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A Common Trext-Book ERROR IN THE THEORY OF ENVELOPES. 
By A. S. Hatwaway. 
The cause of this communication is the recent appearance of several 
text-books on the calculus that embody an error in the theory of envelopes 
that dates at least as far back as Todhunter’s calculus, and is now repro- 
duced in all text-books under the impression, apparently, that it has ac- 
quired the sanction of authority, although Cayley pointed out the error 
nearly forty years ago, while the subject matter is presented in all text- 
books on Differential Equations in its correct form. The error consists 
in defining the envelope of a moving curve as the locus of its self-inter- 
sections, and then proving that the envelope touches the moving curve in 
every position—i. e., proving as true that which is often false—for the 
locus of self-intersections of a moving curve may cut the curve at any 
angle, as at right angles, wherever the two meet. A simple example is 
the curve (y—m)’*=(#—3)*, whose locus of self-intersections, as m varies, is 
the straight line 7=3, which cuts every curve of the given system at right 
angles. The fact is, that the envelope should be defined as the curve that 
touches every curve of a given system. It can then be shown it is a locus 
of self-intersections of the curves of the system, provided such self-inter- 
sections are not the singular points of the given system. The locus of such 
singular points is always a locus of self-intersections, but it is not in 
general an envelope of the system, and may cut every curve of the sys- 
tem at any constant or varying angle. The text-book blunder referred 
to is of the same logical character as would be the attempt to prove that 
a quadruped is a horse. To be sure, a horse is a quadruped, but not every 
quadruped is a horse. Thus a curve that touches every curve of a given 
system is a locus of self-intersections of the system, but not every locus 
of self-intersections of the system will touch every curve of the system. 
The error in the proof arises out of the assumption that if two points of 
a curve approach coincidence, the limiting position of the chord joining 
the two points is a tangent line at the point of coincidence. This is all 
right if the point of coincidence is not a singular point of the curve. But 
at a singular point, as a sharp point like the bottom of a letter V, the 
limiting position of two points that approach the point on opposite sides 
is absolutely indeterminate, and is not necessarily a tangent line at that 
point. 
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