In the sexagesimal system the **numerators** of the successive fractions (the denominators of which were the successive powers of 60) were followed by', ", "', ", the denominator not being written.

- The **numerators** and denominators of the convergents to the general continued fraction both satisfy the difference equation un =anu„_,+bnun_2.

Their **numerators** are denoted by Pi, P2, their denominators by q,, q2, q3, We have the relations p n = an pn-1 +bn pn-2, qn = angn-1 +bngn-2.

The **numerators** and denominators of the successive convergents obey the law p n g n _ l - pn-1qn = (- O n, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.

In fact, a continued fraction ai +a2+ +an+ can be constructed having for the **numerators** of its successive convergents any assigned quantities pi, P2, P3,

" Now for the interpolation of the rest, I considered that the denominators I, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the **numerators** were to be investigated.

ï¿½; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with **numerators** I and denominators continually increasing.