324 



Observations on the Solution 



this proceed as above ; but instead of two terms of the logarithmic 

 series, only take the first, and reject all the second powers. 

 Then ljLb + yLd + M7/ = xLx=La,or7/{M+Li)=La — l'Ll; 



or V= TV— f-r« 



t + HLa 



If we use the hyperbolic logarithms, x= 

 inou tabular logarithms, then x= 



If thecom- 



1 + HL6 • 

 b X -4340945 + La 

 •4S4'J{)45^+ Lb" ' 



Example. Taking, as above, x''= 100, and Z' = 3-59; then by 

 the hyperbolic logarithms x= ;^,^„ '' = 3-5972S8, the same 



as before. 



Having therefore seen that by omitting the second and higher 

 powers of the small addition y we have less labour, and nearly 

 as great accuracv, we shall in future not take them into consi- 

 deration, but employ the first power only. 



Frequently we may approximate faster without finding x at 

 first, but only y, and then also it will be of advantage to have 



X under the form l-^y, in which case L( 1 — r- j= - — r^ 



nearly; and therefore as in the second formula given aboye^ 



y=z ~ ' ■ . If we wished x this way, then x r=: I ^ y ks 



, the same as found before by another mode. 



M + Lb 



The more general form, where y'^=a is capable of solution in 

 the same way as the above, when y is a simple function of x, 

 that is, when it becomes the equation {cx)^=:a. But as the 

 mode may be easily derived from the former, I shall merely state 

 the result. It is this : if ^ be an approximate value of x, then 



_ Mb + La 

 ^~ M+Li-c • 



Before proceeding to the second order of this class, I may 

 mention a simple geometrical construction of the equation .f^ = tf. 

 Let there be any logarithmic 

 curve GH, whose logarithms 

 begin at the given ordinate 

 AG, which is unit. Produce 

 AG till AC be equal to a; 

 draw CB parallel to the axis 

 to cut GH, and BD parallel 

 to AC ; also GK parallel to 

 AD ; through K describe an 

 equilateral hyperbola KF, be- 

 tween the asymptotes AC 

 and AD. Let it cut the logarithmic curve in F; then FE =:.«■. 



For 



