Mr. Hellins on the Rectification 
132 
Now, since v/(i + e) — y/ e is = V(1+ *y— > if Z be P ut — 
H.L. (\/e+</(i'+ e ))> and if the values of A > B > C > &c * and 
of A', B', C', &c. be taken in terms of e and /, and written for 
them in the above two equations, we shall have 
1- * = VT+~e x + Te 
2ei 
— V 1 ~|” 
e x 
— 3 + 
2.4^1 2.2.4c 2 
.,*4:) 2 - 2- 4 e3 
2. 2; 
+ v/l + ‘ ! *(^h + 
\ 2.4.6.8c 2 
&c. 
+ d = ^ 1 + * - h 
3-3 
9 
2. 2. 4.4. 6^2' 
+ 
3-3^ 
2.4.4.612' 
3^5 joo _|_ 
2.4.6.6.8A 2.4.4.6.6.8c » 2.2.4.4.6.6.8c T 
3-5-5 
2. 2. 4.4.6c? 
3-3-5-5 
) + 
3-3-5-5* 
2.2.4.4.6.6.8c 
&C. 
+ 1 + £ X 
__ v/ 1 -f ^ x 
+ 1/1 4-^ x 
2.2.4c 2 
3 
2.2.4c 
2. 4.4.6^ 
3-5 
3-3 
2.2.4.4.6c 2 
3-5-5 
3-3^ 
2.4.4.6.6.8c 
2.2.4.4.6c 5 
_ « S-3-5-5 \ 
V ^ 2.2.44.6.6.8 cV/ 2.2.4.4.6.6.8c 7 
&c. 
3-3-5 -5^ 
1 2,4.6.6.8c 2 
&C. 
Here, on the right-hand side of both these equations, the 
diagonal lines of quantities in which / enters, and the perpen- 
dicular columns which have the common factor V 1 -{- e, the 
first column of the first equation, and the first term of the 
second excepted, are exactly alike, but under contrary signs ; 
so that, by taking the sum on each side, we have 
1 3-5 
r /~~a I 
j v 1 + e x: — 
2 z -j- a = -< 
j_-f- ei V 1 -f- 
2.4c! 2.4.6 c 2 
2.4.6. 8ct 
&C. 
