8 de cosx=(-1)" sin &. sin cos 2. sin cosé x ...sin cos” – 10. d, log"d=x-1.(log,x)-1. (log? x) -1... (log."–1x) -1. 6 DEVELOPMENT OF FUNCTIONS. (8.) d. f(x + y)=dy f(x+y). Taylor's Theorem. Let Ex(u) represent what u becomes when + Dx is substituted for a, then Dx (D x)? (Dx)3 Ex(u)=u+duu + du +d; u + &c. 1.2 1.2.3 error If the n first terms of this series be taken, the limits of (DX)" are the greatest and least values of d"u 1.2.3...n provided there be no value of x between 8, and + Dw that renders u or any of its differential coefficients infinite. (L. C. D. 169.) Maclaurin's Theorem. Let (u)x=0, dou represent the values of u, d" u, respectively, when x=0, then ? u=(u).-0 + dou. + du. + decou. 1.2 1.2.3 + &c. 1 E o de cos” x=(-1)"P, sin cosm–10. d." du log;"x=Pm (log.m – 1w)-1. • Equ=Sm(Dx)m–1.d,m–1:u. Let Exu - U=DzU, then (−1)n-m+1 +S,Deu1+2.5 2n-m+1. d,u where dz*+1:0. n DZU Dx= m n-m+1 nem ag - 1 je u=S,,20–1.dm-:u. 1 1 + В + 1.2.3 (9.) Taylor's theorem applied to two variables. Ex,(u)=u+{d, u.Dx+d, u.Dy} + {d?u(DX)* + 2d_d,u. Dæ.Dy + d. u (Dy)"} 1.2.3{d}u (Dx)3 + 3 dd,u. (Dx)* Dy + 3d,du Dæ.(Dy)? + d; u(Dy)} + &c. dm dou=d”d u. (L. C. D. 25—30.) (10). Laplace's Theorem. Let y=y{x+#.$(y)}, in which x is independent of w and y, then f()(*).d.f4} .d,{py().d.f4(x)} 203 .d{p}().d.ft (x)} + &c. Lagrange's Theorem. Let y=x+w.p(y), then X2 f(y)=f(x) + {p(x).d.f(x)} + d.{P(x)]*.d.f()} 2013 d? {P(x)].d.f(x)} + &c. V{x+x.0(y)}=x+8.0(y). If a third variable z be supposed to receive an increment, then E.,(u)=S»,5,8,(Da)m--. (Dy)**. (Dz)-2. d. *=:2, 97:ds=1:u. Maclaurin's theorem applied to a function of two variables. U=S,2m in-1.dmo:dd:u, = S, S,2m •Y” – 1. -1.dm-":dn:u. (Jarrett, Camb. Phil. Trans. Vol. III, Pt. 1.) $ f(y)=f(ö) +S dm-!{p4(x)}".d.f4(x)}. 1.2 + 1.2.3 m-1. S,ya m x=0 m 8 If f(y)=y, then 2013 y=%+ 1.0(ə) + .d.0(:)]° + .dp()+ &c. (L. C. D. 107–9; Tr. L. Note E; L. D. C. 49–61.) 1.2 1.2.3 IMPLICIT FUNCTIONS. (11.) Let d,u=M, d,u=N, then M +Ndzy=0, from which equation the value of d y may be found. By successive differentiation, the following equations may be obtained, in which u, un, &c. are functions of w and y: Un + ugd y + uz(dzy) + Nd;y=0, U4 + Uzdzy +48(dzy)” + Uy (dzy)* + (Ug + ugdzy) d’y + Ndiy=0, &c. &c. from which the values of d; y, dzy, &c. may be obtained. (L. C. D. 41-8; L. D. C. 1003.) TRANSFORMATION OF THE INDEPENDENT VARIABLE. dy (12.) 0,X = d; a = dx. (dzy)3 3(d;y)? — dxy.dlly da &c. = &c. (dzy)" If x and y are considered as functions of a third variable t, then d, y dey= dex'
(da) 3 diy= (d, *.dog – 3d,.dog.dog + 3 (d? a)”.d,g-d, .doc. (dx) &c. = '&c. (L. C. D. 57---69.) d?y=dw.d?y–dịw.dry (13.) Elimination of an arbitrary function. Let z= f(x,y)=P(t) suppose, then d22 d. f(x,y) d, d,f(x,y) In the same manner, by repeating the differentiation of any given equation, two or more arbitrary functions may be eliminated. (L. C. D. 77–9.) PARTICULAR VALUES OF THE VARIABLE. (14.) Let u=0, when x=a, then u must be of the form v (2 — a)", deu=v,(w - a)-1 dru=vn (x – a)"-", If m=n, then du. u=vn, which is the first differential coefficient that does not vanish, when w=a. If m is not an integer, then dr+lu and all succeeding coefficients become infinite, when x=a. U= v. (x - a)" ve (xa) Let n, r be the greatest integers > m, p respectively, then when x=a, if n>r, u=0; n<r, u, and all its differential coefficients are infinite; U = m<p, = . (L. C. D. 143–53.) $(w) If ur when x=a, the value of u may be thus P(x) found: substitute a +Dx for x in f(x) and $(w), and expand in series of ascending powers of Dx, then a (DX)". + a,(Dx)", + &c. a,(Dx)", -". + a,(DX)",=+&c. 6, +b (Dx)"-", + &c. = 0, when Dx=0. a, +a,(DX)", -m, + &c. If m, <n,u= b(D x)"; – m. + b (D x)", – m. + &c. when Dx=0. 1 U= If u=f(x).f.(x)=0.0, when x=a, put p(x)= fi(x) f(x) then when x=a. when x=a, then before: {f(x)} -1 under which form the value of u may be determined as above. (L. C. D. 12; L. D. C. 107.) Сс UE as |