108 PROFESSOR STOKES, ON THE DISCONTINUITY 



for our purpose, since we may always suppose the amplitude of a included in the range a to ' 

 a + 2;r, by adding, if need be, a positive or negative multiple of 27r, which as appears from 

 (1) or (3) makes no difference in the value of m. 



4. When p is large the series (4) is at first rapidly convergent, but be p ever so great it 

 ends by diverging with increasing rapidity. Nevertheless it may be employed in calculation 

 provided we do not push the series too far,* but stop before the terms get large again. To 

 shew in a general way the legitimacy of this, we may observe that if we stop with the term 



1 .3.5...(2i- l) 



2V*+» ' 



the value of u so obtained will satisfy exactly, not (2), but the differential equation 



du 1.3.,.(2i + ]) 



— - + 2o« = 2 -T-j 



da 2*0'^' 



• + 2a« = 2- • ■'. (5) 



Let «o be the true value of u for a large value a^ of a, and suppose that we pass from a^ to 

 another large value of a keeping the modulus of a large all the while. Since u ought to satisfy 

 (2), we ought to have 



whereas since our approximate expression for u actually satisfies (5) we actually have, putting 

 Af for the last term, 



u 



. + e-'' f\2-Ji)e^'da (6*) 



•'Oo 



If a be very large, and in using the series (4) we stop about where the moduli of the terms 

 are smallest, the modulus of Jf will be very small. Hence in general A^ may be neglected in 

 comparison with (2), and we may use the expression (4), though we stop after i + 1 terms of 

 the series, as a near approximation to u. 



5. But to this there is an important restriction, to understand which more readily it will 

 be convenient to suppose the integration from Oq to (>■ performed, first by putting 



da = (cos 9 + s/ - 1 sia 6) dp, 

 and integrating from p„ to p, 9 remaining equal to 9^, and then 



da = p {- sin 9 + \/ - 1 cos 9) d9, 



and integrating from 9^ to 9, p remaining unchanged. This is allowable, since « is a finite, con- 

 tinuous, and determinate function of a, and therefore the mode in which p and 9 vary when a 

 passes from its initial value a,, to its final value o is a matter of indifference. The modulus 

 of c"' will depend on the real part p^ cos 29 of the index. Now should cos 20 become a 

 maximum within the limits of integration, we can no longer neglect Af in the integration. For 

 however great may be the value previously assigned to i, the quantity p~^'~^e''^ ™'''* will become, 

 for values of 9 comprised within the limits of integration, infinitely great, when p is infinitely 

 increased, compared with the value of e''"'^"'^* at either limit. And though the modulus of the 

 quantity 2e"' under the integral sign will become far greater still, inasmuch as it does not con- 



