24 



35. Thus we have three kinds of solution, viz. : — 

 (I). Particular Primitives. 

 (II). Epicene Primitives. 

 (III). Singular Solutions. 

 86; Suppose that the complete primitive can be decom- 

 posed into factors one-valued, say linear, with respect to 

 the arbitrary constant. Then an epicene solution is a 

 primitive, and particular, when considered in relation to 

 one or more of such factors. But in relation to the rest it 

 may not be a solution at all, or it may be not particular 

 but singular, and consequently not a primitive. 



87. Regarding each such linear factor as a complete 

 primitive, there will be only two other kinds of solution at 

 most, viz., Particular Primitives and Singular Solutions. 



38. The derived equation of course contains the differ- 

 ential coefficient, and it may also contain all, any, either, or 

 none of the three quantities following, viz., the arbitrary 

 constant and the two variables. The cases in which it 

 contains none, and in which it contains the constant alone, 

 are cases in which it cannot be brought under what I call 

 an adfected form, viz., a form containing the arbitrary 

 constant, together with one at least of the variables. In all 

 other cases the derived equation is either adfected as it 

 stands or else may be rendered so ; for the constant, if absent 

 from, may be introduced into it by eliminating a variable 

 between it and the complete primitive or, otherwise, by 

 substituting for the variable in a part only of the expression 

 for the differential coefficient. In seeking a general primi- 

 tive envelope I deal with the differential equation under its 

 adfected form. This form cannot be attained without a 

 knowledge of the complete primitive. 



39. When the complete primitive and its adfected deriva- 

 tive can both be satisfied, for all values of the arbitrary 

 constant, by a system of particular values of the variables 

 we have a general primitive envelope. 



