272 Ohervations on Equation of Payments. 



session of the interest of the sum I. for the time t. between the 

 payments. But here let it be proposed to find an equated time, 

 so as to make the whole at one payment ; —let that time, be what 

 it may, be denoted by z. And I shall endeavour to seek it with- 

 out regard to any rule whatever: and from what has gone before 

 I think it must be evident this time will be the true time, if it be 

 such that our ol)ject is fulfilled. — " By neither party being the 

 loser," it has appeared as an inference iVom the above equation, 

 that at the end of the secrond or last term, A will have acquired 

 the whole interest {(j.xr.xt), and for any intermediate time 



an interest will be derived proportional thereto. Now (Y—i) 

 is the portion of interest arising from :; — - for the interval be- 



tween the first payment and time z ; but by hypothesis the sum 

 a. is also to be retained for the time x. ; and in order that the 

 equation shall be true for x. in particular, we must have 



/q,;c4._^ \ = ( — Y but it may be seen in this 



\ ^l+rt/ \l + r.{l—x)/ ^ 



instance as dependant on the value of x that —- — -^- — 

 hj' .ixr.ii—x) _^.'pi-,gj.gfo|.g adding these equal quantities on both sides 



we get {a.r.x + b.r.x) = h.r.t, as before. Whence a; = (^-^^-j^^ 



the equated time from the first payment : and from which ex- 

 pression, using the same quantities as in our present example, we 

 obtain 



/ '03- X -05x2 N ^ ^,^g ^j^^jg 



V (105+100)05/ ^~ ^' • ' 



equated time: Corresponding exactly with the time as found 

 by the common rule ; and as a proof of the accuracy of which 



we shall find upon trial that the quantity (^a + f--) x f-^ 



will, if exployed at interest for the further time t—x. exactly 

 amount to or eciual {b.r.t) ; but if we take the time as given by 

 Malcolm's rule, tlie former cjuantity with its accretion for the 

 given excess will evidently be less than the latter. Also we have 

 (a.r.x) = l'.r.{t — x) the equation from v>?hich the principle of 

 the rule is directly derived. 



By pursuing a similar mode of inquiry, whatever the number 

 of payments may be, it will lead precisely to the same results. 

 That there is an inconsisteney in the conditioiis of Malcolm's 

 rule must be apparent without any previous examination; since 



* Tl e caiiiciclcncc in this respect is extremely apposite, and besides it 

 supersedes the necessity of a tedious and operose process. 



it 



