310 DR THOMAS MUIR ON A 



The question then is — How can we by transformation set in evidence the 

 fact that 



\b m c n d r \ (a-a)(a-{3)(a-y)(a-S) \ a m c»d r \ (b- a )(b- P)(b-y)(b-S) 

 | b°c l d*\ ' ' (a-b)(a-c)(a-d) ' + | aVd 2 | * (b-a){b~c)(b-d) 



\aH n d r \ (c-g)(c-p)(c-y)(c-S) , \aH"c r \ (d-a)(d-8)(d-y)(d-S) 

 + \a°b l d?\' (c-a)(c-b)(c-d) ' + | a°b^ \ ' (d-a)(d-b)(d-c) 



is an alternating function with respect to the interchange 



/« b c d\ ? 

 \a /3 y SJ- 



Since 



|&o c i rf2 | = £i(b,c,d) = (d-c)(d-b)(c-b), 

 the fractions evidently have the same denominator, viz., 



(d — c)(d — b)(d — a)(c — b)(c — a)(b — a), 

 or £K a bcd)\ 



so that, if we expand the original numerators in descending powers of a, of b, 

 &c, the expression becomes 



^ a -J^ ) [-\ imc '' dr \U i -a^a + a^a/3-a'Sa^y + a/3yS} 

 + 1 a m fi»d r \{¥- b 3 Ha + & 2 2«/3 - bZafSy + a/3y 8 } 

 - 1 a m ¥d r | { c 4 - c*2a + c 2 2a(3 - c2a/3y + a/3y8] 

 + \a">b»c>-\{d i -d^a + d^ap-d'Za(3y + a!3y8}~\ . 



This, when the coefficients of 2a, la/3, &a, are collected and condensed, 



= cj, b c ^ Mfl"^^! - | w»b"<rd s | Sa + I a m b n c r d? \ 2a/3 



- | a m b n c r d 1 2a/3y + I a m b n c r d° \ ?,a(3yS~\ , 



and no farther simplification is possible until the special values of m, n, r are 

 given. 



Taking in order the ten different sets of special values 



0,1,2; 0,1,3; 0,1,4; 0,2,3; 



and denoting the whole expression by F m>nir> we see immediately that 



