38 
Mr . Ivory on the Method of computing the 
and, because r z — - ra . 7 = -§•/* fl — , we get 
i-7 + '' 
J/- 
(r 2 — 2ra . y + fl 1 
3 • 
The last equation certainly proves that, when r = a, the func- 
/ i x 
/ d . 
tion 4- 
~h a 
f 
is evanescent for every value of 7, 
/ 1 V dr . 
between the limits -f 1 and — 1, with the single exception 
of the case 7=1, when the function is infinitely great: and 
I shall now shew that it is to the overlooking of this last 
mentioned circumstance that all the difficulty and paradox 
attending this investigation have arisen. 
Let it be proposed to find the value of the fl uent^ ~ ( ^~j : ~ / ^ 
between the limits x = 1 and — 1 . The indefinite fluent being 
(r— a) . y . V r— ax, we get between the proposed limits, 
f ZSl 7^r 1 = ( r a ) • T • {VF+a - V~a\. 
This fluent is plainly — 0, when r—a. 
Let us next consider the fluent 'J* ~~ ^ 7 —g*r) a ‘ *** > between the 
same limits as before. Here the indefinite fluent being — 
T • tEZc’ we § et between the proposed limits, 
— (r — a) . c . dx 
f 
c r—a 
a * r-fa‘ 
(r—ax ) 1 a 
In this instance the fluent does not vanish when r — a ; for 
it is equal to ~ 
t a -fliiAnf- tTzzL 
(r—ax) z 
case, the indefinite fluent being — — . we get between 
Lastly, let the fluent J * - be proposed. In this 
