276 



WILSON AND MOORE. 



5. Generalization of vector analysis. If we could follow the 

 ideas of Grassraann-Gibbs,^ we should consider the sets of quantities 



^1, X2, 



X„ or.Y('),Z(2),- • ■,Xm 



as components of a vector along the directions dxi or upon the planes 

 perpendicular to those directions. It proves, however, to be impossi- 

 ble to establish here more than an analogy; for it is actually untrue 

 that these elements are such components. 



That the X's may not generally be interpreted as components of a 

 vector is clear from the expression for the differential of work in terms 

 of generalized coordinates, namely, 



dW = Qidqi + Qidqi + . . . + Qndqn. 



The set of generalized forces Qi is covariant under a transformation 

 of the q's, but the generalized forces are not the projections of the 

 resultant force either upon the directions dqt or upon the planes 

 perpendicular thereto in the w-dimensional representative space of the 

 q's; for instance, in polar coordinates in the plane, dW = R dr -\- rOdd, 

 where R and G (not r9) are the radial and tangential components 

 of the force. 



We have therefore to deal not with a generalized vector-analysis 

 but with a generalization of vector analysis when we deal with systems 

 Xs or X^^K A method of converting such a system into one which 

 represents the components of a vector will be mentioned later (note 

 17 to § 12). 



So long as we remain in the vicinity of a particular point and deal 

 only with differentials of the first order, the transformations (1') and 

 (2') are linear with constant coefficients of the type 



y _ V „ i" 



Ifar 



dxs 

 dyr' 

 dyr 

 dxg' 



and the first section of our presentation of Ricci's method will there- 

 fore be strictly algebraic theory of the linear transformation. When, 



8 Grassmann, Ausdehnungslehren von 1844 u. 1862, also Gesammelte Werke; 

 Gibbs, Scientific Papers, Vol. II; Gibbs-Wilson, Vector Analysis; Wilson, 

 Trans. Conn. Acad. Arts Sci., 14, 1-57, (1908). 



