CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 3957 
2(b-c —ny + 2y) (n—1)(b+2y) EMail Parakeet 
3(¢ — 22) (3b-c-ny+4y) (m—-2)(b4+3y) ..... 
’ 4(c — 3y) 4(b—c—my+6y) ..... 
| 
=(n+1)- {n(B+y) —c}{(m— 1) +2) - 2e}{(m- 2)(B + 3y)- 3c}... 
Of these three results it would be interesting to obtain independent proofs. 
(<x) 
(19) In a paper of Hutne’s on Lamé’s functions (Crelle’s Journ., lvi. pp. 77-99) 
there occurs the continued fraction 
041 
EP = 
A= (yy, = CXR 
ies AG 
where the c’s are m in number with the values 
4m(m+1), 4(m-1)(m+2), 4(m—2)(m+3),...., $-1-2m, 
the fractional factor being 4 in every case except the first. From extraneous considera- 
tions the value of the continued fraction was given 
Bea ee — 47). (z — m?) 
EN — - for m even , 
(26), eee CSS 
and (z - P)e- 2 Ce for m odd ; 
(= 2?) . @-aer? 
but the author added, ‘‘ Hinen directen Beweis ftir diese Summirung des Kettenbruches 
habe ich noch nicht aufgefunden.” This, however, is readily obtained by writing the 
value of the continued fraction as the quotient of two determinants, viz. 
= C nN kc Ls Sick tp | 
Cy B= (oh = (th, Cyn Select kpe . 
Cs POR ia, Mune er sath a He 
SSS 
| 4-41 — % O53. 7 Sesh oes 
| oe oe (et. SSD I 
and then using two of the foregoing theorems. ‘Thus, taking the case where m=6, 
and where therefore the c’s are 
Th, Oe ce reas 
3 al 
aa 2 ea ee 
OM Te OK, Dee Ty 213-005} 
(ae ioe, os | 
{ 
3} 2-3 
the dividend of which, changed into 
2-21 6°34 ; 
484-1) 2-2142-12 ~2.(3141) 
DGGE) ceo 1-(3% 4.9) 
62% 3) 2 2il-o-8 
