274 



AVILSON AND MOORE. 



CHAPTER I. RICCrS METHOD^ 



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



xi = xi (yi, 2/2), X2 = X2 {yu 2/2), 



we transform the differential Xidxi + X2dx2 into a new differential 

 in the new variables so that 



Xidxi + X2dx2 = Yidyi + Y2dy2, 



we find 



J-i = Ai- f- A2~ — ■ 



dyi dyi 



-t 2 — Ai- 1- A2T— ; 



dyi dy2 



(1) 



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

 system 



dxi dx2 . dyi dyo 



— mto ' — 



XW X(2) 



yd) 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 PubUc 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, 6, 112-118 (1889); Shidi off. d. Univ. 

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

 Veneto, (7) 4, 1-29 (1897), Ibid., 5, 643-681 (1894), Ibid., 6, 445-488 (1895); 

 Rend. R. Ace. Lincei (5) 4, 232-237 (1895), Ibid., 11, 355-362 (1902). A 

 general sketch of the method is found in Bvll. 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. 

 Relaliiritdtstheorie (with Einstein), Teubner, 1913 (from Zs. Math. Physik, 

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

 notation. 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 

 offered in justification of our reproducing here material which has previously 

 been published. 



