96 REPORTS ON THE STATE OF SCIENCE. 
where 
pay (4m? 12)(4n?@—32) | (An? —12)(4n?— 32) (de? —52)(4e2 7) 
; 21 (8x)? 4! (8a)! 
and 
_4n?—1? _(4n?— 1?)(4n?— 3°)(4n?—5? 2) 
CRP (8a) 3! (8x)3 
was probably true. 
The following is an outline of the proof, which is much shortened by 
adopting the abbreviation (a, a+2s) for the product of the (s+1) factors 
a,a+2,a+4,...(a+2s). 
With this notation 
ee 2n+3) 4 nal (2n—47 +1, 2n+ 4r—1) 
a ae a, Oe 
_(2 2n—1, 2n+1)_(2n—5, 2n+5) _ 
n= 1! (8a) 3! (8x3 
_; (Qn—4r43, 2n+4r—3) 
Powe @rais Gayrot i 
On forming the expression P,P,,,+Q,Q,,, the coefficient of 
(= ee will be found to consist of (27+ 1) terms, of which 
thevhishae sd, et ee 
(2r) ! ‘ 
ste) (2n+1, 2n +3) (2n—4r +3, 2n+4r—3) , 
the second is i r= il : 
the (¢+1)® is 
2n—2t+3, 2n + 2t+1) (2n—4r+2¢+1, Rid Meet! 
(— 1): { 
t! (2r—t) ! 
on\th ig — (2n—4r+5, 2n+4r—1) (2n—1, 2n+1) 
the (27)™ is @r—1)! VW , 
and the (27+ 1)™ is cet 8 at 2n + 4r+ 1) 
2r) 
Tn order to prove that P,,P,,,,+Q,Q,, =1 it is necessary to prove that 
the coefficient of (—1)’ ( i is zero. 
If the rv term in the series, which expresses the value of this 
coeflicient, be denoted by T,, it will be found that 
qty as (—1)'(Qn—2r—1, 2n+2r 41) (2n—27 +3, 2n + 2r—3) 
r+1 ee or Nina r(r—1) ! (r+ 1) ! 
Daa (Tess aU T) +T,., 
— (—1)'2(2n—2r—3, 2n + 27 +8) (2n—IAr+5, In+2r—5) 
KS r(r—2) | (r +2)! on 
