208 Mr. Adams on [Sept. 



/V"-"-' dz (a + e z*) 1 " = 



.m-n( a + c Z* ) m + l — 'ttl + r) tj C ^V"" 1 dl (a + C Zn ) m 



an{r - 1) 



where m = rn — n, P = m, and 6 = c ; in formula C. 



Fluent 32.— Given the fluent of Z T "+ — l d z {a + c «")•; to 

 find the fluent of xC"- 1 dz (a + c jb")" + 1 . 



A"- 1 dz(a + C2") m + ' = 



z r " (a + e z n )'" + l — cn(m + 1 ) y « r "+ * — 1 d z (a + cz n ) 



This fluent belongs to formula E, where m = rn + n, b = c, 

 and P = t$. 



J?tt«tf 33.— Given the fluent of z"—"- 1 <iz(a + c z") m ; to 

 find the fluent of ; r "~ l d z (a + c z") m ~- 1 . 



/V— 4 z {a + c z n ) m - = 



.ri-ifj + c z" ) m — (r n — n) A r "-"~ ' <f z (<r + c:")" 



771 n c 



This fluent belongs to formula F ; where m = r ti, b = c, and 

 P = m. 



Fluent 38.— Given fz r d z (a + c z") m = G 

 Required fz r + n d z (a + c z n ) m = H 

 f*z r+ ' n dz (a + a"f = 



/V+ 1 " Jz(« + c 2")"' = K 



Sac. 

 This class belongs to formula A. 



And J z r d z (a + c z n ) m + 1 = L 

 /V dz(a + cz n ) m+i = M 



fz r dz(a + cz n ) n+3 = N 



&c. 



This class belongs to formula B. v 

 From whence 



z r + ' (a + cz" )"'+ » — a (r + I). G 



H = 



C 777 71 + 77 + r + 1 ) 



jr+n + i( a + cj n ) m + ' - a(r + n + 1). H 

 "~ c (mn + 2 ti + r + 1) 



