THE CURVE ON ONE OE THE COORDINATE PLANES. 



The last two terms in R 1} E 2 , may now take the forms respectively 



—[w — JD >) = u 4 — 



w\ u / w 



Vt( v -h\ =u &. 



v\ U/ V 



So that we have also 



u3R 1 = §M+^? 

 1 m w 



The expression of M 1 now becomes 



m 



_/ SJ0j tt%\/ S&0fc u 4 y\ S 2 j<ftfc 



\ m w /\ m v / 



= —2 [Sj0is/c^ - s?0&s%'] 



771" 



m 



U m w m J 



But the first term is 



vw 



= — 2 S.Vi/vV0%' 



w 



m- 



Si. m(j>~ 1 i= . 



m 



viv 



Substituting this, and multiplying the whole equation by — ^ , we get 



u 4 vwM 1 = 



vw 

 m 



+ u 2 wy *■-'-' 



-+-U2;- 

 m m 



] 



473* 



Of course, if M a = 0, the second member must vanish, the factor vtvw which was intro- 

 duced being never susceptible of vanishing ; in other words, the centre of the ellipsoid 

 being unable to coincide with one of the three planes of the octant. 



We may make further transformations before introducing the hypothesis 6 2 = c 2 . 

 Squaring the expressions of y, z, under the form, 



u(y-v) = -z 1 

 u(z-w)=—y 1 , 



v 2y2 _ 2u 2 vy = z 2 — u 2 v 2 



we get by the first 



the second member being 



Sitjt-ySjf-H - Si(p-HSj<j>- l j = SijV^-H^j 



= — Skd>Jc . 

 m 



