274 WILSON AND MOORE. 



CHAPTER I. RICCI'S METHOD^ 



4. Two types of transformations. If by a change of variable, 



xi = xi (yi, yo), xo = xo (?/i, 7/2), 



we transform the differential X\dxi + Xodxz into a new differential 

 in the new variables so that 



Xidxi + X2dx2 = Yidyi + Y2dy2, 

 we find 



^ dxi dx2 „ ^ dxi dx2 , . 



diji dyi dy2 dy2 



and if by the same change of variable we transform the differential 

 system 



dxi _ dx2 . dy\ _ dy2 



X^) ~ x^ ^^^° y(i) ~ y(2) ' 



7 The lithographed Lezioni already cited is not obtainable either in new or 

 second hand copies and is to be found in very few American libraries; it is to 

 be had, however, at the Harvard library, the Boston Pubhc library, and the 

 library of Washington University (St. Louis). Ricci's first presentation of 

 the essentials of the theory is scattered through a considerable number of 

 papers in different Italian journals, particularly journals of the learned socie- 

 ties. See, e. g., Rend. R. Ace. Lincei, 5, 112-118 (1889); Studi off. d. Univ. 

 Padovana a. Bolognese n. VIII centenario ecc, Vol. Ill (1888); Atti R. 1st. 

 Veneto, (7) 4, 1-29 (1897), lUd., 6, 643-681 (1894), lUd., 6, 445-488 (1895); 

 Rend. R. Ace. Lincei (5) 4, 2,32-237 (1895), Ibid., 11, 35.5-362 (1902). A 

 general sketch of the method is found in Bull. Sci. Math., Paris, (2) 16, 

 167-189 (1892) and a very elaborate outline not only of the foundations of the 

 theory but of many of its applications is given by Ricci and Levi-Civita in 

 Math. Ann. 54, 125-201 (1900). More recently Grossmann, Verallgem. 

 Relativitdtstheorie (with Einstein), Teubner, 1913 (from Zs. Math. Phpsik, 

 Vol. 62), mentions a few of the salient features of the method in a modified 

 notatioii. It is however only in the Lezioni that the treatment of the ele- 

 mentary parts of the theory is given in comfortable detail. Moreover, the 

 Math. Encyc. and the few authors who cite Ricci do so in a manner which 

 suggests strongly that his method is practically unknowh. These facts are 

 oifered in justification of our reproducing here material which has previously 

 been pubhshed. 



