THE CURVE ON ONE OF THE COORDINATE PLANES. 477 



§ VIII. 

 Let us now calculate ^ explicitly. The expression of ^w gives 



Now £ may be written 



Having 



<p~ 1 i= — iSi0 ~ H —jSj<p ~H — JcSk<j) ~ H = — iu 2 —jZj — ky 1 , 



we get 



£=i(y 1 z 1 —x 1 u 2 )—x 1 <p- 1 i . 



The coefficient of i, namely. 



=s.ymy<j>- i j<i>-h, 

 ^-s/vw-y 



= -±8f<j>k, 



m 



where we apply the formulas 



hence 



and introduce 

 Thus we get 



mct>- l Y.(j> 1 X^- 1 fi'=Y.\'/j.' ) 



V.d>- 1 \'d>-' [ u' = —<bV\'u' , 

 m 



X = i, ix =j. 



This gives at once 

 Then 



and 



VI 



m 



S.f^- 1 t = 0. 



Y.U-H=-Y.id>-H.^ 



- X V^= —^(Yi^'H)^. 

 Applying to the second member 



m<p _1 VX/a = V. 0\0/x 

 for 



\ = i, [i = <j>~ 1 i, 

 we get 



0-!V^= -°^Yd>i.i=^Yi(hi . 

 mm 



