Astronomical and Nautical Collections. 14.5 



easier to employ the particular fluent |/* —i = a/b-, dis- 



VhlJL 



z 



covered by Euler, and still more elegantly demonstrated by 

 Laplace, in the form " ^f^ e~** dx — J « : and the applica- 

 tion of this proposition leads us to the integration of several 

 fluents, which may be thus enumerated : 



a. 1 f* dz = J« as V 3. 141592 



Vhl 



z 



, x r z " dz = x / v(*+d d y _ / * 



z y 



putting y = z" +1 . 

 \ r 2 " dz = 1 2"+' _ «+l / » z*dz 



" J h j ia J_ ,»-l # j^^ m-\J hlm . t J_ 



i. y _£i_ =2 _i_- 2Vw; Y-^l = i 



1/0 hi* -L u*± ^ °hi*i. 3 



Z Z z 



4 Z J 4 / 



hi* _L * hit 



z z 



3 hl*-L 3 



z 



2dz _ o z * 



"^ VU- ^ 2 'J° hlli 



z z 



y 2 '^° Uti 3 "hTT 



1 



z 

 8 zz 



3 hi*_L 



3 V 2 



iii. j r q-«)j« « r & _ r 4 _ w 



^° hi* .1 J°w±~ \t* u 



Z Z 



(2(^2 - 1) V«- = - 828427 V^- 



z Z Z 



Vol. XVI. L 



