CURIOrs NUMERICAL PROPOSITION DEMONSTRATED. gOS 



the 2d, 4th, &c. terms will remain the same, except that the 

 signs will be changed from + to — . And as to the 1st, 3d, 

 he. terms of the first two equations, and the same terms of 

 the third, we shall have (by observing that 



(5" _ r") 2 = (5" + r") « — . 4 5" J-" 



(s° ^ r") *= («" + r") -^ — 8 (i^n r" + «" r^") 



&c. &c.) for the sum of the two 



1st terms 2 f"" 



3d terms 2nar"-2" («"+ r")5--2 7ia«""-2"X4*" r" 



5th terms 2 nci""-'»°(s" + r")<— 2«c/''"-*"X85"r"(5*"+r8'-) 



7th terms, &c. 



And, consequently, subtracting from those sums, the Ist, 

 3d, &c. terms of the third line, namely, 



1st term <"" 



Sdtermwa^""-'" (5"4-»^)' 

 Sth term n c <""-*° (a"+ r")* 



the remainders of these particular terms will be 



1st rem. = <"" 



3d rem. :=:.n «/""-«" (5" -f r")^-— 2 wa/""-'-° X 4«" r" 

 Sth rem. n: wci""-"" (5"+r")<— 27ic T"-"" X 8 *" r"(*«''-f r'"] 

 7th rem. &c. &c. 



In short, the whole of the remainder which is equal to 

 zero, will be expressed by 



(^2" _j- r"'")^ &c. 



And hpre it is only ncccessary to observe, that all the terms 

 on the latter sjde of this expression are divisible by T s" r", 

 so that, for perspicuity sake, we may write it thus 



{<"«-«■' -I- r")" —t «" r" A 

 and consequently 



and here, since the first side is a complete nth power, the lat- 

 ter side, which i? equal to itj muft be so likewise, and conse- 

 quently 



