Page:Carroll - Notes by an Oxford Chiel.djvu/25

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ing inversely as T: let A = Able, and E = Enlightened.

We have now to develop ϕ (HGL) by Maclaurin's Theorem.

The function itself vanishes when the variable vanishes:

i.e. ϕ(o) = O
ϕ′(o) = C (a prime constant)
ϕ′′(o) = 2. J
ϕ′′′(o) = 2.3. H
ϕ′′′′(o) = 2.3.4. S
ϕ′′′′′(o) = P
ϕ′′′′′′(o) = J

after which the quantities recur in the same order.

The above proof is taken from the learned treatise 'Augusti de fallibilitate historicorum,'and occupies an entire Chapter: the evaluation of π is given in the next Chapter. The author takes occasion to point out several remarkable properties, possessed by the above series, the existence of which had hardly been suspected before.

This series is a function both of μ and of e: but, when it is considered as a body, it will be found that μ = o, and that e only remains.