Motion and Hyper dimensions. 651 



Thus we have established the relations (viii.) and (x.) 

 which are the generalized forms of vii. (1) and (ix.), the 

 latter being of particular use for the case of three dimensions, 

 viz. of the physical space. We are going to make an appli- 

 cation of it tor the study of the motion of a system of 

 reference in a space of three dimensions, a similar study 

 being susceptible of extension to spaces of any dimension. 



Let us consider the equation 



,, v 3 \ /3vv' v 2 ,\ 7 v 3 

 v" Aa+ jp/16 c. 



Pi J V Pi pi I Plp3 



<PF r „ v*\ /3m/ 

 d? 



If we assume a, b, c to be three vectors normal to one 

 another so as to form a system of reference with the origin P, 

 then the analytical expression of P as function of time, 

 <f>(t), represents the law of motion of the system, and <f>'(t) 

 and <j>" {t) the velocity and the acceleration of the motion. 

 Extending the meaning of cj>', <f>", we may consider <f>"\ 



<f>'"' as the hyperaccelerations of the system of order 



three, four ..., the reality of these quantities depending 

 on the law of the motion itself. 



We can now express the meaning of the above equation 

 through the following proposition : — 



li If a system of three vectors normal to one another, of 

 origin P, is in motion with a given law P = (/>(£), so that 

 the axis of X coincides with the tangent to the path of P, 

 the hyperaccelerations of the system represented by 



d n V 



— -(n = 3, 4, 5,,..w) are analytically expressed with respect 



to a system of axes perpendicular to one another, provided 

 that the dimensions of the orthogonal system of reference 

 are taken in number equal to the number of the order of 

 the acceleration." The meaning of this proposition can also 

 be expressed by saying that "the virtual displacements due 

 to the hyperacceleration of an orthogonal system, moving- 

 with its axis of X in the direction of the tangent to the path 

 described by the origin, are analytically expressed through a 

 system* of reference, of a number of dimensions equal to the 

 order of hyperacceleration." 



