ALGEBRAIC FORMULA. 275 



F = (i + OP', 



If by means of thefe two equations, and the two fimilar 

 equations found in laft Article, we exterminate P' and N', we 

 get . 



5. Let us now aflume two feries of eqviations, fuch, that all 

 the .terms of each feries may be fimilar to one another, and to 

 the equations which we have already aflumed, (Art. 3. and 4.) 

 or fo that 



gi" — I— ■/!—/-" ' ' ^^ g w _ fin if CnV/ 



l + y'i— «*»' ^i+«""+2«"'coi'i>,*»"~ (i+e'") v'i— '"^fin 4*'''"* 



&c. &c. 



each feries being fuppofed to proceed, according to the fame 

 law, as far as we pleafe ; then, by due confideration of the 

 quantities fin 8(p"', fin i6<p'^, S^c it appears that when (p" =. o, all 

 the remaining arches (p'", <p", &c. are alfo each = o, and while 



^^ increafes from o to -, the remaining arches alfo increafe 



from o to - ; fo that, upon the whole, while (p, the firfl: arch, 



increafes from o to a quadrant, each of the remaining arches- 

 ¥■> ^'1 <?'"> &c. will alfo increafe from o to a quadrant. 

 6. Let us next fubftitutc for the fluents as before 



then, following the fame analogy as in Articles 3 and 4. we 

 have 



