i So Singular Solutions. CHAP. 



of what are called singular solutions. We must here 

 state when and how a singular solution arises, for the 

 term is by no means self-explaining. 



A set of curves are drawn which we shall call C, 

 C', C", etc. They are not in general mathematically 

 similar, but constitute a family, varying continuously 

 from curve to curve according to a definite law. 

 They are indefinite in number and indefinitely near 

 to each other, and are so drawn that C intersects with 

 C', C' with C", and so on. A curve S, which is gener- 

 ally of a totally distinct kind from the curves C, is 

 so drawn through these intersections that the curves 

 C are tangential to S ; making the relation of S to 

 the curves C somewhat like that of a circle to its 

 tangents. S is called the envelope of the curves C, 

 and it is " singular," that is to say unique, and not 

 one of a family like the curves C. 



The following diagram shows the relation of the 

 curves C to S : 



