172 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



Let the coordinates be q lt q 2 , q 3 . 



The parameter P is the resultant of the derivatives of Fin 

 any three perpendicular directions. Let these be in the directions 

 of the normals to the level surfaces q ly q z , q s . 



Then, calling these P ft , P, 2 , P Qa . 



(1) = 



= P qi cos (n&s) + P qz cos (n z n s ) + P qa cos (n s n s ). 

 Now Pg it the partial parameter with respect to q l} is ( 16) 



k. 



9^i 

 Hence 



9F , dV , 8F , . , 9F 



(2) - = A! COS (!!) + Aa ^ COS (w 2 ft s ) + A 3 g~ COS (ft^l,). 



If we divide the volume r up into elementary curved prisms 

 bounded by level surfaces of q 2 and q 3 , as in the case of rectangular 



FIG. 42. 



coordinates, we have, at each case of cutting into or out of S, 

 dScos (n s n l ) = dS 1 , where dS l is the area of the part cut by the 

 prism from the level surface q t . 



By 20, 



_ dq, dq s 



** -]t V 



accordingly 



