TRANSACTIONS OF SECTION A. G13 



Tlie vorticity and size of each f^liell must satisfy definite I'elations. When the 

 vorticity of the fluid is everywhere of the same magnitude the ratio of the (n + 1 jth 

 radius to the nl\i satisfies au algebraical relation of the form 



4X„ (.r„- + .vn + 1) = S.v' (.)•„ + 1), 



whereX„ = l-A„_| (l-.r\,_i). _ 



The ratio of the volumes of the shells for the first three are 1, 1'3I3, 1'341. 



4. 0)1 a Dynamical To})} By G. T. Walker, Al.A, 



The author exhibited a top in the shape of a flattened ellipsoid with a central 

 circular portion movable, and arranged so as to be clamped with the lines of 

 curvature inclined to the axes of dynamical symmetry. In this condition a rota- 

 tion communicated to the top when placed ou a sheet of plate glass sets up 

 oscillations which reverse the direction of motion : these reversals may, under 

 favourable conditions, be as many in number as fifteen. A vertical tap administered 

 at the end of an axis of symmetry gives rise to angular velocity, of which the sign 

 depends on the diiierence between the periods of longitwdinwl and transverse 

 vibrations, as well as on the angular deviation of the movable portion. 



5. Suggestions as to Matter and Gravitation in Professor Ilicks's 

 Cellular Vortex Theory. By C. V. Burton, D.Sc, 



On the Graphical Representation of the Partition of Numbers. 

 By Major P. A. Macmahon, F.R.S. 



7. On a New Canon Arithmeticus. 

 By Lt.-Col. Allan Cunningham, E.E., Hon. Fell. King's Coll. Lond. 



This is a series of tables, drawn up precisely like Jacobi's ' Canon Arithmeti- 

 cus,' giving the solution of the congruence 2-^sR (mod. p and mod. m) for all 

 prime moduli (/j) < 1 ,000, and also for all moduli m < 1,000, where m is a power 

 of a prime. There are two tables to each modulus, 7^ or m. The left table shows 

 the remainders (R) to a given index (.r); the right table shows the index (.r) to 

 a given remainder (R). 



Uses of Tables. 



1. To find remainder R after dividing 2^ by p or m {x, p, ni being given). 



2. To find index .v such that 2^-=-;j or in may leave remainder R (R,p, m being 



given). 



3. To find whether a given number N is exactly divisible by a given prime j», 



or power of prime m; and, if not, what remainder (R) is left. 



4. To find whether a given number N is exactly divisible, or leaves a given 



remainder (R) after division, by any prime or power of a prime < 1,000. 



5. To find all the primitive roots of a given prime (/j), and all the roots which 



are residues of a given order e of a given prime p, when 2 is a primitive 

 root of p; and to find all the roots which are residues of a given order e 

 of a given prime ^ when 2 is a residue of order not >e. 



Jacobi's Canon gives the solution of ^r^sR (mod. p and mod ?») as above, ex- 

 cept that (/ is a certain primitive root of ;;. His table is better for case 5 above 

 whenever 2 is not a primitive root of ^ ; but the new canon to base 2 is much more 

 convenient for the more practical uses 2 and 3 above. His canon is well described 

 in Cayley's Report on Mathematical Tables in the British Association Report of 

 1876 : the description applies, with slight obvious modification, to the new canon. 



' The paper will be published in the Quarterly Journal of Mathematics. 



