68- TO] Solution of Equations 69 



This equation ought to be an identity. That it is so is easily seen with 

 the help of the relation 



Now assume that a more general solution of equation (148) is 



in which <f> is any function of x, y, z and X. The equation of the family of 

 surfaces X = cons, is now supposed to be 



Since (j) is so far supposed to be perfectly general, the solution we have 

 under discussion is in reality perfectly general, although written down in a 

 form which is specially applicable to surfaces obtained by distortion from the 

 ellipsoid /= 0. 



Equation (149) is true in any case. If /+ < is a solution of equation (148) 

 we must also have 



f* 8# w r/ 

 ^ (X) V 2 (/+</>)- i/r (X) 4- L = -^Trp (152). 



JO V , /. . , N 



On subtracting corresponding sides of equations (149) and (152), and sim- 

 plifying with the help of relation (150), we. obtain 



SOLUTION OF EQUATIONS 



70. This equation does not admit of direct solution, but may be effectively 

 broken up by assuming a solution 



(j) = u+fv ............................ '..(154) 



in which u and v are functions of x, y, z and X. On substituting this value 

 for (j> and simplifying, equation (153) reduces to 



and this will be satisfied if we can satisfy separately the two equations 



(155), 



^ } = (156). 



