Astronomical and Nautical Collections. 351 



surface of the earth, where y :^ d, the fluxion proposed becomes 



yzz 



; and at an infinite distance, where the 



(2c + 2cd)^{tt-zz) 



fluxion itself vanishes, it does not differ one thousandth part from 



this value : so that neglecting this slight inaccuracy, we may safely 



take for the fluxion 77; — , ,-, ,>, — j}— r, and omitting the 



(2c + 2ca) s/{tt—zz) ° 



constant coeflScient^ — „ , , we may proceed to take the fluent 



yzz 

 of .. — : by means of the 11th proposition [that is, The fluent 



ofrsis = r s — rs + rs — ... = r s — r s +...; the accents 

 denoting the fluent of the quantity marked, when combined with a 

 constant fluxion w, considered as unity.] 



— zz 

 For V (^'— ~") write x ; then x = — 7— , and the proposed fluxion 



yzz . yz , . T. 

 will be — -7- = — ya-, y being also = — : consequently in Pro- 

 position 11, we have r = yz, s = -7, w =: z = . Then, 



taking the fluxions, and dividing them continually by w, we have 



• _ ^ 1 ^ li ■■ ^ ^^^ "^^ _ ^^^^ •■■ 15z* 



* - a;3 + X — x^'^ ~ "T^ + j^ ~ j;5 '*"•" x'^ 



+ + — = + — . . ., and so forth, s [or I 



x'^ jc^ x'' x^ diw" J 



-,l4-l „n— 1 »)l-3 



being =: A + B Z + C + . . . 



j.2n+. ^«-l ^^-^n-s 



or = <* A - — — + B + c _ — + . . . 



For the investigation of the coeflicient A, B, C . . denoting the 

 preceding values of n by n, ?*", 7i'\ and the succeeding values by 

 72,, n„, n,,,, and taking the fluxion of the series, first with respect 

 to X, and then with respect to z, and dividing the results conti- 

 nually by w, we shall have 



