1008 



of the differential equation of their geodetic lines having been at 

 length overcome, the properties of these curves, and also of the 

 lines of curvature, have been carefully investigated by the geometers 

 of different countries. Abroad, the first important steps were made 

 by Jacobi, Joachim, Stahl and Liouville. Profiting by their labours, 

 Mr, Michael Roberts made the remarkable discovery that the lines 

 of curvature of an ellipsoid are related to its unibilics, as a central 

 conic section to its foci. Since then, Chasles, Liouville, Graves, 

 Hart and others, have arrived at various theorems concerning the 

 geodetic lines, and lines of curvature of surfaces of the second order. 



In the theory both of surfaces and curves, considerable advances 

 have been recently made by means of general methods of transfor- 

 mation. Thus, from theorems already established, new ones are 

 derived without the need of an independent demonstration. Mr. 

 William Roberts. Mr. Thomson and Mr. Cayley, have lately fur- 

 nished methods for transformation, by the use of which new and 

 interesting properties of curves have been established. The pro- 

 gress made in the evaluation of definite integrals has likewise led to 

 many discoveries respecting the rectification of curves, and the 

 quadrature of surfaces. 



In ordinary algebra and the theory of equations, no very con- 

 siderable advance has been made during the last tv^o years. I 

 ought not, however, to omit noticing a valuable paper, on Sturm's 

 Functions, contributed by Mr. Cayley to Liouville's ' Mathematical 

 Journal.' Crelle's Journal also contains some useful memoirs on 

 continued fractions, infinite series, and the results of certain sub- 

 stitutions in functions of different degrees. 



The theory of numbers has of late found favour in the eyes of 

 mathematicians. Jacobi and other writers in France and Germany 

 have given to the public several interesting papers on different sub- 

 jects belonging to this department of mathematics; but of recent 

 contributions to it, perhaps the most remarkable has been made in 

 this country by Mr. Hargreave, in a paper " On Prime Numbers," 

 published in the ' Philosophical Magazine.' Sir John Herschel has 

 also very recently v/ritten on the same subject ; his paper, as you 

 will recollect, was read last Session, but is not yet published. 



The theory of La Place's coefficients, so successfully treated by 

 Mr. Flargreave and by Mr. Boole, has been lately made the subject 

 of a memoir by Mr. Cayley, in which he has shown how to extend 

 it to any number of variables. Neumann has also exhibited a me- 

 thod of developing, in a series proceeding according to La Place's 

 functions, the distance between two points expressed by means of 

 elliptic coordinates. In addition to the papers already enumerated, 

 I observe that several relating to the solution of differential equa- 

 tions, the properties of particular series, factorials, faculties, and 

 determinants, have been contributed to the principal scientific 

 journals. 



Before concluding this brief and necessarily imperfect review of 

 the recent progress of mathematics, I cannot abstain from remark- 

 ing, that two of the fields of research which promised best to reward 



