the Lagrangian Theorem of Interpolation, 375 



is known to be satisfied for m l -\-m <i + ... -fmi— 1 (say) /L6— 1 

 assigned values of the system of quantities l v £ 2 , . . . /,-, x ; there 

 will then be /z,— 1 linear equations connecting the fj, coefficients 

 comprised in Up U 2 , . . . V b and therefore the ratios of these 

 coefficients, and consequently of the functions to one another, may 

 be determined. There is no difficulty in representing, by aid of 

 the method of determinants, the result of solving these equations 

 whatever be the number of functions; but for the sake of 

 greater simplicity, I shall suppose three only of the several 

 degrees, e— 1, i— 1, o>— 1 in x, which I shall call U, V, W. 

 Now suppose that / . TJ + mV-f %W = is known to be satisfied 

 for /=//, m=m t , n=n t , x=x t , t taking all possible values from 

 1 to e + i + <D — 1, say t — 1; let the indices 1, 2, 3, . . . t— 1 

 be partitioned in every possible way into 3 groups, containing 

 respectively e — \, i and co indices, as 



ctj 2 . . . e -i ; u e u e +\ . . . u e +i+\i u e+ i . . . v T _ x 



(the terms in any group may be arranged indifferently in any 

 order, but are not to be permuted). Let % h (p, q,r . , ,s) denote 

 in general 



(p-q)x{p-r) . . . x(p-s) 

 x(q—r) .. . x{q-s) 



X (*-*), 



and write 



^f*(* X di XQ 2 ...XQ e _) &{x 9e .X 9e+i ...X ee+i _ i ) &{XO e+i " a erJ 



The mark (?) is used to denote ( — ) raised to a power whose 

 index is the number of exchanges of place whereby the arrange- 

 ment 1,2,... (t — 1) can be shifted into the arrangement 

 lt 2 ,...6 T _ V 



In like manner, let 



K 2 =2? 



] p(x 9i x 6i ...x 9e ) f*(a? ^ fl+1 -..*e e+< . 1 ) ?(«() (+ ,-V,) 

 and 



Then, using c to denote any arbitrary constant, we shall have 



