ON THE ANALYTICAL FORMS CALLED TREES. 283 



the only difference is that the first factors are to be taken without the terms 

 in fi : thus for Table II. column 3, the first factor of the GF 



the second factor being as for Table I. 



■(+'■)■' 



The remaining Tables are Tables III. and IV., oxygen root-trees and 

 centre- and bicentre-trees, followed by a Subsidiary Table for the calculation 

 of the GF's : Tables V. and VI., boron root-trees and centre- and bicentre- 

 trees, followed by a Subsidiary Table ; and Tables VII. and VIII., carbon 

 root-trees and centre- and bicentre-trees, followed by a Subsidiary Table. The 

 explanations given as to Tables I., II. and the Subsidiary Table apply 

 muUitis mutandis to these ; and but little further explanation is required : 

 that given in regard to the Subsidiary Table of Tables III. and IV. shows how 

 this limiting case comes under the general method. As to the Subsidiary of 

 Tables V. and VI., it is to be observed that each* line of the Table is calcu- 

 lated from a column of Table V., rejecting the numbers which belong to f ; 

 thus Table V., column 4, the numbers are 



and taking the sums for the first and second lines only, these are 



1, 4, 9, 17, 29, 45 . . ., 



which, taken with a negative sign, are the numbers of the line *6F, column 5. 

 And so as to the Subsidiary of Tables VII. and VIII., each * line of the 

 Table is calculated from a column of Table VII., rejecting the numbers which 

 belong to t* ; thus Table VII., column 4, the numbers arc 



and taking the sums for the first, second, and third lines only, these are 



1, 4, 13, 32, 74, 155 . . ., 



which, taken with a negative sign, are the numbers of the line *GF, column 5. 

 Referring to the foregoing " Edification Diagram," the effect is that we 

 thus introduce the conditions that in a boron-tree the number of component 

 trees a,b, . . is at most (3— 1 = )2 and that in a carbon-tree the number of 

 component trees a, h, . . is at most (4— 1=)3. 



