32 VARIABLES AND FUNCTIONS. [INT. II. 



differentiating totally 



zdz 



:} 



z* 



{/p2 

 (^+ 



+ 



X) 2 J 



Now if $ A is the perpendicular distance of the tangent plane from 

 the origin, we have by the last formula of 16, 



^ A = l / V (a 2 +X) 2 + (6 2 + X) 2 + (c 2 +X) 2 ' 



so that we may write for the cosines, 



^S. 



(II) C( 



cos 



r\ 



cos 



c 2 +X 

 Now as we move along the normal, we have 



dx = dn cos (n^x) = r -Ar dn, 



-A 

 ~r X 



dy = dn cos (n^y) = 



dz = ^ cos 



2 



C 



Inserting these values in (10), 



+x+x + ' 



so that 



/ \ 7 C^- rk 



(13) Ax = ^- = 2 



/ ^ 2 i/ 2 ^ 2 



V (a 2 + X) 2 + (b 2 + X) 2 + (c 2 + X) 2 



In order to express this result in terms of the elliptic co- 

 ordinates alone we may express #, y> z, in terms of X, JJL, v. Observe 

 that the function 



