VOL. XL-] PHILOSOPHICAL TRANSACTIONS. J 83 



factors as with one single one (n) will make up the index 2a -{■ I, which is an 

 odd number. 



If the common difference a be an unit, it is omitted: 

 Thus, ji^" is = n. 7J — ].n — 2.n — 3. n — 4.n — 5, containing six factors. 

 So of is = 6. 5. 4. 3. 2. 1 , and the like for others. 



If the common difference a be nothing, then the hook is omitted, and it 



. 



becomes the same with the geometrical power: so — i:^ ('" is = 71 _|_ I" according 



to the common notation. 



Proposition 1. — ^n arch /ess than a semicircle being given, with a point in 

 the diameter passing through one of its extremities; to Jind by means of the sine 

 of a given part of the arch less than one half, the area of the sector subtended 

 by the given arch, and comprehended in the angle made at the given point. — Let 

 PNA, fig. 7. p'- 6, be a semicircle described on the centre c, and diameter ap, 

 and let pn be the given arch less than a semicircle, and s the given point in the 

 diameter ap, passing through one extremity of the arch np in p. Then taking 

 any number n greater than 1, let pk be an arch in proportion to the given arch 

 PN, as unity to the number n; and let it be required to find, by means of the 

 sine of the arch pk, the area of the sector nsp, subtended by the given arch 

 NP, and comprehended in the angle nsp made at the given point s. 



From N and k let fall, on the diameter ap, the perpendiculars nm and kl, 

 and join cn and ck. Then let t stand for cp, the semidiameter of the circle; 

 yfor cs, the distance of the given point s from the centre; p for sp, the dis- 

 tance of it from the extremity of the arch, through which the diameter ap 

 passes; and y for kl, the sine of the arch kp in the given circle. 



These substitutions being presupposed, the problem is to be divided into two 

 cases; one when sp is less, and the other when it is greater than the semidia- 

 meter CP. 



Case 1 . — If SP be less than cp, then take an area h equal to the sum of the 



rectangles expressed by the several terms of the following series continued ad 



libitum : 



2 2 2 



^l' __-^|4 '^ '» 



PS _I_ < + "+ ''x/ ^ f , 9<-n + 3lx/ ^ y' , 9_X_25M- n + 5lxf ^ ^ , - 

 1 3j3 t 55 t J.7 t^ 



And the area -^n X h will determine the area of the sector nsp ad libitum. 



For the sector psn, being the excess of the sector ncp above the triangle 

 ncs, will be the difference of two rectangles : -J-cp X pn — 4-cs X nm; but pn 

 is the multiple of the arch pk, namely n X pk ; and nm is the sine of that 



