294 WILSON AND MOORE. 



To differentiate the matrix A of the coefficients a^, and obtain an 

 expression for the second derivative we proceed as follows. ^° As A is 

 self conjugate we may write A symbolically for the analytic work as 

 XX. Then as 



A = VyVy:(A). XX = VyVy:(XX). 



If we use subscripts to indicate the variables to which the differentia- 

 tions apply, we have 



XX =- ViV2yi-(X)y2-(X). 



The symbols yi*(X) and y2*(X) are scalar, Vi and V2 extensive 

 magnitudes. Now 



VXX = [V1V1V2 + V2V1V2 + VV1V2] yi-(X)y2-(X), 



20 Knoblauch in the preface of his Grundlagen der Differential Geometric 

 (1913) lays stress on the necessity of some operation such as his geometric 

 differentiation to illuminate the formulas of differential geometry and while 

 acknowledging the importance of Ricci's work, especially the Lezioni, com- 

 plains that instead of using geometric derivatives he for the most part uses 

 their " Coeffizienten System." A part of this difficulty is obviated by the 

 use of the notations of multiple algebra as here employed by us and more of 

 it by the large use of vectors that we make later in the work. By the combina- 

 tion of these two elements the analysis can be kept measurably simple and 

 interpretable. 



When discussing methods in differential geometry we must not omit that of 

 Maschke; of which an account may be found in the following articles: 

 Maschke, Trans. Amer. Math. Soc., 1, 197-204, Ibid., 4, 445-469, Tbid., 7, 

 69-80, 81-93; A. W. Smith, Ibid., 7, 33-60; Ingold, IHd., 11, 449-474. For 

 the actual use of the method in the theory of surfaces Smith's article is by far 

 the most important of these references. One may say somewhat epigrammati- 

 cally that Maschke's method contrasts with Ricci's in much the same way that 

 the Clebsch-Aronhold method contrasts with Grassmann's. The funda- 

 mental element in Maschke's work is a symbolic treatment of the quadratic 

 differential form. The reason that we have not used this method is because 

 we have a natural preference for the non-symbolic method which is not over- 

 borne, for the simple work that we have in hand, by the gain in simplicity of 

 operation of the symbolic method. In particular in regard to the present 

 question of the solution for the second derivatives and the introduction of the 

 Christoffel symbols we may observe that for Maschke's interpretation of 

 fifki as a Christoffel symbol it is necessary to assume that the symbols fki 

 and/;fc are equal. Under this assumption /i/i; appears as a Christoffel symbol 

 and its appearance in this form may be taken as a justification for considering 

 the symbols /it and fki as equal (for two of the indices in the Christoffel symbol 

 are commutative). A very natural way to arrive at the Christoffel symbols is 

 by Shaw's method (loc. cit., note 1) in which the symbols all have a geometric 

 meaning; but unfortunately in order to follow this method we have to regard 

 the surface as immersed in a space so that ds^ = dT'dr, and for theoretical 

 purposes it is preferable at this stage to remain entirely upon the surface. 



