VOL, XXIII.] PHILOSOPHICAL TRANSACTIONS. 2f 



m —2c.e + I + r.e + l.ba + m— e .2c + 1 + s.2e + 1 .fc 



ma .3c + 1 + na .3e + 1 



m - 2 e.2c+ I + r . 2e + l.bc + Tn'-e,3c+l+s.5e+ 1 ./p 



ma.'ic + 1 + na.4>e+ 1 



&C. 



Concerning this series, the progression of which almost appears by inspec- 

 tion, it is to be observed, 1. That those figures are quadrible, which are defined 

 by the foregoing equation, wheo the exponents m, n, e, c, and the co-efficients 



rt, hi /, have the following relations, viz. when — ^^ , ^^ — is an affirmative 



integer number, which we may call /; and (/ being greater than 2) when the 

 relation of the co-efficients is as follows : viz. 



+^2c^/c_- c -t- 1 + r . fe^ e ±i X — = 

 e — m . Ic + n— s . Ic + I J 



m — 2c . fc - 2c + 1 + ^ « fe - 2e + 1 ^ *p , »w- e.fc — c+ 1 H-r./e — e+ 1 yV 

 m . k + 1 +n.fe+l "• »i.fc+l+». /e + 1 « 



Where v and p denote the co-efficients of two terms, which immediately pre- 

 cede the last term of the area required ; viz. v the co-efficient of the term next 

 to the last, and p the co-efficient of the second term before the last ; as, if 

 Fz^* + 1 i^5c + 1 be the last term of the required area, then v denotes e, and p 

 denotes d. 2. That the last term of the required area is known from the 

 value of the number /; for here also / -|- 1 gives the number of terms in the 

 series, taken from the beginning, which constitute the area required. 3. If 

 / = 1, then the relation of the co-efficients must be this, viz. 



ge — w . 1 — A 4- ^A b e — m.l— A-H*A / 



X /i — X "". 



e— wi.c+1 — 4.C+1 J »i.c-fl-f».c-|-l <» 



But if / = 2, then the relation must be 



w— 2c.c4-l+r.e-Hl ^ *b ^_ 2e -- in . 1 — a + r A . wt — e c+1+j.g-fl fn 

 e— m. 2c + 1 - *.2eH- I / »i. 2c + 1 + n,2e+ 1 m . 2c + 1 -f n . 2c + 1 a* 



Sect. III. Let z*" = ay« + bz'^y^ + « -^-fz^^f^ + « + gz^^y^' + " + &c. be the 

 equation expressing the relation between the ordinate z and the absciss w, 

 and consisting of as many terms as you please ; then the area will be 

 Kzy + Bz" + 'z/*^ + ' 4- cz^« + 'Z*^ + • + P2^* + • /c + « _^ gj-C. 



Which I believe is no contemptible theorem. The co-efficients a, b, c, d, &c. 

 are found by a very easy calculation, as also the conditions of quadribility, and 

 how many terms of the series the area requires. Now the number of these 

 conditions increases with the number of the terms in the equation defining the 

 relation between z and y : and particularly if that number of terms be called 

 N, then N -r- 2 will be the number of the conditions of quadribility ; one of 



■ ■ ' 'k) -isd ^2 ■ ' ■ ' 



