WILSON. — RECTILINEAR OSCILLATOR THEORY. 117 



values of the variables for which two values of the derivatives become 

 equal. In this case the failure is when 



v 2 — 4kvx = 0. 



This equation factors, and there are therefore two surfaces 



v = 0, v = 4kx 



in the space (v, x, t) which are singular in the sense that the funda- 

 mental existence theorem fails. 



Of these surfaces one, namely, v = 0, satisfies the differential equa- 

 tion (6') and is therefore singular also in the sense that it is the 

 envelope of particular solutions. It is possible to put. the particular 

 solutions in evidence. For let x be positive so that v <. 0. Intro- 

 duce a new variable v = — z 2 and let the sign of z be so chosen that 

 2 increases with r. We have then the equation 



Vz 4 + 4:KZ 2 X. 



In cancelling the variable z the double sign goes out. For since / is 

 decreasing with r (see G above) and since 



, dv - dz 



f = -T- = — 2z - r , 

 dr cLt 



it follows that dz/dt is positive. We therefore have to integrate the 

 equations 



dz z . 1 ,— dx . 



rfr = 4« + 4k V4K * + * 2 ' dT=- Z ~> 



with the initial conditions z = 0, x = a < 0, to obtain the particular 

 solutions which are tangent to the singular solution v = 0. For 

 these equations the existence theorems are applicable. As a solution 

 exists, it must be (9) which we found by undetermined coefficients. 



The other singular solution v = 4/c.r is not singular in the sense that 

 it is enveloped by particular solutions but in the sense that it is a 

 cusp locus for particular solutions- (see Picard, loc. cit.). Without 

 following Picard through the individual steps of the work we may 

 write down the result. If vo and xo are two values satisfying the 

 relation v = 4/c.r at r = to, the solution may be expanded into the 

 form 



x = xo + a (r — r) + /3 (r — r) 1 + y (to — t) 2 + 8 (t — t) § + . . . 

 v = vo + a' (to - t) + p (to - t)* + y' (to - t) 2 + 8' (r - t)= + . . . 



