110, ill, 112, 113] CURVILINEAR COORDINATES. 



335 



dn = r sin &dq), .? = . = _ 



' on r sinuate rsin^ 



dcp 



r sin & 



For any function f(r, #, qp), the partial parameters are 



- 



~ dcp V r sin & d<p 



The total parameter, the resultant of these, is given by M S- m - 



, 



" r 2 



sin 2 9\dg>) ' 



112. Cylindrical, or Semi -polar Coordinates. If we take 

 the rectangular coordinate s, the perpendicular distance from the 

 Z-axis, Q 7 and co the longitude, or angle made hy the plane includ- 

 ing the point M and the .Z'-axis, we have the system of semi-polar, 

 cylindrical, or columnar coordinates, for which we have immediately, 



The parameter of a function f(z, p, o) is the resultant of the 

 partial parameters 



= I ' p = + - _ L 



, ay 



' 



113. Ellipsoidal Coordinates. Let us now find the value of 

 the parameter in terms of the ellipsoidal coordinates described in 73, 

 which are defined for a point x, y y z as the three roots of the equation 



1) 



i y 



"^ M 



-1 = 0. 



The three coordinate -surfaces at any point have been proved to be 

 mutually perpendicular at each point x, y y 2. Since the equation 1) 

 is an identity, we have, differentiating totally, that is changing x, y,8, 1, 



zdz 



Now if d^ is the perpendicular distance of the tangent plane from 

 the origin, we have by the last formula of 110, 



