204 THE ROYAL SOCIETY OF CANADA 



Differentiating and using arbitrary multipliers A and B, we have 



Sip 



Sia-\-ASip-{-B ^ =0 

 a~ 



Sja^ASjp-^B^ =0 

 Ska^ASkp + B^ =0 



Sjp 

 Skp 



Sia 



= -A Sip 



1- 



Sja ^ _ 



¥ 



Ska 



= -ASjp 



= -A Skp 



/-2 



Sia 



Sja 



Ska 



(SiaY i^jay {Skay ^^ 



T" T" r^ 



1- - 1- - 1- - 



a- 6^ c^ 



Thence finding r^ and r^ for axes, we have for the equations of the 



axes 



Sip 



Sia 



A 



1- 



ri' 



In the above the differentiated equations, if a be a unit vector, 



are evidently = /x+ .. + .., r^ = x--\- .. + .., 1= [-•. + ••; and the 



a- 



/2 



final equation is 



1- Î1 



^2 



+ ... + ...=0. 



The general scalar equation of the second degree may be written 



aiSipy-\-b(Sjpy+c(Skpy-{- 2fSjpSkp-\- 2gSkpSip-\- 2hSipSjp 

 — 2uSip — 2vSjp — 2wSkp -\-d = 0. 



The term p^ is included in the above since p= —iSip—jSjp — kSkp. 



