IN GEOMETRY, MECHANICS, AND ASTRONOMY. 417 



for we have 



'D„ + „■•(«) = - D„(u'+u"), 

 - Dj('- Dji" 



The operation Z)„. is therefore distributive with respect to u. 



To indicate that />„• is distributive with respect to u\ we shall elevate the subscript index u', 

 and write it in the same line as D, putting a dot between u' and the syuibol on which the operation 

 is performed ; that is to say, we shall write 



Du'.u instead of D^it. 



5. Having thus settled the form of the notation, we shall now interpret the meaning of the 

 expression for Du'. u, namely, 



Dii. u = {zy'- z'y) a + {,vx'- w'x) fi + {yaf- y\i) y ( fi), 



from which, as we have seen, immediately follow the two equations 



Du .u' = — Du . It (7) 



D (ii + u") . u = Dii . It + Dit" . It (8). 



1st. To determine the direction of the line Dii. u, let 



Dii. u = xa + y^(i + z^y, 

 and therefore, by (6), 



a!= zy- z'y I 



y = xz' — x'z^ (y). 



From these equations we have immediately 



x^ X + y^y + z^z = 0, 

 x^ x'+ y y' + z^z'= 0. 



Whence it appears that the line drawn to the point (v^yz^) from O, is at right angles to the line 

 drawn to (xyz) and the line drawn to {x'y'z) ; in other words, Dii. u is at right angles both to u 

 and u . This determines the direction of the line Du' . u. 



2ndly. To determine the magnitude of Du'.it, let i\, r, and r' denote the magnitudes of Du'. u, 

 u, and u respectively, and let Q be the angle made by u and zi : then, by the equations (9), 



X- + y'f + zf = (.r^ + y-+ z") {x- + y"' + «"') - {xx + yy' + zx'f, 



or r^- = rr'' - {rr cos 0)'-, 



and therefore r^=rr'i\x\Q (10). 



Hence the numerical value of the line Dv! . u is the product of tlie numerical values of the lines 

 u and 11 multiplied by the sine of the angle they make with each other. 



6. Since rr' sin is the area of the parallelogram formed upon the lines u and u' as sides, it 

 follows, that Du. M is a line numerically equal to tiie area of the parallelogram formed upon u and 

 7<', and j)erpcndicular to its plane. 



It fi)llow.s from (7) that Du .u denotes a line equal in magnitude to Du'.it, but opposite in 

 direction. 



3H 2 



