THE CURVE ON ONE OF THE COORDINATE PLANES. 473 



From this second form of the equation we deduce 



This parallelism of \\>'j and xfs'k is also the consequence of the condition 



\p-e — , for any e , 



because treating this last equation successively by $.i, S. t /, S.k it gives first 



y . i\pe = S . e\[s'i = , 



which is identically satisfied, because xjj'i is identically zero. 

 Then the two other equations, 



S?>e=S.ei//j = 0. 

 Skxjse — S . exfr'k = 



have to be satisfied for any direction of e . This must be effected by the least 

 number of assumptions, lest we get too many conditions between the three 

 scalar elements of p. The only way to reduce the number of conditions is to 

 assume the parallelism between \j/j and \p'k again. 



§ VI. 



We cannot refrain from sketching briefly a method of development of the 

 equations 



■ty'Vidp — , 



and 



which we have abandoned owing to the complication arising from the rational- 

 ising the expressions containing the radicals u, v, w together. 



If we develop namely xjt'j and xp'k into their components parallel to i, j, k, 

 putting (for reasons easily to be accounted for) 



where with the notations 



we have 



y y — Sk<p ' H 



jti — > r„ — 



V U * W 11 



2 

 It? W ' U 



Rt^-Vl+u-*, 



•R Z \ Z \ , ^ 



1 U 6 V u 



VOL. XXXIII. PART II. 4 A 



