274 DEVELOPMENT of a certain 



redu&ion, our two laft fluxionary equations may be otherwife 

 exprelTed, thus : 



<P* , , f X 0' 



tf i — e l fin 1 y V T ; ^," x ^^.^.^ , 



?fin*p e'( !+<■') r' fin*2ft' , i © « +'«' • 



v/iw'fin'^- ""T^V-^fin"*'^' + :"/i~'Ii^ ~ "7~~ P C0f'2^. 



Let us, for the prefent, denote the fluents of the fluxions 

 which enter thefe two equations, as follows : 



p = f * p' ~ r ? ? 



J/i-j'fia 1 ?' J vA — ^fin^/'* 



N=/l r i' ii °';-, t> N'= A *' finW ,; 



and the relation of the fluents will be exprefTed thus : 

 P=(i-rV)P'> 



2.2 



4. Now, remarking that thefe two equations have been obtain- 

 ed, by afluming 



, 1 — t/i — e % , r ' . fin ip fin zp 



e s — ■ > -, » and fin 20 — .■ ,. , , r -1 = ; — y . „ ->> 



let us farther afTume two fimilar equations, thus, 

 i, = '-/^ fin 4 f = *■* fn 4 '' 



where it is evident that when 9' rz o, ^ is alfo rr o, and that 

 while <p or q> increafes from o to |, qf* will alfo increafe from o 



it 

 to -. 

 2 



If we take the fluxions of this fecond aflumed equation, and 

 ■reafon in all refpects as in Article 3., we {hall arrive at a 'fimilar 



conclufion, that is, putting P' for /— • / and N* for 



J \/i — / /J iin *4f •" 



y F** Vfi n "»4'(p# 

 ^/i — *# 1_ im*4^ ' 



P' = 



