ON THE THEORY OF INTEGRAL EQUATIONS. 305 



of the solid so that it sinks to a given depth. Find the law of density 

 of the fluid in order that the total weight of the solid and weights may- 

 be a given function of the depth to which solid sinks. 



17. Integral Equations in which the Principal Value of the Integral must 



be taken. 



The inversion formula? for integral equations in which the integral 

 has its principal value are often very simple — for instance, Hilbert gave 

 in his lectures the formula 



i i 



p(s) = [/(<) cos tt(s - t)dt + U(t)dt 



o 



/(s) = - f ,p(t) cos *(s - t)dt + [j\t)dt 







particular fornis of which had been known for some time. This formula 

 was used to solve more general equations of the first kind in which an 

 integral has a principal value. The formulae have been derived and 

 extended also by Kellogg. 1 



A general theory of inversion formulae connected with integrals of 

 which the principal values must be taken, 2 has been given by Hardy 3 and 

 many beautiful results have been obtained. 



Integral equations of the type under consideration are of some 

 importance in two dimensional potential problems, as they arise when 

 the point at which the potential is estimated is taken on the attracting 

 curve. 



Additional Bcsults. 



If (f>(x) and its first two derivatives are continuous and l>j>(x)j<K, and 



the integrals 



r 

 f 0(g) lo g x ^ M x ) log (-= x) dx 



J X ' ' J X 



— CO 



are convergent, and if 



bo 



CO 



then ^J.pfSfifck 



J y — x (Hardy) 



— CO 



1 

 Also if \{y) = Pjcosec ic(x — y)<j>(x)dx, 



o 



1 Dissertation, 1901 



- Stokes, On the Highest Ware of Uniform Propagation', PrOC. Cambr. Phil. 

 Soc , 1883, vol. iv., pp. 361-365 ; Math, and l'Jii/s. Papers, vol. v., pp. 110-159. The 

 Correspondence of Hermite and Stieltjes. 



3 Proc. Loud. Math. Soc, 1909, ser. 2, vol. vii., p. 181. 



