350 THIRD REPORT— 1833. 



where 



F'« 



a! F" a F" a 



(F' af ~ (F' of 

 ,„ a' F'" a 3 a' (F" af 



F" a 3 (F" af 



(F af ^ (F af ' 



This series was first assigned by Euler, and the observations 

 which we have had occasion to make in the preceding pages 

 upon hnear approximations will at once explain the circum- 

 stances under which it may be safely applied : it cannot be 

 viewed, however, in any other light than as the analytical ex- 

 pression for the result of the application of such linear approxi- 

 mations, repeated as many times as there are terms of the series 

 succeeding the first. 



The celebrated theorem of Lagrange, which is so extensively 

 used in the solution of the transcendental equations which pre- 

 sent themselves in physical and plane astronomy, will enable 

 us to assign, likewise, a series for the least root, or for any 

 function of the least root of an equation in terms of its coeffi- 

 cients. Mr. Murphy, in a very able memoir in the Transac- 

 tions of the Philosophical Society of Cambridge for 1831, has 

 shown the mode in which such series may be determined, by 

 means of a very simple rule, which admits of very rapid and 

 very extensive application. The rule is as follows : 



" To find the series for the least root of the equation <p {x) = 0, 

 divide the equation by x, and take the Napierian logarithm of 



the quotient which arises ; then the coefficient of — with its 



sign changed is the series which expresses the least root re- 

 quired." 



Thus, to find the series for the least root of the quadratic 

 equation 



x^ + a X + b = 0, 



find the coefficient, with its sign changed, of— in log ^^ 



1 + X \, which is 



j b b^ 4 63 6.5 b^ 8.7.6 _fr' , o 1 



