lis 



PROFESSOR STOKES, ON THE DISCONTINUITY 



Fig. 2. 



can the coefficient of the inferior term alter discontinuously, and the coefficient of the other 

 term cannot change so long as that term remains the superior term. Referring for conve- 

 nience to the figure (Fig. l), we see that it is only at the points a, b, c, at the middle of the 

 portions of the curve which lie within the circle, that the coefficient belonging to the curve can 

 change. 



It might appear at first sight that we could have three distinct coefficients, corresponding 

 respectively to the portions aAb, hBc, cCa of the curve, which would make three distinct 

 constants occurring in the integral of a differential equation of the second order only. This 

 however is not the case ; and if we were to assign in the first instance three distinct con- 

 stants to those three portions of the curve, they would be connected by an equation of 

 condition. 



To shew this assume the coefficient belonging to the part of the curve about B to be equal 

 to zero. We shall thus get an integral of our equation with only one arbitrary constant. 



Since there is no superior term from = - — to 6 = + — , the coefficient of the other term 



cannot change discontinuously at a {i.e. when Q passes through the value zero); and by what 



has been already shewn the coefficient must remain unchanged 



throughout the portion hBc of the curve, and therefore be equal 



to zero ; and again the coefficient must remain unchanged 



throughout the portion cCaAb, and therefore have the same 



value as at a ; but these two portions between them take in the 



whole curve. The integral at present under consideration is 



represented by Fig. 2, the coefficient having the same value 



throughout the portion of the curve there drawn, and being 



equal to zero for the remaindOT of the course *. 



The second line on the right-hand side of (20) is what the 

 first becomes when the origin of 9 is altered by =•=§■"-, and 

 the arbitrary constant changed. Hence if we take the term 

 corresponding to the curve represented in Fig. 3, and having 

 a constant coefficient throughout the portion there repre- 

 sented, we shall get another particular integral with one 

 arbitrary constant, and the sum of these two particular 

 integrals will be the complete integral. 



In Fig. 3 the uninterrupted interior branch of the curve 



Fig. 3. 



is made to lie in the interval — to ir. 



3 



It would have done 



equally well to make it lie in the interval to — tt ; we should thus in fact obtain th^ 



3 



same complete integral merely somewhat differently expressed. 



• A numerical verification of the discontinuity here represented is given as an Appendix to this paper. 



