378 PROCEEDINGS OP THE AMERICAN ACADEMY. 



each family passes through every point and no curve of either family 

 has anywhere a multiple point. At every point, P, in the domain, the 

 two curves (one of the u family and one of the v family) which pass 

 through the point indicate two directions, s 1} s 2 , and if the sense of each 

 of these be determined by any convenient convention, they may be 

 defined by pairs of direction cosines (l lt m^, (l 2 , m 2 ), where k, »*u h, ™ 2 

 are given scalar functions such that at every point 



I 1 * + n h 2 =l, V + « f 1 =l. (1) 



If O is any scalar function of the coordinates which within E has 

 finite derivatives of the first and second orders with respect to these 

 coordinates, the derivative of O at P in the direction s 1 is the value 

 at the point of the quantity 



k ' 9x + Wl Ty (2) 



and this new scalar function of x and y may be conveniently indicated 

 by the expression [D, 0] P . If P' is a point on the u curve which passes 

 through P, taken near P and in the sense of the direction s u [D^ 0] P is 



the limit, as P' approaches P, of j^pj 



If on the curve of the second (or v) family which passes through P, 

 a point Q be taken near P and in the sense of the direction s 2 , the limit, 

 as Q approaches P, of the quantity 



[A.n],- [-P. ,0], ... 



F? (3) 



may be indicated by the expression [ D s _D, Q] P , and this is the second 

 derivative of Q at P taken with respect to the directions s l and s 2 in the 

 order given. 



Thus, if 



2z 1 



Q, = 2 x 2 — y; li= t , m x = , 



, y ~ x n o 2(ix 2 -y) 



L = a , m., = - ; ./>>, fi = — , 



Var« + y a V** + f x V 4* 2 + 1 



A. A, « 



x(S2x 2 y + 8y 2 + 16y + 8a: 2 + 2) 

 V^T7 2 • (4a; 2 + 1)2 



