Introduction. 13 



The obvious uniformity in these results forces the suggestion 

 that the law of formation of the quotient will hold in any similar 

 case. That is, that 



where we have given the — sign to the odd powers of x, n being 

 any even number. Multiplying the right hand side of the equa- 

 tion by a-^x we obtain a"—x'\ thus: 



a"-'—a"-'x-\-a"-^x'— . . . —a'x"-'-\-ax"-'—x"-' 



a-\-x 

 a" — a"~^x-^a"~''x'^''- . . . — a^x"~^-{-a''x""'' — ax"~' 



■i-a"-'x—a"-'x'+ . . . ■j-a^x"-'—a'x"-'-\-ax"-'—x" 



a" -f o -fo -f . . . +0 +0 -fo —x" 

 But multiplying the left hand side of equation (i) by « + -r we 

 obtain a"— x" also. Hence equation (i) must be true, since it 

 reduces to 



a"—x"=a"—x". 



28. Theorem. T/ie sum of like odd powers of two quantities is 

 exactly divisible by the sum of the qua^itities themselves. 



By trial we find this theorem holds in the first few cases as 

 follows : 



{a-\- x) -T-{a-\- x)^= I . 

 {a^ -\- x^) -^{a-\- x)=^a~—ax -\- j(f . 

 (a'-\-x')^(a-\-x)=a*—a^x+a'x'—ax'+.t'. 

 (a'-hx')-^(a-\-x)=a''—a'x-^a'x-'—aKr^+a'x-'—ax^-^x^. 



The simple law in the formation of these results would naturally 

 suggest the general truth of the theorem. That is, that 



'^-^^"=a"-'-a"-'x-{-a"-'x'- . . . -\-a'x"-'-ax"-'-^x'-\ (i) 

 a-f-x 



where the terms containing the odd powers of x have the minus 

 sign, n being any odd number. Multiplying this equation through 



by a-\-x it becomes 



a"-^x"=a"-\-x", 

 and hence must be true. 



