from Equations of Coexistence, 431 



Subsidiary theorem (B). 

 ^ SR (^1 /^2 *•• ^r) 



r/^i . ^2 ••• ^^r 1 



can be expressed by the sum of terms, each of which is the 

 product of series of the form 



it is ahvai/s integral, and when the dimensions of the nume- 

 rator fall short of (m — r) r it vanishes*. 



Subsidiary theorem (C). 

 The only modes of satisfying the equation 



for all forms of the latter factors short of m—r . n—r dimen- 

 sions, is to puty(/?i 7?2 ••• ^r) = 0, or else 

 ^,,7 T ^ constant 



/(^^i ^'^ - ^^) = / h,h,...K 



/ ^1 ^2 ••• K \ 



\ — "r+i • — "r+2 ••• "»j/ 



Theorem (2.) 

 By virtue of the subsidiary theorem (B), the two equations 



(' 



h 



rtr+1 • ^r+2 ••• "»j 1 



— ^1 . — ^2 ..»-/<•«/ 



{«r+l • "r+2 h,,n\ 

 —h^ . -7^2 h„J 



+ 'Z[{x-- k^) {x — ^2) ••• (^' — h) 



{«"r+l • «"r+2 ••• f^n\^ 



-/i, . ^hc,... — h^f 



h 



(' 





are each integer derivatives of the r^^ degree. 



* It may be remarked also in passing, that any term in the numerator 

 which contains ani/ one power not greater than vi — 2r may be neglected 

 and thrown out of calculation. Moreover, an analogous proposition may 

 be stated of fractions in the denominators of which anj/ number of rows 

 are written one under the other : see the last note to page 439. 



