CAPUT 1.

16

39. Problema

Invenire ferient, quae numerum z dato logarithmo y in quocunque fy-Jlemate refpondentem exprimat.

•Solutio.

1. Pro quovis logarithmo y et certis coefficientibus a, b, c, d, etc.independentibus ab y fit

z — i + ay + by- + cy 3 + dy 4 -|-l-py r + qy r + I .

2. Erit differentia numeri z confiderati initar unius funftionis logarith-mi variabilis y per (ZZ. §.):

Az^=(a + 2 by + 3 cy 2 + 4 dy 3 -4- rPy r_I + (r+ i) qy r )Ay

+ x A y 1 +;3 A y 3 + y A y 4 + S A y 5 + etc.

3. Cum autem fit in (2) Ay — y 1 — y — Iz 2 — lz — l(z-f Az) — lz

erit per (3 5. §.)

A A z 3

A Az

Az 3 A A z 4 t-T- —-TTT- + etc,

Ay =

z

4. Quamobretn debebunt dari certi coefficientes k, I, m, etc. inde-pendentes a differentia Az, pro quibus et quovis valore ejusdem diffe-rentiae per (34- §•) in (2) (3) Seret

Az=(a+2by4-3cy 2 + 4 dy 3 -|-f- (r+ i)qy r ) —- Az

-fk Az J + l Az 3 + m A z 4 + n A z 5 + etc.

et ideo etiam

z —Aa + 2 Aby 4 - 3 Acy 2 + 4 Ady 3 H-+ (r + i)Aqy r

+ k.z.A z-f- Lz A,z J 4 m zAz 3 + etc.;

quod eft 'inrpofiibile, quin fit (24. §.)

z=tAa +2 Aby-f 3 Ac} 2 +4Ady 3 -b (r+i)Aqy r .

Ob u; per (29, §.) fiet ergo

A a — i

1

a, ~ 1. A

3Ac —b

1

4Ad = c d “

1,2.3 4 A 4

t I __ __

(r+ i)A q — p 1—P- (r+i)A

Et