536 The Rev. R. Carmichael on the Singular Solutions 

 but stated as indispensable. 



in the case in which 



dy 



><0, 



and 



u=/(^-^'l)=''- 



in the case in which 



d\] 



u=y(.,,,|)=o. 



If we confine our attention for a moment to the former case, the 



JTT 



interpretation of the condition -=— not = appears to be, — if a 

 change of form be attributed to the function y, as well as an in- 

 crement to -y-, — no envelope ; and the geometrical confirmation 

 of this is obvious enough from the accompanying figure. 



To the theory this condition is indispensable; but then an 

 analogous condition or conditions must be regarded as indispen- 

 sable in the investigation of the singular solutions of differential 

 equations of the second and higher orders ; and none such are 

 given. Moreover, in the investigation of the singular solutions 

 of partial diff"erential equations of the first order, represented in 

 general by the type 



U=/(a:, y, 2r,/), g')=0, 



we should expect to find stated, as a condition indispensable to 

 the theory, that the result of the elimination of ^ and q between 

 the equations 



U=0 ^=0 ^=0 



^^' dp "' dq ' 



