26 BOOK OF MATHEMATICAL PROBLEMS. 



I'M. The expression 



ax 9 + 2bx + c 



a'x a +2b'x+c" 

 will be capable of all values, provided that 



(oc f - dcf < 4 (a'b - aV) (b'c - be). 

 Prove that this inequality involves the two 



and investigate the condition (1) that two limiting values exist 

 between which the expression cannot lie, (2) that two limiting 

 values exist between which the expression must lie. 



133. If x lt x a y x 3 ...x n be positive, and if 



then will x^ x y , ... x^ 1 be in ascending order of magnitude. 



VII. Proportion, Variation, Scales of Notation. 



134. If b + c + d, c + d + a, d + a+b, a + b + c be propor- 

 tionals, then will 



ad bc' 



135. If y vary as the sum of three quantities, of which the 

 first is constant, the second varies as x, and the third as x* : and 

 if (a, 0), (2a, a), (3a, 4a) be three pairs of simultaneous values of 

 x and y, then when 



136. A triangle has two sides given in position and a given 

 perimeter 2s : if c be the length of the side opposite to the given 

 angle, the area of the triangle oc - c. 



