116 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



the equations from either. This we will now do. It is important 

 every time that a new quantity appears in dynamics, to have a clear 

 conception of its physical nature. We should make free use of all 

 analogies that our science may offer us, and here geometry aids us 

 readily. The notion of the geometric product and the terminology 

 of multidimensional geometry here furnish us valuable aid. The 

 geometric product of two vectors in three dimensional space, defined 

 by their components X, Y, Z, X', Y', Z', 



xx' + rr + zz', 



is a scalar quantity, symmetrical with respect to both vectors, such 

 that the geometric product of the resultants of two sets of com- 

 ponents is the arithmetical sum of the products of all the pairs of 

 corresponding components. If one of the vectors is an infinitesimal 

 displacement dx, dy, dz, the geometric product is 



Xdx + Ydy + Zds, 



and the multiplier of the change dx is called the component of the 

 vector in the direction of the coordinate x. In like manner 

 let us speak of a quantity defined by components P 1; P 2 , . . . P m as 

 a vector in m- dimensional space. The geometric product of two 

 such, of which the second is an infinitesimal displacement compatible 

 with the constraints, and defined by the quantities dq i} dq%, . . . dq m , 

 may be, by analogy, defined as 



P l dq l -f P 2 dq 2 -\ h P m dq m . 



If now the vector P 1? . . . P m is equivalent to the system of vectors 

 X r , Y r , Z r , we have equations 39), 40), 42), and the latter, 



P,= 



serves to define the component of the vector -system with reference 

 to the coordinate q s . Thus we have spoken of P s as the force- 

 component of the system for the coordinate q s . It is to be observed 

 that we do not insist here on the idea of direction, and that our 

 terminology is merely a convenient mode of speaking. Nevertheless, 

 the notion of work gives a means of realizing by the senses the 

 meaning of our term component, for, if we move the system in such 

 a way that all the g's except one q s are unchanged the work done 

 in a change of the coordinate dq s will be P,^,. 1 ) 



Let us now find the component of our velocity -system according 

 to our generalized coordinates. We have, according to our equation 



1) For a further elucidation of the nature of the geometric product, in 

 connection with multidimensional geometry, see Note III. 



