122 RELATIONS BETWEEN SEVERAL [BOOK I. 



must necessarily be a positive quantity, it is evident that no uncertainty remains 

 here. So far as relates to equation 15*, we observe, in the first place, that L is 

 necessarily greater than 1 ; which is easily proved, if the equation given in article 

 89 is put under the form 



,- __ - I cos 2 i/_i_ tan&quot; 2 at 

 cos/ I cos/ 



Moreover, by substituting, in equation 12*, Y^ (L x] in the place of M, we 

 have 



and so 



and therefore Y^&amp;gt; ^. Putting, therefore, Y= $ -\- Y , Y will necessarily be a 

 positive quantity; hence also equation 15* passes into this, 



r + 2 rr + (i //) r + ,\ f //= o, 



which, it is easily proved from the theory of equations, cannot have several posi 

 tive roots. Hence it is concluded that equation 15* would have only one root 

 greater than i,f which, the remaining ones being neglected, it will be necessary 

 to adopt in our problem. 



93. 



In order to render the solution of equation 15 the most convenient possible 

 in cases the most frequent in practice, we append to this work a special table 

 (Table II.), which gives for values of h from to 0.6 the corresponding loga 

 rithms computed with great care to seven places of decimals. The argument 

 h, from to 0.04, proceeds by single ten thousandths, by which means the 

 second differences vanish, so that simple interpolation suffices in this part 

 of the table. But since the table, if it were equally extended throughout, 

 would be very voluminous, from h = 0.04 to the end it was necessary to proceed 

 by single thousandths only ; on which account, it will be necessary in this latter 

 part to have regard to second differences, if we wish to avoid errors of some units 



t If in fact we suppose that our problem admits of solution. 



