f 277 ) 



From what is said above we may conclude nol only that in 

 the spontaneously pulsating hear! there appear still other actions 

 than (hose which we find expressed in the contraction, bul also 

 that those actions are to some extent independent. The actions 

 not visible to the eve and characterized by definite electric pheno- 

 mena, suggest the results of stimulation-processes as they can 

 be shown also in the nerve without accompanying change of form. 

 Though, however, the electrocardiogram may possess a certain inde- 

 pendence of the form-cardiogram, the above communication does 

 not in the least afford a reason to conversely come to the conclusion 

 of the independence 1 of the latter with respect to the former. 



Mathematics. — "On groups of polyhedra with diagonal planes, 

 derived from polytopes". By Prof. 1'. II. Schoute. 



Introduction. 



1. By "diagonal plane" of a polyhedron we understand any plane 

 having- only edges in common with the boundary of that body. 1 ) 



There are two regular polyhedra admitting diagonal planes, the 

 octahedron and the icosahedron. Through any edge of the octahedron 

 passes one diagonal plane, containing the centre and bisecting the 

 dihedral angle of the two faces passing through the q(\^q. Through 

 any edge of the icosahedron pass two diagonal planes; the angle 

 formed by these planes and that formed by the two faces through 

 the edge have the bisecting planes in common, and the cross-ratio 

 between the couple of diagonal planes ami the couple of fares has 

 i (3 — y/b) for one of its six mutually connected values. 



The fact that only the two mentioned regular bodies possess diagonal 

 planes is closely connected with this that through each of the vertices 

 pass more than three faces. If we take away from the triangular 

 faces meeting in a vertex the sides passing through that vertex, so 

 as to retain of each the side opposite to this vertex, we find in the 

 case of the octahedron a square adjacent to this vertex, in the case 



') Iu the last memoir of Dr. Fit. Sohuii with the title "Over de meetkundige 

 plaats, etc." (On the locus of the points in the plane, the sum of the distances 

 of which to )) given straight lines is constant, and analogous problems in space 

 of three and more dimensions, Verhandelingen Kon. Akademie Amsterdam, first 

 section volume IX, no. 5, 1908) occurs a series of polyhedra with the property 

 that through any edge passes <.nc diagonal plane. By extension to polydimensional 

 spaces polytopes with diagonal spaces also make their appearance. 



