I»APER feY MAX MARGITLES. 



311 



then we obtain the following - equations for the determination of the 

 constants 



0+j>-(*+!>-*°- 1 



(s+ 3 )a,- ( Jh-+ 4 +2 )«,+ ifca 1 = 



V 7/ v 5 7 y 5 



,-2 + _f_ W - ( °.k + _5_ + i - 3 ) a, 

 i H- 2 7 l i i + 2 ' 



4 

 + 



-.+ A-«,- 4 = 



(11a) 



i = 5, 7, 9, . . .3 



Apparently a x remains undetermined ; for the computation of the 

 others, following the lead of Laplace, we write 



a, 



4 k (i + 2) 



a,i ~ i 3 k (i + 2) + (i -2 ) i (i + 2) - (%- 1) i (i + 1 ) 0i 



By the interchange of i with t + 2 a similar expression ir. formed for 



^i— and then ^± 2 and in a similar manner for the subsequent terms of 

 a { _ 2 a t 



the series, and by substituting these values in the above equation we 



obtain a continued rapidly converging fraction. 



N 1= 3 k. 7 + 3. 5. 6, 



• • ■ 



#3=3 /»•. 9+5. 7. 8, 

 Z 3 =4 jfc. 4. 5. 6. 9 



# 5 =3A\11+7.9.10, 

 Z 5 =4 k. 6. 7. 8. 11, 



a 3 



If in the second of equations (11a) we pat a 5 =q 3 a 3 , then will - also 



be determined, and the quotient has the same value as if it were com- 

 puted from the serial fraction 



a 3 4fc. 7 



N3-& 



o O 



