108 CAIil'MAHL OX TilK DETERMINATION IN TERMS OF A 



whence in this case also 0=0, and, consequently, e<]iiation (xv) will hold with /v + 1 

 written for />, whatever integer y^ may be. But we have already seen that the equation is 

 true when // = 1 ; it is, therefore, true for /; = 2, or /; = 3, &c. 



_, — |-, COS (/• H — • Wi — ) 



If then we write I„, ibr / ^ ' 2 ,;w we have bv (xiii) and (xiv) 



H 



- - _ / f dr 



'I- v/ 



n 







whence by (xv) 



IT I 6 y 



^^ — x/ "TT j 72 J ^ * + 216 * j 







These results do not agree with Cauchy's. His M; _j_ ,* should be equal to _ . c-'' . Ij ^ ,. 











with — r written for /•, but it differs from it, in every case, by terms which do not contain 



either e 'î ' or / g ^ '^ dr 

 ' 

 Again, from the equation 



I, = /■" f ^ cos r t) JH = ^/l- .fa'' 







we may, by successive differentiations with respect to r, obtain 



!_, = - /^.s^.r^'^' 



- — _ 1 ■ •' 



I_, = 3 y'-^,^ . (3 r - 1) . 6 ' ' 



-4r' 



I_, = -27 y^^^^ .r{r-\)l 



I_, = 27 y^ (3/-^ — G /•'+!) f - 



• !<eo Juunial Ji- I'Krulu I'ulytcfhiiiqiie, iS° Culiier, p. L'3S. 



