VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 205 



the ordinate bd = z ; and let the equation expressing the natures of the curves 

 be z = rv^y^, in which r denotes any given determinate quantity, and n the in- 

 definite exponent of the indeterminate quantity y. I say the area is abd = 



^ qv + V ^2ay - yy mto ==^^y + — — =_^ + 



„+l - H- -r - ■ -^ yy ,im\'^ ' nx^T+ii 



2n 



OA X 2w — 1 n + 2 , OB X 2« — 3 n — 3 , flc X 2« — 5 n - 4 <, Iraa n + 1 



n-l y + n-1 y + n-^ V &«. - ===.,,/ 



2ra3 X 2n + 1 n a^A x 2/1 — 1 n - 1 a^B x 2» — 3 n - 1 „ „^ i • i ^i 

 , ?/ ^-^^=;^ — V ,, — y &c. Oi which theorem 



it is to be noted, 1st. That it consists of two infinite series ; of which the first, 

 connected by the sign +, is multiplied into v ^ 2ay — yy : but the terms of 

 the latter, affected by the sign — , are absolute. 2. That, in the former, the 

 capitals, a, b, c, &c. denote the coefficients of the terms which precede them ; 

 also in the latter the same values obtain as in the former. 3. That the quadra- 

 ture is exhibited by a finite quantity, when n is a positive integral number, or 

 equal to nothing, or when In is an odd number ; for in these cases both the 

 series break off. 4. That 2q is equal to the last term breaking off in the for- 

 mer series. 



Example 1 . — Let z = -. Because m this case w = O, r = - : therefore it will 



be the area abd ■= v^ ■{■ 1v ^ lay — yy — lay. 



Carol. — The whole figure afe is equal to a double square, the side of which 

 is Acp, taking away the square of the diameter. 



Example 1. — Let z = ^. Because in this case ra = 1, r = — ; there- 



fore the area will be abd = •^ — -^v^ -{- v v lay — yy Y. ~ -\- \ — 

 ^y'- — ^ay. 



Example 3. — Let z = •^. Because in this case n ■= 1, r = -, the 



area 



will be abd = y^l ,- iv^ + V ^^2ay - yy X ^£ + 'I + I 



2y^ 5j/^ 5ay 



Wa ~ Ts ~ ~' 



On a large Piece of Ambergris thrown on the Shore at Jamaica. By Mr. Rob. 



Tredwey. N°232, p. 711. 



Two years ago, a man found ISOlb. weight, dashed on the shore, at a place 



