THE CURVE ON ONE OF THE COORDINATE PLANES. 487 



the last but one equation becomes 



w? r _ w 3tt 3 + m2 s/ k< i ) -i i _x]=o. 



u 6 L w -I 



As the factor Wi= — a^CiA vanishes only for particular directions of a, fi, y, 

 the general solution will be either 



yi±u3 = Y = u*ty<tf</> -H + X, 



or 



w 



«3 = Z = w 2 SZ;0Z;0- 1 i-X. 



W, _ 



But the equation 2 — L = is a consequence of these two, because 



(Y + Z) = u z S(Vj<f>j + Yk<f>k)<j>-H 

 = -zt 2 St0{0- 1 = -W 1 ?j 2 ; 



W m 3 W ?t 3 



we may therefore look upon the expressions of — — and — — as con- 



stituting the two equations sought for. 



§ XII. 



If we express the quantities Y and Z in function of the nine coefficients 

 a v b v c v a 2 , b 2 , &c, we will find that the expressions contain the same con- 

 stant factor A which we have found to affect the quantities W 1} W 2 , W 3 . 



First as to X we have 



a 2 



* =s/#=s^ 



t a 2 



v 1 - 1 



a 2 a 2 b 2 c 2 



x = «i + -f(m 2 + Si0i) 



lib 



= 2a 2 « 3 |a 2 + ^(l-«t 2 ) + c 2 (l-V) + & 2 (l-c 1 2 )} 

 = 2a 2 « 3 1 a 2 + b 2 + c 2 + ^(1 - a x 2 ) + ( - c 2 V - 6 V) | 



