PEIRCB. — SPACE DIFFERENTIATION OF THE SECOND ORDER. 385 



no /2 <? 2 ft , a 3 2 ft , 2 3 2 ft 

 A"° = /l 3* 2 + Wl W + " l "'d* 



^'2f~\ 02/~\ ^2/~\ 



+ 2Z 1 m 1 •^— ^-+ 2m lWl • — — - + 2^% • ^ =-, (22) 



ox ^ dy dz dx • dz v y 



whereas, if *! is not fixed, • 



n "o ;2 3 2 fl t 2 5 2 n , 2 3 2 ft , _. 3 2 ft , 3 2 ft 



o; 9 °" Q (j % ^i , cVASQ 



+ 2 «! Wi ■ ^ =- + UY + »iV + %Vhr 



dx ' dz \ dx dy dz J dx 



\ x 5x 3y l dz J dy 



( dn x 3% 3«i\ 3ft 



H 1 -9x- + nh '-9j + rh ^)d7' (23) 



All the coefficients in (22) are constants; all those of (23) are in 

 general variable. If s x is defined by any infinite system of straight lines 

 of which just one passes through every point of space, and if the direc- 

 tion Sj at all points of any one of the lines is that of the line itself, the 

 coefficients of 3ft / dx, 3ft / dy, 3ft/3zin (23) vanish. In particular, 

 if the direction s x is that of the radius vector from a fixed point (a, b, c), 

 (23) takes the form of (22) though the remaining coefficients are not 

 constants. In any case if the coefficients of two of the three quantities 

 3ft / dx, 3ft / dy, 3ft / dz vanish, the third must vanish also. 



If the gradient, h, of ft does not vanish at any point of R and if s is 

 the direction in which ft increases most rapidly, 



D, ft = h , 



(24) 

 j)2 _ r3/?.3ft 3A.3ft to.ftf\l h 



\_dx dx dy dy dz dz J / 



If A' is the gradient of the scalar point function which gives the value 

 of h, and if (ft, h) represents the angle between the directions in which 

 the point functions ft and h increase most rapidly, 



, 7N f3h 3ft dh 3ft dh 3ft "I / , ,, ,„„ 



and Z> 3 2 ft = h! • cos (ft, h) , or h [D a h] (26) 



vol. xxxix. — 25 



