528 The Rev. R. Carmichael on the Singular Solutions 

 the equation 



(du du. du\ / \ 

 ^ dx ^ du dz)\ du 



dy dz 

 is equivalent to 



I du du du I 



\dx dy dz / 



\cosa cosp cos 7/ 

 Hence we infer the known theorem, that the surface 



Vx+ Vy+ \^ z=- Vn 

 is such that the sum of the intercepts upon the axes made by 

 any tangent plane is constant and equal to n. 



(e) When m= —2, we find ti»at the singular solution of the 

 equation 



du ^ du du a \ f 1 1 1 o"l^-i 



dyj 



©•-(1)'-©'— ■ 



If the number of variables x, y, &c. be reduced to two, and 

 a = b = n, we find that the singular solution of the equation 



1 



+ ■ 



(du du\ 

 "^di'^y'd^j)^ (duy^ /duy 



x^ + y^=n^, 



the equation to a hypocycloid. As this curve is known to be the 

 envelope of a right line of constant length, whose extremities 

 move on two rectangular axes, we have thus arrived at the geo- 

 metrical interpretation of this differential equation, an interpre- 

 tation readily confirmed a posteriori. 



More generally we see that the singular solution of the 

 equation 



du du du _ r 1 1 1 1 -i 



^:7Z+yx.-i--:7;-J JduV'^~Jd^^~ndu^ 

 n 



'dx'^dy^'dz \ JduY ' o(duY „/duV' 



\dx/ \dy/ \dz/ 



is 



2222 

 x^ + y'^ + z^^n^ ; 



and that this again is, as is known, the equation to the surface 



