526 REPORT — 1901. 



Euler, Legendre, Gauss, Eisenstein, Jacobi, Kronecker, Poincar6, and Klein are 

 great names that will be for ever associated with it. Who can forget the work of 

 H. J. S. Smith on homogeneous forms and on the five-square theorem, work which 

 gave rise to processes that have proved invaluable over a wide field, and which 

 supplied many connecting links between departments which were previously in 

 more or less complete isolation ? 



In this connection I will further mention two branches with which I 

 have a moi-e special acquaintance — the theory of invariants, and the com- 

 binatorial analysis. The theory of invariants Avas evolved by the combined 

 efibrts of Boole, Cayley, Sylvester, and Salmon, and has progressed during 

 the last sixty j'ears with the co-operation, amongst others, of Aronhold, 

 Olebsch, Gordan, Brioschi, Lie, Klein, Poincare, Forsyth, Hilbert, Elliott, and 

 Young. It involves a principle which is of wide significance in all the subject- 

 matters of inorganic science, of organic science, and of mental, moral and 

 political philosophy. In any subject of inquiry there are certain entities, the 

 mutual relations of which under various conditions it is desirable to ascertain. 

 A certain combination of these entities may be found to have an unalterable value 

 when the entities are submitted to certain processes or are made the subjects of 

 certain operations. The theory of invariants in its widest scientific meaning 

 determines these combinations, elucidates their properties, and expresses results 

 when possible in terms of them. Many of the general principles of political 

 science and economics can be expressed by means of invariantive relations connect- 

 ing the factors which enter as entities into the special problems. The great 

 principle of chemical science which asserts that when elementary or compound 

 bodies combine with one another the total weight of the materials is unchanged, 

 is another case in point. Again, in physics, a given mass of gas under the 

 operation of varying pressure and temperature has the well-known invariant, 

 pressure multiplied by volume and divided by absolute temperature. Examples 

 might be multiplied. In mathematics the entities under examination may be 

 arithmetic, algebraic, or geometric ; the processes to which they are sub- 

 jected may be anj^ of those which are met with in mathematical work. It is 

 the jmnciph which is so valuable. It is the idea of invariance that pervades 

 to-day all branches of mathematics. It is found that in investigations the 

 invariantive fractions are those which persist in presenting themselves, even when 

 the processes involved are not such as to ensure the invariance of those functions. 

 Guided by analogy may we not anticipate similar phenomena in other fields of 

 work ? 



The combinatorial analysis may be described as occupying an extensive region 

 between the algebi-as of discontinuous and continuous quantity. It is to a certain 

 extent a science of enumeration, of measurement by means of integers, as opposed 

 to measurement of quantities which vary by infinitesimal increments. It is also 

 concerned with arrangements in which diflerences of quality and relative position 

 in one, two, or three dimensions, are factors. Its chief problem is the formation of 

 connecting roads between the sciences of discontinuous and continuous quantity. 

 To enable, on the one hand, the treatment of quantities which vary per 

 saltuni, either in magnitude or position, by the methods of the science 

 of continuously varying quantity and position, and on the other hand 

 to reduce problems of continuity to the resources available for the manage- 

 ment of discontinuity. These two roads of research should be regarded as pene- 

 trating deeply into the domains which they connect. 



In the early days of the revival of mathematical learning in Europe the subject 

 of ' combinations ' cannot be said to have rested upon a scientific basis. It was 

 brought forward in the shape of a number of isolated questions of arrangement, 

 which were solved by mere counting. Their solutions did not further the general 

 progress, but were merely valuable in connection with the special problems. Life 

 and form, however, were infused when it was recognised by De Moivre, Bernoulli, 

 and others that it was possible to create a science of probability on the basis of 

 enumeration and arrangement. Jacob Bernoulli, in his ' Ars Conjectandi,' 1713, 

 established the fundamental principles of the Calculus of Probabilities. A 



