352 REPORTS ON THE STATE OP SCIENCE. 



f(x) which is subject to certain restrictions can be expressed in the 

 form 



CO CO 



f(x) = - f [sin (w + 6) sin {u* + b)f{V)duclL 



O 



A somewhat analogous result has been obtained recently by Orr. 



If C(\) and S(X) are two polynomials such that the values of X for 

 which C(\) + i S(\) in zero have negative imaginary parts, and f(x) is 

 a function which satisfies Dirichlet's conditions and is such that 



CO 

 



is convergent, then — 



CO CO 



fC(.\) cos \x + S(X) sin \x [>„,r/., V 7 



u 



= l\f(^ + 0)+f(x-0} . x>0 

 = x ^°°) /(a, + 0) . x = 



C(oo) + tS(oo)*' v ' 



4. Abel's Integral Equation. 

 In 1826 Abel gave the solution of the integral equation 



/( ' T) -J(^-^ f(a) = ' ■ (1) 



a 



in a form equivalent to 



u{z)= s\nX^U'(x)dx ... (2 ) 



7T j(z — xy~ x 



He obtained the particular case of the equation in which X = ^ in the 

 solution of the dynamical problem of finding the form of a curve such 

 that the time a particle takes to slide down the curve from an arbitrary 

 point to the lowest point may be a given function of the arc described. 1 

 This problem contains the problem of the tautochrone as a particular 

 case, and is analytically equivalent to the problem of finding a surface 

 of revolution on which the geodesies satisfy certain conditions. The 

 case in which the surface has an equator and all the geodesies are closed 

 curves has been solved by Darboux. 2 



Another form of the integral equation had been obtained previously 

 by Poisson 3 in some researches on the distribution of temperature in 

 a conducting sphere, but was left unsolved. It was subsequently solved 



1 Collected Works, p. 11. The paper was published in Christiania in 1823. See 

 also Orelle,\o\. i., 182G, p. 153. s Theorie generate dee surfaces, t. 3, pp. 5-7. 



1 Journal de Vtcole Poly technique, 1821, 19, p. 299. 



