Royal Society. I33 



not better mark their sense of the value of this discovery than by- 

 awarding the Copley Medal to its author. 



The Council have awarded the Royal Medal for Mathematics to 

 H. F. Talbot, Esq., for his two memoirs entitled, " Researches in the 

 Integral Calculus," published in the Philosophical Transactions for 

 1836 and 1837*. 



Nothing perhaps tends more directly to bring the correctness of 

 refined theoretical investigations in physics to the test of numerical 

 results, than improvements in and extensions of the processes of 

 integration. Any advance therefore which is made in this difficult 

 branch of analysis must be viewed not merely in the light of a diffi- 

 culty overcome in the progress of abstract science, but likewise as ha- 

 ving an important bearing on the advancement of physical intjuiry. 



The branch of analysis to which Mr. Talbot's researches belon"- 

 is one which is connected with a long series of valuable investio-a'^ 

 tions from the time of Fagnani and Euler to that of Legendre, Jacobi, 

 and Abel : it relates to integrals under the same form which are 

 separately nonscendental, but which furnish, under particular con- 

 ditions of the variables, an algebraical result when two or more of 

 them are connected together with the signs -|- or — . The celebrated 

 theorem of Abelf, which may be made to comprehend some of Mr. 

 Talbot's results, is the most comprehensive and most important of 

 all the general conclusions which have been arrived at in this de- 

 partment of analysis : but the process adopted by Mr. Talbot is more 

 allied to that followed by Fagnani than by Abel, and is equally re- 

 markable for its great simplicity and for the vast number of novel 

 and interesting results which it furnishes, including not merely several 

 of the most remarkable of those which are already known, but like- 

 wise many others which are apparently not deducible by other 

 methods. 



The Council have awarded the Royal Medal for Chemistry to 

 Professor Thomas Graham for his paper entitled " Inquiries respect- 

 ing the Constitution of Salts; of Oxalates, Nitrates, Phosphates, 

 Sulphates, and Chlorides," which was read to the Society on the 24th 

 of November 183Gi, and since published in the Philosophical Trans- 

 actions. This paper they have 'considered as being the last of a 

 series on a general subject of great importance : and as the sequel 

 of Professor Graham's researches on the Arseniates, Phosphates, and 

 modifications of Phosphoric Acid, read to the Society on the 19th 

 of June 1833, and published in the Philosophical Transactions of 

 the same yearj. He has therein shown that, by considering the 

 M'ater which enters into the comjjosition of the different classes of 

 salts, whicli the phosphoric acid forms with the several bases, and 

 which has been considered as water of crystallization as standing in 

 a basic relation to the acid, a very sim])le. view might be taken of 



• Abstracts of Mr. Talbot's papers appeared in l.oiul. and Ecliiib. Phil, 

 Mag., vol. viii. p. ^A\), and vol. xi. p. 210.— Ejmt. 



\ See Lond. and Kdiiib. I'bii. Mag., vol. vi. p. 116. — EmT. 



: All abstract of Prof. Gralumi's papers appeared in Lond. and Edinb. 

 Phil. Mag., vol. iii. p. 451, 451), and vol. x.p. 31C.— Edit. 



