Astronomical and Nautical Collections. 353 



before. The integral, h n, n, is obtained by the well known 

 rule. " Let the increment be reduced to the products of arithme- 

 tical progressionals, whose common difference is the quantity by 

 which the variable magnitude is increased at every step, and the 

 integral of each increment will be found by multiplying it by the 

 preceding term in the progression, and dividing it by the number 

 of terms thus increased, and by the common difference." Wood's 

 Algebra, N. 429.] 



From the third term we find C, := 7)iC + riB. I suppose 



C:^-^5,andC, = ^5„thatis ^•''"'" B ^ !^B + n^B, 

 R R, R,n R ' 



Qmn, , mn,, /-> m^Q , , ,,, , 



or —u- + — -L. Q — — TL + n . We may now make 



nR nR, • R •' 



JOu = _, that IS — Ji-L^ — — '-, in order that . " Q mav 

 nR, R R, R nR, • 



be = 71 . J3ut (LJ- is a new value of ; .which is there- 

 in, R 



fore constant, and we may make R :=: nin,n, [and R, =z mnji,, ], 

 whence Q = n,nn\ and Q — ^ niiriri\ Consequently C == . 



B, and^ C, s: ^^ C. 

 n" 



By the fourth term we have D, =: vi''D-\-n:"' C. Hence in the same 

 manner we find D = _ — t — C, and D, — . "' ^" D. From these 



terms the manner of forming all the rest is easily inferred : and if we 

 denote each term with its sign by the letters A, B, C. . . ., we 



shall have s =: 1.3.5 ... (2n - 1) ^^ + -J^ ^ A +'— 



ar""+' 2tn zz 4m 



^B + ... = 1.3.5...(2«-1)£:^ + (!L±i>^^ + 



(w -" 1) (ra — 2) XX ^ {n - 3) {n — 4) xx ^ 

 4(2n - 3) zz 6(2)1 - 5) zz 



