ON MATHEMATICAL TABLES. 305 



viz. in A the columns 2, 3, 4, and the last column are the totals given by 

 the Tables IV., VI., VIII., and II., and the remaining- numbers of columns 

 5, 6, 7, 8 have been found by trial ; and in B the several columns are the 

 differences of the columns of A. The signification is obvious ; for instance, 

 if the number of knots is =9, then Table A, if valency, or maximum number 

 of branches from a knot, 



is = 2, 3, 4, 5, 6, 7, 8 or any greater number, 

 No. of trees = 1, 18, 35, 42, 45, 46, 47 : 



viz. with 9 knots the tree can have at most 8 branches from a knot, so 

 that the number of trees having at most 8 branches from a knot is =47, the 

 whole number of trees with 9 knots ; and so the number of knots being as 

 before =9, Table B shows that the number of 47 is made up of the numbers 



1, 17, 17, 7, 3, 1, 1 ; 



viz. 1 is the No. of trees, at most 2 branches from a knot, 



17 

 7 

 3 

 1 



1 



I annex also a Plate showing the figures of the 1 + 1 + 2 + 3 + 6 + 11 + 23 + 47 

 trees of 1, 2, 3 . . . 9 knots, classified according to their altitudes and number 

 of main branches ; and as to the bicentre-trees, according to the number of 

 main branches from each point of the bicentre. The affixed numbers show 

 in each case the greatest number of branches from a knot ; so that when this 

 is (2), the knots may be oxygen, boron, carbon, &c. atoms ; when (3), boron, 

 carbon, &c. atoms ; when (4), carbon, &c. atoms; and so on. 



Report of the Committee, consisting of Professoi' CayleYj F.R.S., 

 Professor Stokes, F.R.S., Professor Sir W. Thomson, F.R.S., 

 Professor H. J. S. Smith, F.R.S., and J. W. L. Glaisher, F.R.S., 

 on Mathematical Tables. (Professor Cayley, Reporter.) 



The present Eeport (say Keport 1875) is in continuation of that by Mr. 

 Glaisher, published in the volume for 1873, and here cited as Report 1873. 



Ueport 1873 extends to all those tables which are at p. 3 included under 

 the headings : — 



A, auxiliary for non-logarithmic calculation, 1, 2, 3 ; 



B, logarithmic and circular, 4, 5, ; 



C, exponential, 7, 8 (but only partially to C 8), other than those tables 

 of C referred to as "h .1 tan (45°-|-|^) ;" and also partially (see art. 24, 

 pp. 81-83) to the tables included under the heading "E. 11, transcendental 

 constants e, tt, y, &c., and their powers and functions." 



A future Report will comprise the tables, or further tables, included under 

 the headings : — 



C. 8. Hyperbolic antilogarithms (e'') and h.ltan (45°+|^>), and hyper- 

 bolic sines, cosines, &c. 

 1875. X 



