An elementary treatise on differential equations and their applications . Fig. 15. Thus the ^-discriminant may be an envelope of the curves ofthe system, and if so, as shown in Ait. 55, is a singular solution. 61. The tac-locus. The envelope is thus the locus of pointswhere two consecutive curves of the family have the same valueof p. But it is quite possible for two non-consecutive curves totouch. Consider a family of circles, all of equal radius, whose centreslie on a straight line. SINGULAR SOLUTIONS 73 Fig. 16 shows that the line of centres is the locus of the pointof contact of pairs of c
An elementary treatise on differential equations and their applications . Fig. 15. Thus the ^-discriminant may be an envelope of the curves ofthe system, and if so, as shown in Ait. 55, is a singular solution. 61. The tac-locus. The envelope is thus the locus of pointswhere two consecutive curves of the family have the same valueof p. But it is quite possible for two non-consecutive curves totouch. Consider a family of circles, all of equal radius, whose centreslie on a straight line. SINGULAR SOLUTIONS 73 Fig. 16 shows that the line of centres is the locus of the pointof contact of pairs of circles. This is called a tac-locus. Fig. 17. fig. 16. shows circles which do not quite touch, but cut in pairs of neigh-bouring points, lying on two neighbouring loci AA, BB. Whenwe proceed to the limiting case of contact these two loci coincidein the tac-locus XT. Thus the y-discriminant may be expected tocontain the equation of the tac-locus squared.
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