of Edmlurgli, Session 1883-84. 
589 
qz of the cycle of partial quotients shall he 1 less than the unique 
partial quotient is 
••• + 5 •••» = ^.- 2 )- 
16. + D he expressed as a continued fraction with unit- 
numerators, and the middle element of the cycle of partial 
quotients he equal to A - 1, then 
D _ fe > • - > 9!z- lA) + (^2 > ♦ ■ • > g.-i ^ A - 1 ) ^ 
fej •••5 <iz-l) 
This and the preceding proposition are established exactly as the 
corresponding propositions regarding culminate cycles have been 
established (§§ 2, 3). 
17. The condition (q^, qz-i) + (q 25 •••5 92 -i) = ^(qu •••, qz- 2 ) 
being satisfied, the general expression for all integers whose square 
roots have penecidminate cycles is 
i[( - 1)'(2 m - 1) fe, . . . , g..,) - ( - l)'2fe , , g..,) (g„ . . . , g.„) + l]^ 
+ (-1)'{(2m- 1) (g^, g..i)- 2(g2, g.-sF} • 
From § 16 we have 
T-i, + qz-i, A-1) 
(?i j • • • ) T-i) 
_ (2A-l)Qz_, + 2Q._, 
R_. 
whence, as in § 4, 
DR_, = (2A-l)Qz_,+ 
(-1W2A-1) + 2R_A-. 
R-i 
Now since (§ 15) P^.i + Q^_i = 2Pz_2, then and Qz_i are both 
even or both odd : but they are mutually prime, therefore they are 
both odd. We thus have the foregoing fractional form = an odd 
integer = 2m - 1 say. This leads, as in § 4, to 
2A = ( - 1)^(2m - 1)R_, - ( - 1)^2R_2Q._2 + 1 
and D = (-1)^(2m-1)Q,_i-(- l)^ 2 Qt 2 
whence the desired formula. 
