X - ^'ilzi

uo) 17^ "'^ n^ '"

FORMULES

eHo

eHs

^%(nv)= JJ^a/p + ^-i-^

dJL, V)

civ)

oAo^^r-^^

'/; = ri, Tî, .... /-„!

Tz tu>, „ mil

*(a) = V A„^A e2«mi' = V A„e"/'"> " "^ w, "^

r *(« + 2Wi) = <ï>(m),

Au/,, {'t + tù,)

(3) { *,.(«) = \ ^ ^ e2j7ti«/i+/-) = ^/i e2r»'TO2r3(Ap H- r-c | hz),

(4) <Ï>o(m) = ^3(2(^|2t), *,(M) = 2r2(2Pl2T).

I 3r|(p) = A-'2rf(p) + /:2ri(p),

2r|(o)&2(^'-4-c)2r2(^-c) =-&|(c)2rf(p) +âl(c)3r|(,;), &l(o)2r,(p + c)2r4(P-c) = 2r|(c)2r|((^) +2r|(c)^|(t^),

= + 2r3(c)2r4(c)2ri(0 2r2(^') + ^i(0^2(c)2r3((;)2r4(p),

2r3(o)Sr4(o)2r3(<; + c)2r,(ç.-c) = -2r,(c)&2(c)2ri((>)2r2(p) + 2r3(c)&,(c)2r3(t^)&4(i'),

2r2(o)&4(o)2ri(p + c)2r3(P-c)

= -2ri(c)Sr,(c)2ri(p)2r4(p)4-&2(c)&3(c)S2(03^3(0-

FORMULES. "279

/ Tp=&p(a + 6)2rp('a — è)2rp(c + c?)&p(c — ^),

(i) I Tp = &p(a + c)2rp(a-c)&p(c?+6)Srp(rf-6),

( Tp = &p(a + d) &p(a — d) 2rp(6 + c) 2rp(6 — c),

(2) Ti + T; + T'i = o,

STsCa^I aT) 2-3(7 ip-u) /• = a+|3— 1 = 2 ^^'■'^"■'''^2r3[a?+7 + rpT|(a-|-P)T]

(i) { x3r3[a7— Pa^-+-/-apT|aP(a + P)T]

X 2^3 [ pa7 — ajK + ra^T I ap (a + P)x],

^ii^) ^2(7) = ^2(37 +J I 2t) 2r3(a7 - JK I 2x)

+ 3r3(a:- -+-7 I 2T)2r2(a^— jK I 2t).

2Bo CALCUL DIFFÉRENTIEL.

•iriU

a II

çoa" —

, . . I ^ ^^0 ?pa c t t £

(2) Uo=r-' ÇpT= F" == ï — = çpoÇoy— ç^aÇay,

çoa Çyo Çy»

(3) pa = ea+ Vào= ep+ ^^|o = ^Y+ ^yo'

(4) 3p«i = ^o-f-^,lo+^YO'

ey— «a «a— ep <?3— «y

(5) pa = ea+ -— -T- = ^P+ ^^17^ = "T+ TUTT'

1 " ?ya ' ?a,3 ' -=;iy

(6) ioy

ep— gj/ «a — ^^

^aowp = /ep— ea,

(0 < s/ea— es /«p J Ipyco -

/ea — «y s/e^—e^

(2) A- = ^2lW3, ^■'=?23«Jl;

|ao(" + 2wp) =— ^ao«^ ^py(M + 2wp)=— ^pyJ^

$ao(" + Wa) =— s/ea— ep /«a— Syloa" = — v/ep — «a y/^y — ^a ?oa ">

V«a— «y V^y— ea

FORMULES

(i) $âo(«) = — ^po(")^r«(")'

(2) $;«(«)= ?pa(«)$ra(")'

(3) $py(«)= — (ep— ey)?OY(w)$ay(«).

(0 raV«)=[«a-ep-l-?lo(")][^a-e-r-+-aio(«)]>

(2) ^o?c(«) = [' + («a- ep)^o'a(«)] [1 + («a- e^^)kU{u)],

(3) ?^'y (iO = [i - ^.ly («)] N- ^a-+- (^a- «y) ^|y(">]-

(ep-.a)(e,-..)^ ^^^

(3) (ep— ea)(ey— ea)^la(") = — ^> — ^a,

(4) (e^— ea)%(«0 = — CfJ« — «^a,

(5) epo(«)aya(«) = ^,^) = ^"-^«"'

(6) (ey — ep)^a3(")loy(")= ?3" — î^y",

(7) 2r'i(p)3r^+i(p)-^,(p)2?'^+.(p) = 7r2r2^i(o)Srp+i(p)2ry^i(p),

(8) yy^,(i')^^^i{^)-'^t+r{v)y^+t{^) = T.?;l^i(o)^i(i')?J<,Mr

^oy ^ ^g- g ^^v g + loy g ^gv « ^py M

" I— (ey— ea)(ey— ep)|o\"lor«

^pyt^^gyal^yg — ^oyCT^gyM^gy^

oy"

^^yg^pvCT — (ep— ey)$oyM^gvagoya^aya

I — («y— ea)(ey— ep)||y«ÇoY«

?,'yW + (eg-ep)p^

^Py(M+ a)$py(M — a)= -

i + (ea— ey).-r— ^3y«

sày '^

3[|oy(" + «)-H?OyC" — «)] = ^4oy"|gy«$py«, 3 [ ^oy (m + a) — ^oy (« — «)] = 2 ^oy « lay «* ^jîy ", 5l[^Py(M +-a) + ^py(«— «)] = a^pyw^pyflt,

5V[|py(a + a) — |py(M — a)] = — 2(e^ — gy) ^oy« ?ay "^oy« lay«r 2, = 1— (ey— ea)(ey— ep)?o\«?oy«-

Wl, W3 I,

^ao(«| >^Wi,>^W3) = jlao f y ^1,0)3),

COi, W3

FORMULES

cn(M,A)= ^13

\/ei— ^3

dn(i<, A')= ^,3

\/ei-

Wl, W3 ,

wi, W3 ,

sn VM v'ei — 63, A.; = , ;

V^pM — es

/ / j\ y/p ?^ — ei

cn(^a V^i— es, A-; = =:

y/pti — 63

P" = es

ei— e3

sn2(M y/ei — es)

sn'a = en a dnw, en' M = — sn a dn M, dn'M = — A^snw cnw.

sn^if -\- cn-u = I, dn^M-i- A-2 sn^M = I.

(i) sn'^u = (\ — sn^u)(i — k^sn^-u),

(2) cw'^u = (i — cn2it)(' — A-2+ A:2cn2M),

(3) dn'2a = (i — dn2M)(dn2M — n- A-2).

LXXT.

(i) a>i \/ei — es = K,

(2) ws/ei— 63 = iK',

(4) tK' = ^2r|(oiT),

^'\iK

(7) en M

(8) dau=^^k'^j^^

Fig. I.

z = sn (m, k).

Fig. 2.

5 = cn(a, A').

Fig. 3.

(Jn(M, W).

Fig. 4-

s = P(m! w,, Ws).

sn(

— u. A) = — sn(M, A-),

cn(

— u, k) = en(M. k),

dn(

— u, k) = dn(M, k),

(« -+- 2K) = — snw,

(a •+• aK) = — en w,

(m -1- aK) = dnii,

'm + 2tK') = sn M,

{u -f- 2tK'; = — en M,

[u + liVJ) = — dnM,

(m + aK + 2tK') =: —

sn a,

(w -t- aK H- 2tK') =

cn?i,

[ i< + 2 K -1- 2 iK' ) = —

dn u,

cn«

dn a

dn ?i

M + tK') = y

a + ili') =

k snii i dn M A sn «^

ili') = — i

T- -i^/N t dnw k en it

A- en M

, sna en II

FORMULES

( i) sn(a 4- a)

sn«cnadna-i-snacnadn?<

I — /i- sn- u sn^a (2) cn(w-i-a)

en M en a — pn u dn 11 sna dna

(3) dn(«f-f-a)

i — X:^ sn2 u sn'^a dn M dna — A'^ s,nu en m sna en a

(4) sn(i< -t- a) -+- sn(M — a) =

(5) sn(i^ + a) — sn(?f — rt) = (6)

(7) cn(M + a) — cn(w. — «) =

(8) dn(«-i-a)-i- dn(i/ — a) =