of Differential Equations. 529 



such that the sum of the squares of the intercepts upon the axes 

 made by any tangent plane is constant and equal to n^. 



(f) ^Tien m= — 3, we find that the singular solution of the 

 equation 



du du du „ C 1 1 1 ^ 



and the geometrical corollary, analogous to those in the two pre- 

 vious cases, is obvious. 



{r)) Hitherto the values assigned to m have been integers ; but 

 if we suppose »2 = f, we find that the singular solution of the 

 equation, taken for simplicity in three independent variables, 



du du du r/ du\% / , du\i ( du\^'\ f 



is the symmetrical surface of the thii-d order 



a^ rfi z^ 



1- — H — =1. 



c? b^ c^ 



Indeed we might have easily proved independently, that the 

 perpendicular upon the tangent plane at any point of the surface 



a3^-^3 + c3~ ' 

 is given by the equation 



3 3 3 3 S 3. 3 



p^=a^ cos^ \ + b^ cos'' fjL + c'^ cos^ v, 



where \, /i, v are the angles made by the perpendicular with the 

 axes of coordinates. 



(6) Again, if we suppose m = ^, we find that the singular 

 solution of the equation, taken, as before, in three independent 

 variables, 



du du du f / du\i {,du\^ ( du\h'\ ^ 

 '"d^+yTy+'Tz'^iVTj -^yTy) ^VTz) J 



is given by the equation 



a b c ^ 

 - + - + -=1. 

 X y z 



II. Let it be proposed to investigate the singular solution of 

 the equation 



du du du o . {du\:= /du\-=: /du\- 



