19^ 



» OC 00 



r t» r tn r e-i 



I «-' dt^ Xe' — dt -I- I (L\ (t) dt. 



J (1+0" J (H0'+' jn+ty^"^^ 



o 



The first integral of the second memher we convey to the first 

 membei', and we find : 



00 00 



r r+' r e-t 



|g-' —dt=] (fv(t)dt (17) 



J (1+0"+^ J(i-t-0' 







Tf we apply the same process lo tiiis, and again to the losnlt, 

 etc., we find at last after ?»-fold api)liance : 



I e-t dt = tn! \ — (r>„ (t) dt . . . . (1 8) 



J (i+0"+i J (1 + 0'"+' 







We can render this formnla still more general by summation 



from ?i = to ?z =■■ oo after division by ( — 1)'" ???/; we get: 



00 00 



r V' re- f 



e-'^^ dt — r/„ (t) dt (19) 







We apply the process explained above to this again and by 

 summation again after division by (—'])'"?>?/ etc. we finally find: 



C , ^"+'" C^ ffn it) 



\e-h -— - dt = m! \e-' ^ ' , df. . . . (20) 



J (1+0"+' J («+/t)-+i ^ ^ 







in which k and m represent positive integers. 



Of course a formnla analogous to (121 may be deduced fi-om Ihis 



00 



(l+n)"+l 

 



.(21) 



- j e-^' t/2"+i [e--^yHi^{e-^!/'') + e^!/Hi{e '^■'^] di/ Ce - 2'.'/ r/ „ ( - « -) 







dt. 



By summation, formula (13) is, however, found again. 



The formulae (4) ajid (5) may also be used in order to expi-ess 

 the polynomia //o„ in </'s. Foi- this pui-pose we multiply the two 

 members of (5) by 



1 ^ 11 1 1.3 1 1.3.5 



]/{l — m =1 6» . — <9^ . — 0' 0' — ... 



2 2! 2' 3.' 2' 4! 2" 



By equalizing the coefficient of 0" in the second member of the 



14 

 Proceedings Royal Acad, Amsterdam. Vol. XVll. 



