1818.] Lacroix's Differential and Integral Calculus. 20? 



x* d x 



Fluent 17. — Required the fluent of v ^ j7 ;. 



This fluent being of the same form as fluent 13, it may be 

 compared therewith ; by making 2 n — 1 = 3, or n = 2, a = a% 

 is 1, and^/* = 1. 



Then f , * 3d * = *~** (a 4 + x 2 )£ ; the same as Mr. 



J V a 2 + x' J 



Dealtry's when reduced. 



Fluent 20. (Case 4.)— Required the fluent of ,— — - . 



By comparing with fluent 13 ; we have 2 n — 1 = 1, or n = 

 1, b = \, f = 1, and a — — a. 



m . /* xdx 4 a + 2 x = 



Ihen / ■ . = — ~ V x — a ; the same as Mr. 



»/ V j — a •* 



Dealtry's when reduced. 



Fluent 23. (Case 2.) — Given the fluent of — : re- 



v ' V 2 a x + x» ' 



quired the fluent of — 



1 V 2 a x ■ + 



*3 



/x* dx /* , „ 



= /V"dx(2 a + x)~i 



x i-2 (2 a + *)3 — 3 a^xi ix(2o+ x)-i 

 _ 



_. r (2 a x + *»)* _ If C xdx 



2 2 »/ (2 a x + x")A ' 



By comparing with formula A, m = •§-, « = 1, a = 2 a, b — 

 1, andP = - i-. 



Fluent 28. — Given the fluent of (a + c 2")" . d 2 B ~* d z ; to find 



the fluent of (a + c **')"' . d z* n ~ l d z. 



dCz 1 "- 1 dz(a + c z n ) m 



dz" (a + c s n ) m + l — a n fz n — 1 d z(a + c z n ) m 



c n(r + 2) 

 d z» (a + c s» ) m + i ad /» T , 



'Hie same as Mr. Dealtry's, when L is put instead of 



- — , m + |) + n ' ( ; — . In making the comparison for this fluent, 2 n 



is written for m, c for b, and m for p. 



Fluent 31.— Given the fluent of z rn ~ l d z (a + c z n ) m ; to find 

 the fluent of z Tn ~ n ~ l dz(a + c z n ) m . 



