ALGEBRAIC FORMULA. 



*75 



F= (i-f-^P", 



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

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

 get 



?= (i+ < ^.( 1 +^)p j 



5. Let us now affume two feries of equations, 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 



J» = L=V^=2T fin g ,„ _ A" W fia^ 



&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> IV , &c. it appears that when <ff — o, all 

 the remaining arches <?>'", <p lv , fee. are alfo each = o, and while 

 f increafes from o to ~, the remaining arches alfo increafe 

 from o to - ; fo that, upon the whole, while <p, the firft arch, 

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

 ?', <p\ <p'", &c. will alfo increafe from o to a quadrant. 



6. Let us next fubftitute for the fluents as before 



P""= f *'" &c N'" - r t fin '*»'" ' & 



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

 have 



