420 Mr. Adams's Mathematical Problem. [Dec. 



series which are always supposed to be o{some magnitude, covld 

 not have been produced from nothing, that is, the expression, 



""" ~ ""1 cannot, by any operation whatever, produce a given 



V — w 



mao'nitude when v is equal to w ; but when i; — w is indefinitely/ 

 small, as before remarked, we may then, without any sensible 



^rror, consider - or - equal to unity ; on this supposition, the 



righthand side of Equation I. would become m u™~', and the right 



hand side of Equation VIII. would become - v'^ . Hence it 

 (follows that when u — ?« is indejinitelij small, the expression 

 " ~ ""', will be indefinitely near to m u'""' or »««;'""' ; but there 



V — w 



never can be an absolute equality between these expressions.* — 

 {See Analyst, Section 15.) 



The following example is answered according to Equations 

 marked (E) and VIII. 



Example. — Given v = 11, w = 10, and m = -J, to find the 



true and approximate values of ^^ _^q • 



First answer from Equation (E). 

 1 = + 1-00000000 



m-l r_ _ 3 J_ _ Jl^ 



2 ' w~ 'i '10 60 



i-2 

 «i-2 r_ J_ _ 3 J_ J_ 1_ 



~~^ • It • 60 3 ' 10 ■ 60 2700* ' ' 



4 



- — 3 



■ 2700 "^ ~r • To • ~ 2700 = M800 = + 0-00001543 



= + 0-01666666 

 = - 0-00037023 



w 



The sum of four terms = + 1-0163118(5 



rmo—' = I 10^"' = \ \(f = I X 2-1544347 = 2-8725796 

 And 2-87-25796 x 1-01631186 = 2-919432, which nearly agrees 



4. 4. 



Tvith^^^J^^?- = ^^* - ^^^ = (121-12lf - (lOOOOf 

 = 2*91942 = true value. 



* From Equation 1 , we have Dm — 2e,„ = (y — iti) A u"" — ' = r A u™ — ' , where 

 ^ — (2) = r may represent any given quantity ; but if we suppose r = o, or v = Wf 





 -w shall then baveO = x A u'"- ' or A w™ - ' ^ n ~ ^' 



