[112] PROFESSOR DE MORGAN ON SOME 



which, beginning from A (not its centre) spreads without touching the singular ellipse, until 

 at last it touches it in one point, with a contact of the second order. Immediately afterwards 

 two points of contact of the first order separate, which move in opposite directions as the wave 

 proceeds, until they meet on the opposite side, and there is again a contact of the second order. 

 The contact then disappears, and the wave diminishes, vanishing at B. Now the more near 

 the ellipsoid is to a sphere, the smaller is the duration of the period of contact : and when the 

 ellipsoid becomes a sphere, there is but an instantaneous contact, the circular wave arriving at 

 complete coincidence with the singular circle, as to a terminal position, and then receding. 



This geometrical view may fail in one particular. There may be horizontal projections with 

 no sections belonging to them, owing to c being imaginary. The two surfaces, <p (», y, c) = 

 and (b (x, y, •arc) = 0, have the same tangent cylinder, but meet it in different curves : they 

 have the same horizontal projections, but a given projection has not sections in both, unless c 

 and isc be both real for the value of c. 



I now proceed to the consideration of the test derived from the differential equation itself. 

 This is usually stated as follows : — The singular (meaning extraneous) solution is to be looked 

 for among those relations which make y_ y infinite, y = X (*» V) being the differential equation. 

 Remember that (p (x, y, c) = and <p x + <p y ^ = are equations of identical meaning, and 

 treat c as a function of three variables, ,r, y, ^, obtained from the second. Substitution of 

 this value of c in the first will make it identical, if for ■% we write its value in terms of a? and y. 

 That is, 



(dc dc \ , , jdc dc \ 



* + Hb*5*)"* ^ + Hdy- + ^ =0 ' 



dc (<p x dc\ dc _ (<^y dc\ 



K^'K^^dx}' Xl 'd x -~{$ c + 'dy') , 



dc (dc dc \ 



If, as will most usually happen, the differential coefficients of c be of ordinary form, then 

 y, and v- are finite or infinite with 0, : (p c and (p y : (p c : that is, y, and y^ may be substituted 

 for <!? x and <£ y in the test already examined. But the possibility that dc : d% may vanish, or 

 that dc : dx or dc : dy may become infinite, must immediately strike the mind. If indeed we 

 can be assured that the concourses of the singular curves with primitives take place at points 

 of finite curvature, so that y" or y, + y^ y or (c x + c y ^) + c„ is ordinary and finite, we much 

 increase the presumption that c x and c y are ordinary, but still we have not ptoved it. This 

 difficulty puts the complete meaning of the test in question beyond the reach of ordinary 

 analysis. The following mode of examination will succeed completely. 



When two solutions of y = y (x, y) meet (and therefore touch), it follows that y_ x or y^, 

 one or both, must be infinite, except only in some of the cases in which one at least of the 

 curves is of infinite curvature. The following sketch of a proof may easily be reduced into the 

 language of algebra and of limits. 



