398 SECTIONAL TRANSACTIONS— A*. 



holds for each of a set of eight functions satisfying a certain set of eight 

 linear differential equations. 



Let li = ckj^— ,(i=i,. . ., n), be n linear differential forms with con- 

 oxj 



stant coefficients such that 



where the Uik are constants and the ik any of the numbers i , . . ., m. The 

 numbers n for which such relations exist can be completely determined by 

 properties of (not necessarily associative) hypercomplex systems over the 

 real numbers. The best known non-associative hypercomplex system — 

 the Cayley numbers — is closely connected with the set of eight linear 

 diflFerential equations mentioned above. 



Laplace's operators in two and four variables are special cases of a class of 

 differential operators which are connected in the following way with hyper- 

 complex systems over the real numbers. Let 5 be a hypercomplex system 

 with n base elements e^, . . ., Cn and let x-ie^ + . . . + XnCn be a general 

 element of S, where the xi are any real numbers. The norm of XiC^ -\- 

 . . . + XnBn, if defined by means of the regular representation of S, is 

 a homogeneous function /(xi, . . ., Xn) of the nth degree in. Xi, . . ., Xn. 



If the co-ordinates are replaced by the differential operators s — , ■ • ., k — a 



^, . . ., s — j is obtained. Let .5 be the system 



of complex numbers or of quaternions. The operator which is so obtained 

 is Laplace's operator. 



Mr. Garrett Birkhoff. — Lattice forms (12.15). 



Wednesday, August 24. 



Mr. J. H. C. Whitehead. — A generalisation of groups (lo.o). 



The starting point is the equivalence of simplicial complexes under three 

 kinds of elementary transformations and their inverses. The first are 

 elementary sub-divisions, giving combinatorial equivalence. The second 

 are of the form K -> K + aA, where aA, but not aA or A, belongs to 

 K, A and aA being k- and (k + i)-simplexes for an arbitrary value of k, and 

 A being the boundary of A. The third consists of these, together with 

 transformations of the form K -> K + A, where A, but not A, belongs to 

 K, and the dimensionality of the simplex A exceeds some fixed m, which 

 may be arbitrarily chosen in the first place. Complexes which are equiva- 

 lent under the second and third kind are said to have the same nucleus and 

 the same 7w-group respectively. The justification for the term Tw-group lies 

 in the theorem that two complexes have the same fundamental group if, and 

 only if, they have the same 2-group . The m-group of a complex is seen to 

 be a topological invariant, for each value of m, and the nucleus is a topo- 

 logical invariant provided the fundamental group satisfies a certain condition, 

 which is stated in terms of the ' integral group-ring.' An immediate 

 application is that certain invariants discovered by K. Reidemeister, and 

 shown by him to be combinatorial invariants, are actually topological 



