116 



PROFESSOR STOKES, ON THE DISCONTINUITY 



The complete integral of this equation in ascending series, obtained in the usual way, is 



« 



-1 



...)J 



(19) 



These series are always convergent, and for any value of x real or imaginary assign a 

 determinate and unique value to u. 



The integral in a form adapted for calculation when x is large, obtained by the method of 

 my former paper, is 



u 



.( 1.5 1.5.7.11 1.5.7.11.13.17 U 



[ 1.144a;* 1.2.144a^ 1.2.3.144V j[ 



^ f 1.5 1 .5.7.11 1 .5 .7.11 .13.17 1 



+ Dx-ie'^ {l+ i+ + r— + —\ 



I 1 . 1440?* 1.2. 144'a' 1.2.3. 144V J 



(20) 



The constants C, D must however be discontinuous, since otherwise the value of u deter- 

 mined by this equation would not recur, as it ought, when the amplitude of x is increased by 

 2n-. We have now first to ascertain the mode of discontinuity of these constants, secondly, 

 to find the two linear relations which connect A, B with C, D. 



Let the equation (20) be denoted for shortness by 



«= Gr-i/,(<p) +Z>a?-i/2(.i7) ; (21) 



and let f(jv), when we care only to express its dependance on the amplitude of x, be denoted 

 by F{d). We may notice that 



-F'i(0 + |t) = /'.(0); F,{e + %^)^F,{e) (22) 



16. In equation (21), let that term in which the real part of the index of the exponential 

 is positive be called the superior, and the other the inferior *''^' ^' 



term. In order to represent to the eye the existence and 

 progress of the functions fi(x), /j,(*) for different values of 

 d, draw a circle with any radius, and along a radius vector 

 inclined to the prime radius at the variable angle take two 

 distances, measured respectively outwards and inwards from 

 the circumference of the circle, proportional to the real part 

 of the index of the exponential in the superior and inferior 

 terms, 6 alone being supposed to vary, or in other words 

 proportional to cos ^9. For greater convenience suppose 

 these distances moderately small compared with the radius. 

 Consider first the function Fi(9) alone. The curve will evidently have the form represented 

 in the figure, cutting the circle at intervals of 120", and running into itself after two complete 

 revolutions. The equations (22) shew that the curve corresponding to F„{9) is already 



