530 The Rev. R. Carmichael on the Singular Solutions 



where 



a-\-b-\-c+ &c. =OT. 



The general solution of this equation is, as we have seen, 



a b c 



«,r + /3(/ + 7^ + &C. = Aam . /3m . ym . . . . , 



where «, /3, y, &c. are arbitrary constants. 



Differentiating this expression with respect to u, and for bre- 

 vity putting the right-hand member equal to K, we obtain 



aK bK cK „ 



x= — , y= —5, s= — , &c., 

 7nu mp my 



whence 



«K r, bK cK , 



a= — , /3= — , 7= — , &c., 

 ?na? my mz 



and the singular solution desii'ed is 



a b c a b c 



m . X m , ym . zm ... =1 Afl"' .bm.cm. . . . , 



or 



m"" . x" y'' z' . . . = A"' . a" 6* c" . . . 



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



we see that the singular solution of the equation 



du du ^^ _ \ fdu\i /du\i /duXi 

 '^di^y'd^^''Tz~^\di) •\dy) '\Tz) ' 



{ 



du du du~\^_.Qdu du du 

 dx dy dzj dx dy dz' 



27xyz=h?. 



It is known independently that this is the equation of the 

 surface such that the product of the intercepts made upon the 

 rectangular axes of coordinates by any tangent plane is constant, 

 the volume of the constant pyramid being to the volume of the 

 parallelopiped formed by the coordinates of the point of contact 

 as 27 : 6, or as 4|^ : 1. It is easy to identify the two results by 

 dividing both sides of the last equation by 



J /duy /duY /duY\i 

 \\di) +\d^) ^\Tz) J ' 



III. The singular solution of the partial differential equation 

 du/ du du du „ \ . /«>/'^"\^ g/^^^\^ , o n 



'di{'Tx^ydy-''-di+''v+'\wi) ""yd-zj +^*^-=^ 



is easily determined in a similar manner. 



