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DESCRIPTION OF FIGURES. 



Fig. 1. Median longitudinal section of the infundibular organ of a Branchiostoma 

 of 52 m.M. in length, 600: 1. 



Fig. 2. Gross section through the same of a Branchiostoma of 54 m.M. in length 

 600: 1. 



Fig. 3. The same as Fig. 1. Neurofibrillae stained with chloride of gold. 



Fig 4. Cells of the infundibular organ, a of a Branchiostoma of 22 m.M. in 

 length, b of 50 m.M. in length, c cross-section of the upper ends of 

 the cells. 



Fig. 5. Median section of the brain of a Branchiostoma larva of 3,4 m.M., recon- 

 structed from cross-sections. 



Fig. 6. The same of a specimen of 10 m.M. long. 



Fig. 7. The same of a specimen of 21 m.M. long. 



Fig. 8. Median section through the brain of a Branchiostoma of 28 m.M. in 

 length. 



Mathematics. — "About difference quotients and differential quo- 

 tients". By Dr. L. E. J. Brouwer (Communicated by Prof. D. 

 J. Korte weg). 



(Communicated in the meeting of May 30, 1908). 



Different investigations have been made which are very completely 

 summed up in the work of Dim : "Grundlagen für eine Theorie 

 der Functionen einer veranderlichen reellen Grosse" Chapt. XI and 

 XII, on the connection between difference quotients and differential 

 quotients, particularly on the necessary and satisfactory properties 

 which the difference quotients must possess in order that there be a 

 differential quotient. One however always regards in the first place 

 these different difference quotients in one and the same point x 

 together, forming as a function of the increase of x the derivatory 

 function in ,r . The existence of a differential quotient means then, 

 that that derivatory function has a single limiting point in .v i.o.w. 

 that in x t the right as well as the left derivatory oscillation is 

 equal to zero. 



Other conditions for the existence of a differential quotient are 

 found when in the first place the difference quotient for constant 

 ^-increase A is regarded as a function of a? and then the set of 

 these functions for varying A is investigated. Let f(x) be the given 

 function which we suppose to be finite and continuous and let 

 <fA (?) be the difference quotient for a constant ^-increase A. The 

 different functions </a {%) form an infinite set of functions, in which 

 each function is continuous. We shall occupy ourselves with the 



