140 Mr. Hellins on the Rectification 
same as those in the original equation ; so that, if we can find 
herefrom another equation in which the numerators also shall 
be the same as those which were there found, we shall obtain 
an important point. Now this may easily be done as follows : 
Multiply the last equation by a , and take the fluents, and there 
will be 
d — f — = 157 &c. a x : — — -f- — 
J a u ' 2 “ 2. 2. a 
3-3« 5 3-3-5 -5^ 7 
, &C, 
2.2.4 2.2.4.46’ 
Divide this equation by a 1 , and take the fluents again, and we 
shall have 
= 7&c.x:-4 + 
• flrf 
a 
3-3-5^ s 
2.2.4 2. 2. 4. 4.6 
, &C, 
which is evidently . 
This new equation will be fitted for our purpose by taking 
the fluxion on both sides, multiplying by a 3 , and then taking 
the fluxions again; which operations being performed, (re- 
membering that a must still be made constant,) and the terms 
brought all to one side, and properly arranged, the result will 
be add -f* ( ™ — a ) ad — ( aa -f* 1 ) d = o. 
32. The values of d , d, and d, are now to be taken in terms 
of the pair of series obtained in Art. 30, and substituted in 
the equation last found. And, although the coefficients of the 
logarithmic series, and the law which they observe ad infinitum, 
be already discovered, yet, for the sake of brevity, I denote 
the first, second, third, &c. of them by «, £, y, &c. respec- 
tively. In this notation We have 
f + 
c 
a s 
y 
« 5 
j i 
o a i 
_E_ 
,7 9 
a 9 
■, &C. 
■, &C. 
