vapour, in the experiments of Professor Magnus, does not, therefore, surprise me.

I have at present no means of judging of the validity of the assumption by which Professor Magnus accounts for the effect observed when his air, instead of being passed through cold water, was urged through water at a temperature of 60° or 80° C. He here assumes precipitation, though there is none visible. By a similar assumption he explains the experiment of Professor Frankland, in which aqueous vapour was discharged along the axis of a cylinder of hot air and carbonic acid, and protected from precipitation by its gaseous envelope.

With regard to the formation of dew, the amount deposited depends on the quantity of vapour present in the air; and where that quantity is great, a small lowering of temperature will cause copious precipitation. Supposing 50, or even 70 per cent. of the terrestrial radiation to be absorbed by the aqueous vapour of the air, the uncompensated loss of the remaining 30 would still produce dew, and produce it copiously where the vapour is abundant. Attenuated as aqueous vapour is, it takes a good length of it to effect large absorption. I have already risked the opinion that at least 10 per cent. of the earth's radiation is intercepted within 10 feet of the earth's surface; but there is nothing in this opinion incompatible with the observed formation of dew. A surface circumstanced like that of the earth, and capable of sending unabsorbed 80 or 90 per cent. of its emission to a distance of 10 feet from itself, must of necessity become chilled, and must, if aqueous vapour in sufficient quantity be at hand, produce precipitation.

I should willingly leave to others the further development of this question, feeling assured that, once fairly recognized by field meteorologists, the evidence in favour of the action of aqueous vapour on solar and terrestrial radiation will soon be overwhelming. An exceedingly important instalment of this evidence was furnished by Lieut.-Colonel Strachey in the last Number of the Philosophical Magazine. It is especially gratifying to me to find my views substantiated by so excellent an observer and so philosophical a reasoner.

Let me say, in conclusion, that nothing less than a conviction based on years of varied labour and concentrated attention, could induce me to dissent, as I am forced to do, from so excellent a worker as Professor Magnus. Hitherto, however, our differences have only led to the shedding of light upon the subject; and as long as this is the result, such differences are not to be deprecated.

XVI. On the Evaluation of Vanishing Fractions, with some Supplementary remarks on Newton's Rule. By J. R. YOUNG, formerly Professor of Mathematics in Belfast College*.

F,(x), F2(2), &c., in ƒ1(x) ƒ2(x)'

succession, till we arrive at a fraction in which numerator and denominator do not both vanish for x=a. And it is plain that these fractions will remain unaltered though for F2(x), F3(x), F(x), &c., and at the same time for f(x), ƒs(x), f(x), &c. we substitute

This being so, I think that whenever F(x) and f(x) are rational polynomials, it will be more systematic and at the same time more easy, to compute the value of the vanishing fraction as in the following examples, a step of the work on the left and a step on the right being taken alternately. It will be readily foreseen. that, although no expressions of higher degree than the third are here taken, these being fully sufficient for the purpose of illustration, yet the higher the degree, and the larger the number a, the greater is the facility of this method of calculation as compared with that usually employed.

1. F(x)=x3—2x2−x+2, f(x)=x3-7x+6: to find

F(2)

f(2)

2. F(x)=x3-5x2+3x+9, f(x) = x3 — x2 — 21x+45, x=3.

3. F(x)=x3+2x2-4x-8, f(x)=x3+3x2-4, x=-2.

When the general symbol a is under the radical sign, the usual method is to substitute a+h for x, and then to develope the terms containing a+h by the binomial theorem, a being the value of x for which the fraction becomes

0

0

But it will fre

quently be more convenient and simple to proceed as in the following example.

the result being

5(x-1)

X

-1

; hence the value of the proposed frac

tion for x=1 is 5 divided by these factors when 1 is put for x

in them also; that is, the value is

In

5

842

my demonstration of Newton's rule, the case in which the sign of inequality in any of the criteria is replaced by the sign of equality is not adverted to; and Newton makes no mention of it. I propose here briefly to examine what inference may be drawn from this circumstance.

1. And first, let it be the leading triad of terms that furnishes this equality. Then if the roots of the equation be diminished or increased by that quantity which will cause the second coefficient in the transformed equation to be zero, it will follow, from the theorem established at the end of my paper in the May Number of this Journal, that the third coefficient in that transformed equation must be zero also. For, as there shown, the condition. 2nAn An—2— (n−1)A2_, =0

necessitates the condition

2nA'„A'n—2—(n−1)A'2_¡=0,

which, if A'n-1=0, can have place only when also A'n-2=0. Now if all the coefficients, after this third, in the transformed equation turn out to be zero, the roots of the primitive equation must all be equal, their common value being the nth part of the second coefficient (when divided by the first) taken with contrary sign. But if all the coefficients do not vanish, then, as is well known, the equation must have at least one pair of imaginary

roots.

The inference, therefore, is that the equation must have either one pair of imaginary roots at least, or else that the roots must be all real and equal.

The same conclusion follows when it is the final triad that supplies the equality, since we may reverse all the coefficients.

2. But let it be an intermediate triad which furnishes the equality. Then by taking the derived equations in succession, we shall at length arrive at one of these in which the intermediate coefficients enter the final triad; and consequently, as just seen, this derived equation, if all its roots be not equal, must have at least two which are imaginary. In the latter case, two at least must occur in the primitive equation. In the former case the derived polynomial (after division by the first coefficient, if this be other than unit) will be a complete power; its three leading coefficients must therefore fulfil the condition of equality,

which it could not do unless the three corresponding coefficients of the primitive fulfilled that condition: hence in this case also the roots of the primitive must all be equal, or else two at least must be imaginary. The general inference, then, is this: whichever triad furnishes the condition of equality, the equation must have either two imaginary roots at least, or else all its roots must be equal. When, therefore, the equation is not a complete power, the sign, in the application of Newton's criteria to the several triads of coefficients, is to be regarded as significant of the same thing as the sign >.

It may be observed here that we know that the equation m(x-a)=0 will have all its roots equal to a; and that by developing the first member by the binomial theorem, the abovementioned condition of equality will have place throughout; and further, that when this condition has not place throughout all the triads, the first member of the equation cannot be the development of m(x-a)",-in other words, that the roots of the equation cannot be all equal. Hence the theorem, that, if the roots are all equal, the condition of equality has place for every triad, holds conversely; that is, if the condition has place for every triad, the roots are all equal.

In no subject is greater caution needful, in dealing with the converse of established theorems, than in the Theory of Equations; for the most part they are inconvertible. It has been regarded as questionable, or "more than questionable," whether the converse of Newton's rule is true; but there ought to be no doubt at all on the matter. Equations, in any number, may be proposed, for which not one of Newton's criteria of imaginary roots holds, and yet of which the roots shall be all imaginary. To be convinced of this, it is sufficient to refer to the theorem marked (II.) at the end of my paper in the Journal for October 1865, to consider A, as negative, and A, as a comparatively small coefficient. From the theorem marked (I.), too, it is at once seen that if p in the cubic be negative, two roots of it may be imaginary, though Newton's rule could never make known the fact. And Newton himself was fully aware of the inconvertibility of this rule; for, when speaking of the marks by which to distinguish the entrance of positive roots from the entrance of negative roots, he says (I still refer to Raphson's translation), "And this is so where there are not more impossible roots than what are discovered by the rule preceding. For there may be more, although it seldom happens."

Note. In the opening paragraph of my paper in the Journal for May last, the word "holds" should be replaced by "holds or fails:" the next following paragraph, however, sufficiently