VOL. LXIX.] PHILOSOPHICAL TRANSACTIOXS. 485 



theorems, by writing for x, in the correcting quantity, successively its values ir, 

 f, 0-, T, &c. 



2. For the correcting quantity sought may be assumed the quantity 



x' — =t' .X' — fi' .x' - v' . x' - ^' . &c. . .1' . .i^ — e' .^ — a-' . .!•' ~ t' . &c. 



_^ , \y nr v 



„' — a' . ■X' — /i' . z' — y' .3' - i'' .&c. .i'' . ■:' — 5' . a' — o-' . a' — t' .&c. 



X' — a' .x' — /^' . x' —y' .x' —h. Sica'' . x' —■i' .x' — o' .x' — ■>' .&c. 



' ^- — «.' . i' — li' . i' — y' . ^' - S' . &C. .f . J' _ t' . 5' - <r' . f' - I' .&C. "^ '^' "^ 



3. In general, let z be any quantity which is = O, when x becomes either a, 



j3, y, S, i, &c.: let z become successively a, b, c, d, &c. when x becomes tt, ^, 



a-, T, &c. respectively. When:?' is either = j, o-, t, &c. let n = O; but \( x = tt, 



let n = />■• in the same manner when x is either = tt, o-, t, &c. let P = O; but 



when a; = f let P = r: and similarly, let 2 = O when x is either tt, f , t, &c. ; but 



when X = (T let 1 = s: and likewise, when x is either tt, ^, <t, &c. let t = O; 



but when a: = t let T = <: &c. then for the correcting quantity sought may be 



. z n ,zP ,z£ ,zT .' 



assumed -.-.T'^ + -.-.T + - . - . t^ + - . _ .t^ + &c. 



Theorem. Assume n quantities a, |3, y, S, i, &c. then will the sum of all the 

 n quantities of the following kind 



-I ^! L 



a, — fi . a. — y.« — i\a — i . &C. ,8 — x.B — y ./i — i'.ii — s . &c. v — «.y — /3.7 — iJ\ y - f . &c. 



+ &c. = 0, if m be any whole number less than n — I ; but if ?« = n — j 

 then will the above-mentioned sum = 1. In general, the sum of the n terms 



«" (fiv^Scc. + /3y« &c._+ M + y.5'. &c. -|-_&c.) /S" (ayJ'&c . + «y^ &c . + ^i &c. + yJ'i &:c. + &c.) 



X 



— /i.x — y .» — S'.ec — i .kc. ' /3 — a./3 — y . /3 — ^.;3 — 6 . &c. 



+ &c. = O, if m be less than n, and m + r not equal to « — l ; where r is equal 

 to the number of letters contained in each of the contents above-mentioned 

 QyS &c. (3y£ &c. (iSi &c. ySt &c. &c. &c. respectively: but if to -j- r = w — ] 

 then will the above-mentioned sum = + 1 ; it will be + l if / be an even 

 number, otherwise — 1. 



Fill. On the Periodic Time of the Co/net of the Year ] 770. Bi/ J. A. Lexell, 

 of the Petersburg JrMclemjj of Sciences, p. 68. From the Latin. 

 Mr. Lexell having, from the observations of this comet, computed its ele- 

 ments, and particularly its periodic time, which he states at 5^ years, or nearer 

 3 years and 7 months; he proposes here, reversely, to show that these elements 

 agree very well with the best observations that have been made of it; but the 

 contrary if those elements be much varied. These elements, as Mr. L. has 

 deduced them, are the following: viz. 1. The longitude of the ascending node 

 4' 12° O'; 2, the inclination of its orbit to the ecliptic 1° 33' 40"; 3, the elonga- 

 tion of the descending node from the perihelion 44° 17' 4"; and therefore the 

 longitude of the perihelion 1 1' 26° 16' 26"; 4, time of passing the perihelion in 

 the year 1770, Aug. IS'* 15*> 5"" nearly, or Aug. 13.5450; 3, the comet's dis- 



