at most: to fix the ideas, 

 say the carbon-trees, where this number is =4. The distinction as to root- 

 trees and centre- and bicentre-trees is as before, and the like theory applies 

 to the two cases respectively. Considering first the ease of the root-trees, 

 and referring to the former figure for obtaining the trees of altitude N from 

 those of inferior altitudes, then the trees a, h . . .oi altitude N — 1 must be 

 each of them a carbon-tree of not more than (4—1 = ) 3 main branches : this 

 restriction is necessary, inasmuch as if for any such tree the number of main 

 branches was =4, then there would be from the root of such tree 4 branches 

 jihis the new branch shown by the dotted line, in all 5 branches ; and simi- 

 larly, inasmuch as there is at least one component tree a contributing one 

 main branch, the number of main branches of the tree c must be (4—1 = ) 3 

 at most : the mode of introducing these conditions will appear in the expla- 

 nation of the actual formation of the generating functions [see explanation 

 preceding Tables III., IV., &c.]. The number of main branches is =4 at 

 most, and the generating functions have only to he taken up to the terms in 

 f ; the first factor is consequently in each case affected Avith a symbol [*' ' ' • ^], 

 denoting that the only terms to be taken account of are those in f, f, f, t^ ; 

 hence as there is a factor t at least, and the whole is required only up to t\ 

 the second factor is in each case rec[uired only up to t^. 



As regards the centre-trees,' the generating functions have here the same 

 expressions as for the root-trees, except that instead of the symbol [<^-'*], 

 we have the symbol [<^ ••''], denoting that in the first factor the only terms 

 to be taken account of arc those in f, f, t* ; hence as there is a factor f at 

 least, and the whole is required only up to t\ the second factor is incachcaso 

 required up to f ; and wc then complete the theory by obtaining the bi- 

 centre-trees. The like remarks apply of course to the l)oron-trces, number 

 of branches =3 at most, and to the oxygen-trees, number =2 at most ; but, 

 as already remarked, this last case is so simple, that the general method is 



