526 The Rev. R. Carmichael on the Singular Solutions 



which have been at various times proposed as examples, but a 

 large variety of new and interesting results besides. 



(«) Thus, let ??i = 3, and we find that the singular solution of 

 the equation 



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

 that the singular solution of the equation 



du du f o/du\^ ,^/du\^\^ 



^^+.x.4+^e.={..(|)VK|)V4S)^^4* 



(/S) Again, let m = 2, and we find that the singular solution 

 of the equation 



du du 



'Ta^-^y-di, 



is 



When the number of variables x, y, z is reduced to three, we 

 have the known theorem that the envelope of the series of deve- 

 lopable surfaces whose general partial differential equation is 



du du du r a(du\^ Ta(du\^ a(<ly\^\ i 

 ^Tx^yTy^'Tz = Y\-dx) ^Ady) +^%^) > ' 

 is the ellipsoid 



It is worth remarking, that the differential equation which first 

 suggested to Taylor the existence of a solution not involved in 

 the general integral, " singularis qua?dam solutio problematis," 

 as quoted by Lagrange from the Methodus Incrementum (p. 27), 

 namely, 



comes under this example, as a particular case. Tn fact, if we 



