Vector Differentials. 193 



7. As an application of some of these expressions, let us 

 examine the criterion that the state of affairs contemplated 

 in Dupin/s theorem may exist : in other words, find the 

 differential equation which must be satisfied by the unit- 

 normal to a family of surfaces in order that there may be 

 two other orthogonal families. 



One form of the condition is that SeVc and S?yV^ shall 

 both vanish, e and rj having the same meaning as in Art. 1. 

 Furthermore. 



Ve=Vfav) 



= V?7 . v — rjSjv — 2^?7, 



and by operating with Se we obtain for all families of 

 surfaces 



SeVe=S97V*7 (1«) 



Hence, if the condition just mentioned is f ulfilled, 



SeVe + S97V?? = (19a) 



It is here not essential that e and tj shall be of constant 

 length. We may, therefore, put for them any other vectors 

 to which they are respectively parallel. If g and g' be the 

 roots of the quadratic equation 



X 2 ~ m 2X + ni i=Q, 



so that (%— i/)e=0 and (%— g / )y = 0, and if we operate on 

 any vector at right angles to v with %—g ! and with %— <y, 

 the two results will be parallel, in order, to e and to rj. 

 Choosing as a convenient operand the unit-vector //,, that is 

 UVVf, we shall have 



S(x-/>V(%-^> + S(^-y>V(x--^>=0,. (196) 



and by expanding and rearranging 



^(2%— ^.JmV^ + S (m 2 2 —2m 1 — m,x)^V/i'- - S^Xi u Vm 2 =0. ( 19c) 



From (10), by writing ^ for (/> and fju for t and d^u — O^lp, 



The form of (19«) shows that we are concerned only with 

 that part of VSJyji lying in the tangent plane. The vector 



part of - V" is tyfju ; the terms X^^X :um f^X^i-tf 1 h* ve n0 

 Phil Mag. S. 6. Vol. 5. No. 26. /VA. 1903. 



