THE CURVE ON ONE OF THE COORDINATE PLANES. 489 



We have now for Y = u 2 Sj<f>j4>~ 1 i + ~K 



We have 



Substituting this into the [ ] we get 



a 2 a-fb 2 c 2 



+ b 2 b 2 a 1 b 2 c 2 — b 2 b 1 2 c 2 c 3 



+ c 2 c 1 2 a i b 2 c 2 + <?cft)<}> z . 



The terms in the second line are 



= b 2 b 2 c 2 {aj> 2 -c.^ , 

 and as 



_ e 3 = Syk = SVaPVij = BajB/3i - SaiSffi = afo - a x b 2 

 we get 



»A — % = «2&1 > 



so that the terms in & 2 become 



b 2 b x *c 2 a 2 . 

 The terms of the third line are 



c 2 c 1 \(a 1 c 2 + b s ), 

 and as 



i 3 = — Sj3k = Sayij 



= SajSyi — SaiSyj 

 = fl 2 Cj &iC 2 , 



hence 



fJ l C 2 T ^3 = a 2 C l ' 



the terms in c 2 become 



c\ 3 b 2 a 2 . 

 We have now 



Y = E« 2 ( tj — 2 )(a 2 a l 3 i 2 c 2 + b 2 b x 3 c 2 a 2 + <?c x 3 a 2 b 2 ) 



And if we put 



Y' = 2a 2 «i 3 & 2 c 2 , 



and consider that by § VIII. we have 



namely, there we have 



-a 2 a 2 & 2 5 2 c 2 c 2 

 ( + A)- &2 + c2 c2 + ft2 a S+j2» 



we have thus 



Y=-Y'A. 



With the notation already indicated in § VIII. as to W A , the equation 



■2&-T.0 



VOL. XXXIII. PART II. 4 C ■ 



