294 ELEMENTARY ILLUSTRATIONS. [CHAP. XVIII. 



Let X = oo, then Vis positive and infinite. 

 Again, 



= (l+p-q-r)(l +q-p-r)[(l + r-p-q)\-l} 



+ similar positive terms, 



which expression is positive between the limits X = 



and X = oo. 



If then we construct a curve whose abscissa shall be measured 

 by X, and whose ordinates by F, that curve will, between the 

 limits specified, pass from below to above the abscissa X, its con- 

 vexity always being downwards. Hence it will but once intersect 

 the abscissa X within those limits ; and the equation (16) will, there- 

 fore, have but one root thereto corresponding. 



The solution is, therefore, expressed by (9), X being that 

 root of (13) which satisfies the conditions (15), and s, t, and u 

 being given by (12). The interpretation of c may be deduced 

 in the usual way. 



It appears from the above, that the problem is, in all cases, 

 more or less indeterminate. 



