574 



and even numbers as exterior differences, combined with every pos- 

 sible and available middle difference ; for negative differences may be 

 rejected, inasmuch as, if the roots be put according to their algebraic 

 value, all the differences must be even ; thus the roots and differences 



of 15 above were 



2, -3, 5 

 -1, 1, -2,3; 



if the roots be placed according to their algebraic value, they would 

 be 2, 1, 1, 3, and with the differences above 



1,2,2 

 -2, -1, 1,3; 



15 will therefore be found in the column above 5, and in the fourth 

 place. The Table (extended indefinitely) would therefore contain 

 every possible odd number the sum of whose roots may equal 1 . 



It is possible that this connexion between the roots of the squares 

 into which 4n+l maybe divided, with the exterior differences of 

 the roots of the four square numbers into which 2n + 1 may be 

 divided, formed part of the mysterious properties of numbers to 

 which Fermat alluded when he announced the theorems of the poly- 

 gonal numbers. 



III. " On Cyanide of Ethylene and Succinic Acid." Prelimi- 

 nary Notice. By MAXWELL SIMPSON, Ph.D. Communi- 

 cated by Dr. FRANKLAND, F.R.S. Received August 1, 1860. 



Succinic acid bears the same relation to the diatomic alcohol 

 (glycol) that propionic acid bears to ordinary alcohol. Propionic 

 acid can be obtained by treating the cyanide of the alcohol radical 

 with potash. Can succinic acid be obtained by treating the cyanide 

 of the glycol radical with the same reagent, or is it an isomeric acid 

 that is formed under these circumstances ? 



C 4 H 5 , 



Cyanide of ethyl. Propionate of potash. 



Cyanide of ethylene. Succinate of potash ? 



