140 RELATIONS BETWEEN SEVERAL [BOOK I. 



102. 



Since y must necessarily be positive, as well in the hyperbola as in the ellipse, 

 the solution of equation 16 is, here also, free from ambiguity :f but with respect 

 to equation 16*, we must adopt a method of reasoning somewhat different from 

 that employed in the case of the ellipse. It is easily demonstrated, from the the 

 ory of equations, that, for a positive value of H\, this equation (if indeed it has 

 any positive real root) has, with one negative, two positive roots, which will either 

 both be equal, that is, equal to 



ly/ 5 1 = 0.20601, 



or one will be greater, and the other less, than this limit. We demonstrate in 

 the following manner, that, in our problem (assuming that z is not a large 

 quantity, at least not greater than 0.3, that we may not abandon the use of the 

 third table) the greater root is always, of necessity, to be taken. If in equation 

 13*, in place of M, is substituted Y\J (L -\-s\we have 



)^&amp;gt;(l + z)^, or 

 4.6 4. 6.8 



whence it is readily inferred that, for such small values of z as we here suppose, 

 Y must always be &amp;gt; 0.20601. In fact, we find, on making the calculation, that 

 z must be equal to 0.79858 in order that (\-\-z]Z may become equal to this 

 limit : but we are far from wishing to extend our method to such great values of z. 



103. 



When z acquires a greater value, exceeding the limits of table HI., the equa 

 tions 13, 13* are always safely and conveniently solved by trial in their un 

 changed form ; and, in fact, for reasons similar to those which we have explained 



t It will hardly be necessary to remark, that our table II. can be used, in the hyperbola, as well as 

 in the ellipse, for the solution of this equation, as long as h does not exceed its limit. 



I The quantity H evidently cannot become negative, unless f &amp;gt; ; but to such a value of f would 

 correspond a value of z greater than 2.684, thus, far exceeding the limits of this method. 



