204 ' PHILOSOPHICAL TRANSACTIONS. [aNNO l6g7 . 



and the parabolic arc ac = v. Then let the general equation expressing the 

 natures of an infinity of irrational curves, be z = rvtj", in which r denotes a 

 given determinate quantity, and n the indefinite exponent of the indeterminate 



quantity y. I say the area abd = ^^ — f]v -\- V lay + yy into 



r n + I TO „ raa x In + I n— 1 OA x 2n — I- n-3 ^^ 



« + 2 X ra + l'^' ~ « + 2 X n + ll'^ n X k + 2 x ^~+l*'-^ «— 1 -^ "" 



"° ^ "<>~ i/" ~ — ^'^- Where it is to be observed, 1st. That the great letters, 



Aj e, c, &c. denote the coefficients of the respective preceding terms. 2. That 

 if 7! be a positive integer, or equal to nothing, or if 2/< be an odd number, then 

 the quadrature may be exhibited by a finite number of terms, by the series in 

 these cases breaking off. 3. That + 5 is equal to the last term breaking off. 

 4. That, of the terms multiplying the quantity '^ 'lay + yy, the last term 

 breaking off is to be doubled. 5. That all those figures in which n is a positive 

 integer and odd number, or more generally, all the figures in which the last 

 term breaking off has the affirmative sign +, have a portion geometrically 

 quadrable, and v/hich is easily assignable by the series, by taking the absciss as 

 in note 4 of the preceding series. 



Example 1. — Let z = v. Because in this case r = 1, n =■ O, therefore the 



term last breaking off is — ., ?/", whence + 5 = ^a by note 3. 



And because in this case — \a is the last term breaking off, therefore — a is 

 the last term to be multiplied into v' q,ai) + j/j/j by note 4, and therefore abd 

 •=■ vy -\- ^ av ■\- V 'lay -V yy Y. — ^y — a. 



Example 1. — Let z =. ~. Because in this case r = -, n =. ], therefore 



• a a 



the last term breaking off is — '"" ^ ~" _ =^^,ti" ~ = ■ a. Hence q = -la, 



and so ^a, is the last term to be multiplied into V lay + yy. Therefore abd = 



1 - ? + V'^^l^ X - £ - ^ + I- And if we take y = 

 1/ \aa, the area belonging to this absciss will be quadrable, and = 

 tIj. V v'2a'' + ia' X 5a — ^/ \aa. 



The following Part is added from N" 235, p. 785. 



Thirdly, let acp, fig. 1, be a semicircle, ade a curve geometrically irrational, 

 of which the ordinate bd cuts the semicircle in c. Let the quantities be de^ 

 noted as before, viz. the diameter ap = la, the absciss ab = y, the arc ac = v. 



