117 



dz dndz ■*" " af« rf.,^ "^ Wi / ' 

 „ dVd?V . (d^V\^ , , dVd'V , , fd^V\2 , ^dl 



dri dtl^ 



f._dV d*V , ^ t^=r d^V . dVd*V . _ dWdW 



(L") 



dz dr?dz 



dr^dz dn'dz + ^'' </« rf,* + ^^^ dv" dn' 



^^'^ d^ d^ + ^''U.v ' 



and, making ;? = 0, we find by (I") the following equations in ordi- 

 nary differentials, from which V'-^\ V'-^, V^^\ are to be deduced : 



dvn 



= ^4- 2FW»; 

 dz ' 



= 



rfFW 



dz 



+ 8^« »''^> + ^ (^)^ 



(M") 



= 



dF(3) 

 dz 



dvmdvv^ 



+ 12 FU) F(3) 4- ^ ^ + 8F«2 



These three differential equations, when divided respectively by 

 K'*'*, F'"*, F"^^, can easily be integrated, and give, when combined 

 with the integral of [K"), 



(N") 



Voi ^1 J *2 > "J^s > being the four arbitrary constants introduced by the 

 four integrations. The functions F'"-^ , F"^ , V-'^ , V^'^ , are therefore 

 of the form 



VOL. XVI. 



u 



