of Exponential Equations. 325 



For by construction BD = a, and AD = La, whence ADKG = 

 1 X La; but by the property of the hyperbola AK= AE x EF= 



EF log, EF = Lff, whence EF"=/7, and EF = .r. 



Tliis construction might be shortened by making AD = La, 

 from a table of logarithms of the same modulus as the above 

 curve. But upon the whole, 1 would prefer the former method. 

 Were AC less than AG, x and the hyperbolic curve would lie 

 on the opposite side: but 

 AC is then liable to a li- 

 mitation, which evidently 

 happens when the hyper- 

 bola touches the logarith- 

 mic curve. The limitation, 



though it shall be found af:r K^ 



terwards by the differential 

 calculus, yet admits of a F^ 



very simple geometrical in- 

 vestigation. Let F be the 

 point of contact ; then by 

 completing the construction AC = a, when it is the least possi- 

 ble; draw a tangent MF at F, which must touch both curves at 

 that point, and also ME be the subtangent. But it is a prin- 

 ciple of the hyperbola, that the dtmble tangent, or line touching 

 the hyperbola and King between the two asymptotes, is bi- 

 sected in F; whence also ME=:EA; but since ME is a subtan- 

 gent to the logarithhiic curve, and as all these subtangents are 

 equal, if a tangent were applied at G, AE will bo the subtangent or 

 modulus of the system. Hence the construction is manifest: for 

 we have only to make AE a subtangent to the logarithmic curve, 

 raise the perpendicular EF, wliich gives x ; through F between 

 the asymptotes MA, AG, describe the hyperbola FK; draw then 

 GK parallel to AM, and KD to AG; then BD = r/. 



2d. With regard to the second order, one mode may be suffix 

 cient; I shall thevefore choose the simplest: 



X X 



Let X =0, to find x. Since x^ = a, then x'^Lx'^ha, and 

 xLx 4- L"x - L"ff. Let x = b + y = l(^\+ j-\ then Lx = 

 LL + L(^1 + ^\ =: Lb + ^^ nearly: and therefore V'x => 

 I (Lb + ^^) = V'O + ^ nearly. Also, xLx = {b -h y^ 



(Lb + ^) = ILb + 7jLb + My. 



Whence from these we get xLx+ L"xz:^L"a^L"b + ILb-k- 

 X3 y 



