472 DR G. PLARR ON THE DETERMINATION OF 



so long as the true nature of the limiting curve is ignored. This our inability 

 of establishing the proper limitations between the three scalar elements of p will 

 force us into the fear that, instead of an outermost limiting curve, we will 

 deduce a possibly self-intersecting curve with many branches, but certainly 

 all evidently contained in the outermost limiting curve. 



From the equation Ydp\pe=0 we draw the three scalar equations — 



&>idp\Jse = S . eiJs'Vidp = , 

 Sjdp\{,e=S.exls'Vjdp=0, 

 Slcdpxjse = S . exfs'Vkdp = . 



As we have assumed 

 we get 



dp=jdy+kdz , 



Vjdp = idz , 

 YJcdp = — idy . 



The two last of the scalar equations will give 



Se\Js'i = . 



But we easily find that xfj'i is identically zero, because substituting i for w in the 

 expressions of i// 'w , V'/w we get 



\Js i = ^ — 1*= -\ ^ 



r u u u 



and 



which owing to M 2 = S^ _1 i become 



so that xjj 'i + xjjii = xp'i = identically. 



There remains the first scalar equation 



S e ^'Vid P = 0, 

 or 



S^dyxf/k — dzxjs'j] — . 



As this equation is to be satisfied for any direction of c we must have 



yf/Vidp = , 

 or, under another form : 



dyxjs'k — dzxjs'j = . 



