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The result may be easily verified. The given series may be thrown successively into the forms.

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The second example in my former paper was the series

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and a particular solution of px = p(ax) is x = C = vx,

(a − 1) (x

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- 1)'

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This result may also be verified; for the original series developed

1

1

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term by term in powers of and and corresponding series of

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a

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and resolved into series of positive powers of a and x, with a similar subsequent process, it yields

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Let us now apply this method to examine some of the more common series of analysis: let us take

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Multiply every term by x, and it will then appear to be

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Like all other results of strict methods of passing from the sum of n terms to the sum of an infinite series, this expression is infinite when the series is infinite. But my object here is to remark, that owing to ax = a having only the root = 0, there can be no discontinuity among the values which correspond to arithmetical values of the above series.

x

If we consider the series

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we find the value of the preceding to be as follows,

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No more simple value of μx can be found than the finite integral of

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If μ could be expressed, the value of the preceding series would be

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What we have here to observe is, that in consequence of ax=x+1, ax increases without limit for all values of a, and there is no discontinuity.

I shall only further remark, that the preceding results confirm, so far as they go, an opinion which I have long entertained, namely, that series which may be divergent, or which may be brought as near to divergency as we please, such as that for e", require much less circumspection than those which can never be made to diverge. In the first, generally speaking, the arithmetical value (between the limits of convergency) is the analytical value throughout; in the second, there is frequently discontinuity in the arithmetical values, and the general equivalent of analysis is not easily expressed. I will not however enlarge upon so general a topic, but beg to remain,

Dear Sir,

Yours very truly,

AUGUSTUS DE MORGAN.

5 UPPER GOWER STREET, April 30, 1836.

. P. S. Some time ago, I communicated to the Society what I consider a failure in the proof of the celebrated theorem of M. Abel, on the expressibility of the roots of equations which are values of a periodic function. As I have since printed my objection in the Calculus of Functions alluded to in the preceding paper (§. 90, 302, 303,) I take this opportunity of referring to the subject.

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VIII. Piscium Maderensium Species quædam novæ, vel minus rite cognitæ, breviter descripta. Auctore R. T. LowE, A.M. Iconibus illustravit M. YOUNG.

[Read November 10, 1834.]

VEL prudentissime cunctanti fugit inexorabile tempus; et qui rem dubiam, nimio suadente metu erroris, semper in crastinam horam differt, superbiæ forsan suæ potiusquam scientiæ commodo consulit. Piscium nempe Maderensium species quasdam insigniores, pro novis habitas, prorsus stabilire, aliis comparatis speciebus affinibus in Museis Britannicis tam servatis quam editis in libris, ipse de die in diem frustra cunctatus speravi. Quum autem rei certioris me fefellit spes, icones perpulchras saltem, cura vel exquisitissima pictas, pro erroris culpa in re qualibet momenti levioris indulgentiam impetraturas credens, animum recepi. Si enim opiniones et nomina falsa, veteresque pro novis species ponuntur, icones bonæ nunquam non utiles; minus tantum quam ex votis auctoris evadunt. Eum si culpa rodit, scientia vix ulla, leviore certe, afficitur injuria: immo ipsius periculo potius augeatur!

ORD. ACANTHOPTERYGIANÆ.

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1. S. fusco-nigricans, luteo maculatus, maculis evanescentibus: pinna caudali, dorsalisque analisque parte molli, postice rotundatis, nigris, candido fimbriatis: spinis pinnæ dorsalis analisque distincte filamentosis:

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