142 
Mr. Hellins on the Rectification 
- — + 
a. • 
-f- / <^ 
+ 
2 a 
a 
JL JL ^ 

» a 3 
3-4 e . 
+ 
2a 
v 
a s 
Jr. 
3 ^ 
+ 
i 
T 
a? 
7 » 
- -4, &C. 
a 9 
-21- & c 
J S — 
t a 7 
a 9 
7 ^ 
,9 ’ 
&c. 
5.67 7.8J , 9.IO! 
''rfS 
7 
4- 4lf- — ±il -L. 44Z _ Z41 & c 
T a 3 as ^ a 7 a 9 7 
__ J^L 4. 
a 1 • 
5.67 
, &C. 
Now, unless each of these parcels of quantities, as well as 
their sum, be universally = o, this equation will be of no use 
to us. And if it can be proved, that the coefficients of the 
terms in the one series are, each of them, = o, then it will 
follow, that each of the coefficients of the terms in the other 
series is also = o. But each of the coefficients of the terms 
in the logarithmic series is = o ; which may be proved thus : 
when the like quantities in this series are added together, the 
first term will vanish; and the coefficients of the second, third, 
fourth, fifth, &c. terms (without the common factor /,) will 
be 3« — 2.46, — 3.56 + 4 .67, 5.77 — 6 .$$, — 7.9$ + 8.ioe, 
&c. respectively ; and the law of continuation is obvious. But 
by the law (see Art. 29 and 30;) which the coefficients a, 6, 
7, See. are known to observe, ~ is = £, ~ = 7, p— ^ 
2^2i — £> & c . . an d therefore, 3a — 2.46 = 0, 3.3s — 4 . 6 y 
— °> 5-7y — ffi8£ == °y 7-9 $ — 8.iog = 0, &c. 
The value of a, viz. i, which was discovered in Art. 29, is 
found also in the algebraic series, as will presently appear. 
For, adding like quantities together, the first term of this 
series also will vanish ; and the coefficients of the second, third, 
fourth, fifth, sixth, See. will be as follows : 
