378 PROCEEDINGS OF 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, Si, Sg, and if the sense of each 

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

 defined by pairs of direction cosines (/j, Wi), (I2, m^^ where li, mi, l^, m^ 

 are given scalar functions such that at every point 



If O is any scalar function of the codrdinates which within R has 

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

 coordinates, the derivative of 12 at P in the direction Si is the value 

 at the point of the quantity 



Z • ^ + m • ?^ ^2^ 



^' 9x^'''' 9y ^^^ 



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

 by the expression [Z), ^'\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^, [Z), W]p is 



the limit, as P' approaches P, of ^^Wi ' 



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 $2, the limit, 

 as Q approaches P, of the quantity 



[A,o]^-[A,o], 



PQ ^^^ 



may be indicated by the expression [D, D, fijp, and this is the second 

 derivative of 12 at P taken with respect to the directions Sj and s^ in the 

 order given. 



, 2x 



m., = 



V^x'+l V4^x'+1 



^ . n o_ 2(4ar--y) 



T, ' •' / f, ■ ■ ' 



'' ~ Vx^ + f ' ■'' "" V 4 a;2 +■ I 



X (32 x'y + 8y- + 16 y + 8 x- + 2) 

 y/x^ + y"^ • (4a:2 4- 1) i 



