116 Proceedings of the Royal Irish Academy. 



Vectors of variable magnitude. 



"RTien the magnitude B of the vector is variable, the Curl-vector and 



the Divergence cannot be expressed by means of ciu-vature and torsion 



alone. We require in addition to know R and the vector-gradient of iJ, 



. ^, ^ , . dR dR dR 



i,e. the vector whose components are — ;— , - ^^, — 5- • 



ax clij dz 



There is one vector associated with each point in space, and thus the 

 whole system of vectors is represented by a Il-family and a one-parameter 

 system of surfaces, R = constant. The magnitudes of the components of the 

 vector are IR, mR, nR, and therefore the magnitudes (!«, v, w) of the curl, 

 using the special axes, are — 



d , -ns d . ^^ ^ ^ dR m' ^ dR 

 dy dz ' dy p dy 



d ,,-n^ d , ^ ^ dR I' dR 



- --J(SJ^(fJ^(Sj-S. 



where v is the arc of the orthogonal trajectory of i? = const. Then ^ is a 

 linear quantity and represents the magnitude of the original vector divided 

 by the magnitude of the gradient. In order to represent geometrically the 

 du'ection of the Cui'l, take the nonnal of the orthogonal trajectory of the 

 n-family for axis of 0:, the binormal being axis of y ; then V = 1, m' = n = 0. 

 If 6, (p, \p are the direction-cosines of the direction of the vector-gradient 

 oiR, 



„ g dR q d.R , g dR 



'''^ = -R^:' ''"t' = -Rd^' '"^'-^--rTz' 



Thus, the magnitudes of the cmivponents of the Curl along the normal, hinorm/il 

 and tangent line of the orthogoiud trajectory of the H-family are 



T, cos A ^ / 1 cos d\ ^ / 1 1 \ 



ff \P 9 f Vi-i -TzJ 



where 6, <f) are the angles made by the direction of the gradient of R with the 

 normal and binormal of the orthogonal tra.jectory. 



