276 WILSON AND MOORE. 



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

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



Zi, X,, • • •, .Y„ or XC), X(2), • • •, Z(«) 



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 = Qdqi + q-idq-z + . . . + Qndqn. 



The set of generalized forces Qi is covariant under a transformation 

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

 resultant force either upon the directions dqi or upon the planes 

 perpendicular thereto in the 7i-dimensional representative space of the 

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

 where R and 9 (not rQ) 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 A"(*). 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 



V — ^ Y — — 



oyr 



OXs 



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 Atialysis; ^^'ilson, 

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



