11 



at a finite distance from the origin and from the perpen- 

 dicular, and we have a case of Symptotic Envelopement. 



20. But we may conceive the kiw of construction as 

 changing, and the point of contact as passing off to an infi- 

 nite distance, and becoming the point at infinity in the 

 tangent. We shall then have a case of Asymptotic En- 

 velopement. 



21. Again, we may conceive that the foot of the perpen- 

 dicular also passes off to an infinite distance, and so that the 

 tangent lies altogether at infinity. In this case, the line at 

 infinity is an envelope. 



22. In each of these three cases all the primitive curves, 

 non-consecutive as well as consecutive, are, or are conceived 

 as being, in mutual contact. And in whichever of the three 

 senses above indicated the word envelope is used, in that 

 same sense the word tac-locus might also be used. In all 

 the three cases the envelope is also a tac-locus. Such an 

 envelope may be called a General Primitive Envelope. 



23. When the primitive curves are not all envelopes but 

 a certain particular primitive curve is an envelope, we have 

 a Special Particular Primitive Envelope. Such an envelope 

 I shall call an Epicene Envelope. 



24. When an envelope is not a primitive curve, it is a 

 Singular Envelope. 



25. There are three species of envelope, viz. : 



I. General Primitive Envelopes. 

 II. Epicene Envelopes. 

 III. Singular Envelopes. 



26. There are three varieties of the first species, and a 

 general primitive envelope may give rise to — 



(1) A Symptotic Envelopement. 



(2) An Asymptotic Envelopement. 



(3) An Envelopement on the line at infinity. 



27. Suppose each particular primitive curve to be capable 

 of being represented by its respective particular primitive 

 equation, or, briefly, by its particular primitive. Then an 

 equation wherein the parameter, now regarded as the 

 arbitrary constant, is left undertermined in value, but by 



