130 Mr. W. Williams on the 



mation variable at, we get, by putting (v-~ $)=6 and inte- 

 grating between + 7T, 



ft- 1 rVtf. - sin(2^-H)^ 



e^f*F(0f*)30-h-§ f "f (<? + ;*) cos «0B<9; 



so that the function under the sign of integration becomes 

 infinite only when = 0. 



12. In the particular case when F(0 + #) has a constant 

 value c all the terms on the right in (11) vanish except the 

 first, the value of which is c. Hence in this case Sqo = c. If, 

 in addition, the limits of integration are from — ir to 0, or 

 from to 7r, instead of from — ir to 7r, we get Soo = 2 c » 

 These results will be required later. 



13. Since the function under the sign of integration becomes 

 infinite when = 0, we have to break up the integral into 

 three portions A, B, C, taken respectively between the limits 

 — it to —h, —h to A, and h to it*. We shall now show that 

 A and C vanish when n=zo for values of h as small as we 

 please, and therefore that the value of So, depends only upon 

 the infinitely thin strip B within which the function integrated 

 becomes infinite. 



14. Consider first the portion C. Let (2n + l)|0=$, and 



Whatever n may be, we can always choose h | so that (2n + 1) ^h 

 is a multiple of it. The integral can therefore be broken up 

 into a number m of elements in each of which the range 

 is 27r, and one element at the upper limit in which the range 



is ^ or -^-. This latter element will have a finite value a. 



For a given value of n let p be the value of the numerically 

 greatest of the remaining m elements. Then the sum of the 

 {m + 1) elements lies between + mp -J- a ; and therefore C lies 



between + p-\ — ^ t\1 since ~ T is <1. But when n 



— r 9r(2w + l)' 2?i + 1 



* The reasoning is precisely the same if the limits are — it to —A, 

 — h to g, and g to tt, h and g being independent small quantities. 



t Or, if (2w+l)-|A is not a multiple of n, each element of range 2tt 

 can "be broken up into four portions in each of which sin <p preserves the 

 same sign, so that the reasoning of (14) is still applicable. 



