m n a + +&c.} -n 2n {iminop) &c.} an ma + xan (15.) Development of Sq.com- } (a + bx^)by series. 62 p(p-1).mn + m +- 2n p(p-1)(p-2) 2.3n + + &c. m + 3n a? p(p-1)x - 2n + Im+np b'm+n(p-1) 1.2.62' m+n(p-2) 20XP+1 (m, n) ax (m+npb (m+np) (m+n.p-1)62 (m,n)(m - 2n)a^x – 3n + . &c. (m + np)(m + n.p-1)(m + n.p-2)68 2 XP+1 (m+n.p +1) 6 m(m + n)a &c. m+n(p+1) 2011 XP+1 + (m + 0.0 +1)(m + m.p+2) X2 + &c. (P + 1)(P + 2)(P + 3) noa? 1 OM XP pna X-1 X-?+ &c. p(p-1) n26 &c. m(m+n) x m(m + n)(m + 2n) 1 X oom n -XP+1 (p+1) nb (p+1)(p+2) n°62.00 (m, n)(m — 2n) X2 + &c. (P + 1)(P+2)(P + 3) n°63 (Tr. L. 175-80; Hirsch, Int. Tab.) &c.} + &c.) m n {c+1 EE b + cy ez=0; becomes loce+ezyje +exy?)?? or R(x) (16.) Reduction of (a + a,& + a, x2 + azw3 + aq&") } Assume a= then the denominator, which may be 1+ y represented by X, becomes (e +e, y +egy + ezys +exy')} (1 + y) in which e, e,, &c. are functions of a, a,, &c. b, and c. Let the values of b and c be determined from the equations en=0, Rw) R(y) R(y) then X Y this again may be separated into R,(y) y.R,(yo) Y R(x) becomes ' which may be integrated by preceding methods. R,(yo) Y Soor/ If R, (y) is a fraction, it may be decomposed, and each separate fraction may be integrated, or reduced to the form fua (y® + a) Y R(Q) may be reduced to the form X by multiplying the numerator and denominator of the fraction by the denominator. Sizle+exx +exo)] у? WY: X ume R(x) S. * (a + a, iv* + a,&* + agx6); may be separated into two parts R(2) #Spe by separating R(x) into R2(x*), and x. R,(x”), and putting wʻ=y. R(x) 8.R(0%) (a.+ ay.x2 + &*)&=Xy; to reduce the second, assume (a + a,m? + 2qw*)&=x; then by substituting for æ its value, we obtain respectively yo.Rz(y) z?.R_(x4) and Sota-4a2, + 4 ay)} (a' — 4 a f_R(w).a + a, v2 + a, x** and SR(a).a + aqx++ a&^%** may be transformed in the same manner as the preceding formula. To reduce SR(x).a + a,x + aqx* + a 3x37** assume a + ax + a? + az 203 aš + xz, or=aztw + %, by either of which assumptions, the term dzw will disappear from the radical. R(x) , separate 126 + ax8)* R(x) into R, (x) and X.R2(x), and put y=x?: the part corresponding to R2(x) may be further reduced by assuming 1 y + Y (L. C. D. 406—11.) Rw) b.b2 + R(2) (17.) Integration of (a + a, x2 + a264) =b.6,(1 +e2.x)(1 +eg.x?). R(Y) Sad becomes Sy *(a + a,x2 + +)? ei y 1+ and since ( 1 may beexpanded by the binomial theorem in a series which converges rapidly, if oz is small, the required ei R(y) (1 )+ [2] If ei and e, are nearly equal, put the given function under the form R(2) en ez 1+ e, integral will depend on So T– ay“).1 + y 1 1 = p? - k, and ea = pi + k, then by substituting 2 2 these values we obtain /: {(+202)2 – k* } R(20) 1 1.3 = S.R(w). + 1 + + &c R(2) a b + 0,20914 then b, b, +0202 R,(yo) 2 2 where p*yand q*y* are either both positive or both negative, and p=(+0,62)} + { + (0,62 – b,c,)}t, q=(+0,6)}-{+Cb– b,c,)}}, or q={ +(czbą – b,c,)}}-(+0,ba); + or - according as C, and b, have the same or different signs: and the second value of q must be taken, if c,b, and b,c, have different signs. The same transformation may be repeated by assuming yQ Pi=p+ (p? -99), 1+q*y 9.=p- (p? – 9')}; 9, ૨, Y, , Pe=P. + (p,' – 9,3)}, 92=P2-(P-9)}; 1+9iyi &c. &c. where Qm represents {(1 P.y)(1+ny)}: or the values of P1, 91, &c. may be thus calculated ; let P + 9 =?, P-9 =s, P. +9=19 P1-91=812 &c. &c. then r=r+s, and s, =2(rs). &c. by this process we obtain a series of the form S,R(9°) +S4R(99) + &c. R(Y) (10 in which, since pq continually increases, Px - 9n may be made as large as we please. Similarly by assuming &c. 9-1=(pq), &c. &c. + or |