THE CURVE ON ONE OF THE COORDINATE PLANES. 475 



The cubic relative to i// becomes a quadric, namely, 



where 



M x = - Sifjyjr'k( = E, R 2 - Q 2 ). 



M 2 = - Sjyfr'j - Skfk = + Q - Q = . 

 The quadric can also be put under the form 



^> s fi> = V[iV(*a>)] . 



Putting <y || Vidp we get 



^(Yidp) = Y. [iY( K Yid P )] 



= Y. i(dpSici — iSicdp) 



= Yidp(-M 1 ). 

 Hence by the above : 



M 



yfr(dp) = YidpX^- + 1 J( -, 



and 



M x = by ■^'(Yid P ) = 0. 



This shows that when the point of tangence is on the limiting curve we have 



+(d P )=Q, 

 but this remark is unessential as to the sequel. 



§ VII. 

 Let us examine \jj<o more particularly. First we see that 



ylr'6=-2Yid>-H— 



TO r U 



= 2Yi<p~H, 



owing to 6 = iu+jv + kw. As ^ and <£ _1 are self-conjugate-vector functions, it 



follows that 



2Yi<j>-H = 0; 



thus we have 



f o '0 = O. 



This equation at once gives us the certainty that the primitive function xji (o will 

 vanish also when w takes a particular direction. (We have shown this neces- 

 sity in a paper published in the Proc. Roy. Soc. Edin. of the year 1881-82, 

 p. 342.) 



* Proc. Rotj. Soc. Edin., 1882-83, p. 342. 



