PEIRCE. — SrACK DIFFERENTIATION OP THE SECOND ORDER. 385 



^■■^fi (S^fi 9H2 



+ 2hm, . ,,^+ 2;/.,Hi •,^^+ 2/1/^1 •,^^, (22) 



whereas, if s^ is not fixed, 



, , p-n ,5^0 , 9-Q. ^ , 5'f^ ^ 5^0 



1 ^ dx^ dy dz dx- dy 9y dz 



+ 2/i«i . 



— o" + ( <i • 7^ + »«r 0- + ?«i • 7^ 7^ 



a; ■ cy^ \ cya^ cyy dz J dx 



—4— I / ■ .i jii • I j^ • 1 



\^ 9x ^ 9y ^ 9z J 9y 



( 9n, 9ni 5hi\ 50 



\ dx dy dz J dz 



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

 general variable. If Si 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 (9f2/ 5a:, 90 / 9y, 50/(92 in (23) vanish. In particular, 

 if the direction Sj is that of the radius vector from a fixed point (o, 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 

 5f2 / 9x, 90 j 9y, 5n / 9z vanish, the third must vanish also. 



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

 the direction in which 12 increases most rapidly, 



\_dx dx dy dy dz az _j / 



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

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

 the point functions O and h increase most rapidly, 



cos(n,/0 = I ^ • ^ + ^ • ^% ^ • ^~^ / h • /.' (25) 



5n 9h ^50 9h ^ 9n 

 9x 9y 9y 9z 9z 



and DJ" Q = k' ■ cos (yl, h) , or h [Da h] (26) 



VOL. xxxix. — 25 



