THE CURVE ON ONE OF THE COORDINATE PLANES. 481 



Further, owing to dp - l^dt', the fundamental equation 



yf/Yidp=0 

 becomes 



t'Vi? =0. 

 Now 



Vt£>V(*V*fc) 



as \\i'i = identically zero, the equation becomes now simply 



As we have 

 it follows that 



and the fundamental equation also takes the form 



= W 3 '+'lc-W 2 '^j. 

 If we replace xjj'k and (—*/>)') by their expressions (already stated in § VI.), 



-^';=iPi+jQ +Mfc 2 , 



the equation will decompose itself into three vector components parallel respec- 

 tively to i, j, k, which have to be annulled separately, so that we get 



0«=-W 8 'P a +W ! , < P 1 - I 



O^W^ + W^Q, 

 = W 3 'Q + W 2 R 2 . 



These reproduce the equations of the same form stated in § VI., in which dj/and 



dz are replaced proportionally by W 3 ' and by W 2 ' . These three equations are 



equivalent to only two distinct ones, because as, for example, the necessary 



relation 



BjB,-Q" = 



will help to transform the third of the equations into the second one, 



W 3 'Q + W 2 'R 2 = |- (Wa'Ri + W 2 'Q) = . 

 For this reason the equation 



can yield but two distinct scalar equations. We will get them by treating 

 ^(« = 0byS.t( )andbyS.#( ). 



VOL. XXXIII. PART II. 4 B 



