of the Hyperbola. 139 
found in a diagonal line,) be taken from the first of these 
products, all the coefficients of the remaining terms ( since x 
= H. L. 2,) will be known quantities; and consequently all 
the remaining terms in each perpendicular column may be 
added together, and to their like quantities in the second pro- 
duct ; so that the new pair of series expressing the value of d, 
will be this, viz. 
T a , b c a 
a — 7 -, &c. 
1 a 1 a 3 a 
[ ~Ta + 
3 l 
3-3-3 1 
See. 
2.4. 2.4.6.42 
where the values of A, B, and C, are 0*44,314718, 0*03680519, 
and o*o2 183137, respectively. The law which the coefficients 
of the logarithmic series observe is evident ; the law, which 
the coefficients (A, B, C, &c.) of the other series observe, will 
be discovered by the following process. 
31. It appears by Mac Laurin’s Fluxions , Art. 808, and by 
the Philos. Trans, for 1802, p. 462, that d = 1*57, &c. x : 
T-&+ A - UPA &c ' ; where it is evident that 
the value of d depends entirely upon that of a , and that these 
two quantities must be constant or vary together. Therefore, 
supposing these quantities to vary, and taking the fluxion on 
both sides of the equation, we have 
d = 1*57 &c. a x : a — 77 + 
3-3-S^ 3-3 
, &C. 
2. 2. 4.4 2,2.44.6.6’ 
This equation divided by a, and the fluxion taken again on 
both sides, making a constant, gives. 
ad. 
3-3-S'* 3 
3 • 3 • 5 • S* 7-* s 
&C. 
= 1*57 &c. aa * : x — — + 
a aa u • 2 * 2.24 2.2.44.6 ’ 
Here we find the denominators of the fractional coefficients of 
the terms on the second side of the equation to be the very 
T 2 
