12 



means of which any particular primitive may, by an appro- 

 priate determination (accompanied or not by a decomposi- 

 tion into factors), be constructed is the complete primitive. 

 General primitive and epicene envelopes are represented 

 by particular integrals, singular envelopes by singular 

 solutions. 



28. Epicene and singular envelopes possess a common 

 geometrical property. But an epicene envelope is, and a 

 singular envelope is not, a primitive curve. So a particular 

 integral is, and a singular solution is not, a case of the com- 

 plete primitive. And whether we say that an envelope is 

 at once epicene and singular, or that a solution is both a 

 singular solution and a particular integral, the logical prin- 

 ciple of contradiction is alike violated. In the geometry 

 the contradiction is plain. In the analysis it is as real 

 though less plain. It is less plain because, if we decompose 

 a complete primitive into factors, each factor will represent 

 a distinct part of a curve. The curve is therefore no longer 

 represented as a geometrical whole, but as a synthesis of 

 parts, each having its own analytical representation. The 

 several factors will give rise to distinct differential equa- 

 tions, two or more of which may have a common solution. 

 But such solution may be a particular integral of one dif- 

 ferential equation, and a singular solution of another. Illus- 

 tration of this may be drawn from my paper " On Particular 

 Integrals " in the Quarterly Journal of Mathematics (Vol. 

 XIII., No. 51, see pp. 240-1, art. 5). For if, instead of look- 

 ing at its separate parts, we view the parabola as a whole, 

 then the epicene primitive (therein called a singular inte- 

 gral) possesses the geometrical properties of a singular 

 solution. 



29. Nevertheless, if we confine our attention to the pro- 

 duct, and suppress all reference to its factors, we may regard 

 the common solution either as particular or singular. It is 

 only in this sense that a solution can be both a singular 

 solution and a particular integral. 



12, St, Stephen's Koad, Bayswater, 



London, W., Oct. 9, 1884. 



