of Differential Equations. 527 



reduce this to a symmetrical form, we obtain 



Vdi^yTy) ~\dy) \dx) ' 

 the singular solution of which is 



while the general integral is 



where « and /S are arbitrary constants. 



More generally, the singular solution of the equation 



is 



c2 a^ b^ ^^' 



(7) When m = l, we must have recourse to the original equa- 

 tion, and we find that the singular solution in this case is the 

 aggregate of the equations 



x — a = 0,- 



y-b=o, 



&c. 



In the case in which the number of variables x, y, z, &c. is 

 reduced to three, the geometrical interpretation of this result is, 

 that the envelope of all the cones, whose common vertex is the 

 point whose coordinates are a, b, c, is that point itself. 



(8) When m= —1, we find that the singular solution of the 

 equation 



(du du du \ / Ji 1 1 0. \ _ 1 



dx dy dz ' / I du ,du du I ~ ' 



\ dx dy dz / 



G)*+(l)^e)^^- 



It a = b = c = n, and the number of independent variables be 

 reduced to three, it is obvious that if P be the perpendicular 

 from the origin on the tangent ])lane to the surface indicated, 

 and a, /3, y the angles made by it with the axes of eooi'dinates, 



