tJjie InJimUjimal Calculus, ^t 



lily ^^f which may render itfo. Tliis quantity A>" can only 

 be infinitely fmallj but I foon find that it is abfolutely no- 

 thing; becaiife the othtT terms of the equation are free from 

 infinitely fmall quantities. For, by bringing all the terms of 

 the equation to one fide, 



this equation, (tP X ~^) + ^" = <^> 



can have no place in the Method of Deteraii nates, unlcfs 

 each particular term be equal to o; conlb^uently $'' = o, 



and TP = — X ~—, as before. 



aa a — x' 



62. It appears, in general, from what has been faid, thai, 

 if we put P for the fubtangcnt of any curiae whatever, we 



ihall have the impcrfc(3: equation P — y —j- ; and confe- 



quently,(by article 34) the equation P = y X Ihn. \-jAi will 



be rigoroufiy accurate. 



If we put Q for the angle included between the tangent of 

 the curve, in any point whatever, and the correfponding ordi- 

 nate, w€ fliall evidently have 



P y 



the Tangent of Q-==. *, and xhit Cotangent ofQ = -jj- ; 



hence we have the imperfe6t equations, 



Tang, Q = -^, SiVid CoLQ =: -^, 



<jr the rigoroufiy accurate equations. 



Tang, Q = lim. (-^) and Cot. Q =^ I'm. (-^). 



63. Probl£7n IL Required the value, which niufl be at- 

 tributed to Xj in order that it's function \^2ax — xx may 

 be a maximum^ that is, greater than if any other valu© 

 whatever were attributed to x» 



Make ^%ax — xx = y, that \^ yy — 2^.r— xv, and con- 

 ilrudl a curve, whofe abfcifle is x^ and its ordinate _;-; and the 

 problem will then he. To find the grcatell ordinate of that 

 pufve. Now, fince from the point M the ordinates dccreafp 



* The i:adius is here CQpfidercd as unity ; and therefore, 

 Tang, ®^: 1 : : P : >/, &c.— VV. D. 



G z both 



