28 KEPORT— 1876. 



are 3-f -, 3-1-^, 3+^, ^vliere the deiiomiuator we begin with is the first integer (j rente f 

 than the half of 7 : similarly, that before we como to 



^ + 7 + 15' 

 we hav* 



^ + 7 + 8' 



3 + ^ 



1 



7 + 9' 



^+7 + 1(5' 



and so on. When an even partial denominator occurs, we take as the partial deno- 

 minator to begin with, either its half or the first integer greater than its half, accor- 

 ding as the partial denominator following is greater or less than that preceding, or, 

 these being equal, according as the next following is less or greater than the next 

 preceding, and so on. 



Another improvement, though verbal, is important, viz. in regard to the term 

 convergent, the present definition of which seems arbitrary and unreasonable. With 

 great convenience it may be defined as follows : — A convergent of a fractional 

 ■number is a fraction ivhich is a closer approximation to the given mimher than any 

 other fraction icith a smaller denominator; so that Lagrange's problem is simply to 

 find all the convergetits of ang fraction. 



On the Belation between two continued Fraction Expansions for Series. 

 By Thomas Muiu, M.A., FJi.S.E. 



On the Use of Legenclre^s Scale for Calculating the First Elliptic Integral. 

 By Professor F. W. Npavman-. 



Denoting the first elliptic integral by F(f, w), and taking x such that x:^k 

 =:F(c, a>) : F(c, ^n-) ; then, in Lagrange's scale, fi-om a we deduce successively aj^, 

 (Oj, ojj . . . . by a given law, with the aid of c,, c^, Cj . . . . previously determined from c. 

 Then .r is the limit to which co, 2"^(B[, '2'-ai,^, 2 -'0)3 .... converge. If c is mode- 

 rately small, the convergence is rapid. But if e^ is very near to 1, it may be ex- 

 pedient to reverse the direction of the new amplitudes and moduli, viz. to calculate 

 c backwards c', 0", c'", so as to make c'", c", c', c, c,, Co, . . . . a series continued by a 



single law; and similarly from to calculate backwards co', w" , a'" Then a, 



a", (o'" .... are proved to converge to a fixed limit co' and F(c, a>) : F(6, ^7r) = Nap 

 log tan {\ 7r-|-5 io') : i tt. The function Nap log tan (\ n+l w) involves but a single 

 element co, and was calculated by Legeudre. Gudermanu has since published a far 

 ampler table. In practice the limit a is quickly reached : often it sufiices to make 

 ca=co') at worst Q)' = a)". Tims for very large values of c^ Lagrange's scale prac- 

 tically suffices, presuming that we have at hand tables of F(c, ^tt) and F(h, Itt). 



But Legendre, who discovered a new scale after completing his principal calcu- 

 lations, regarded his new scale as having much advantage in finding F(c, co) at 

 once rapidtv and accurately. In it .r is the limit of co, 3"iuj, 3'^a>.,, S-^a>.^ . . . ., and 

 the convergence, generally excellent in Ivagrauge's scale, is far more rapid in 

 Legendre's. In Lagrange's scale the relation of ojj to co is tan (a>^ — io) = b tan to. 

 The relation in Legendre's scale is to the eye as simple, viz. tan f (mi — <»)=A tan w; 

 but in the constant A, = /^(l— c" sin ^/3), the value of (i k detennined by the equa- 

 tion F(c, /3) :=f F(f , Itt). a practical difficulty arose in the very considerable trouble 

 needed to obtain A (or its logarithm) numerically when c was given. Legendre 

 showed how ^ was obtainable from c : tlie cubic equation arising can be solved by 

 a mere extraction uf the cube-root ; but there are also two quadratics involving two 

 extractio)is of the square-root, Then from /3 we have to calculate >^(l—c^ sin ^^) 



