f)18 Dr. 0. J. Lodge on the Variation of 



and the function of 6 on the left-hand side of this equation 

 increases in geometrical progression for an arithmetical in- 

 crease of x. 



08. So far we have been practically exact ; but it is now 

 necessary to introduce several approximations. Notice that r 



is a large constant and s a small one ; so that - and sd are 



small quantities whose squares may be neglected for all pro- 



1 /r 



bable values of 6. Moreover observe that — a / -, though it 



may be imaginary, is never large and is often fractional. 

 Write, therefore, the denominator of the left-hand side of [19] 

 in the following successively approximate forms, 



r — /-, 2<9\~|^ v ^ 



1 + s6+ n/?W^1+— )] 



— Jl 



1 jr 



1 /- 1 - Lj (T\-i 



£=(1+n/?'s)™ v *+-0(vWl)(l+vV«)" KJ 



m 

 7)1 + 0, _ l^/l 



Hence we have as the quantity which is to go in geometrical 

 progression in equation [19] 



m+0 m L J 



and we may conveniently write equation [14] to the curve of 

 temperature down the rod in the simple approximate form 



°' r +e **+e > • ■- ^ 



w r here it is to be remembered, see equation (39), that 



^ = ^^,™^ = ^ ORa= ^o_p arologa; [21] 

 n% k vn k /lq /Cq 



* The approximation in [21] consists in writing mn, the largest term of 

 A, instead of A. It is unnecessary to work with this approximation j but 

 it is useful as showing how naturally the various constants occur in fx. 



