This work, which is represented in fig. 13 by the shaded rectangle, is equivalent to the heat

Here p=62.58, u=210, l=80, r=606·5; and on substituting these values in the above equations we get them in the forms

If, for instance, the whole mass were at first liquid, so that 1=0, we have in this case

x2=0·1165, 2-v1=24·5, g=361.

In the adiabatic change of a kilogram of water into a mixture of ice and steam, therefore, 0·1165 kilogram will vaporize and 0.8835 kilogram freeze; 3'61 calories of the intrinsic energy will be turned into external work; and the increment of volume represented in fig. 13 by the length MN will be 24.5 cubic

metres.

§ 8. Conclusion.

As a result of the above discussion, there follows the theoretical possibility of representing the behaviour of water in its three different states of aggregation by a solid geometrical figure, even though considerable difficulties would be encountered in the exact practical execution of such a model by reason of the insufficiency of the experimental results already in hand*.

The same procedure as this we have followed for water may also be adopted for representing the behaviour of any other body by a model of its temperature-surface. As the ground of such a model we may take a hyperbolic paraboloid, which represents the behaviour of the so-called perfect gases, and pieces may be stuck on it to represent the differing behaviour of other bodies.

On the model for such bodies as expand in melting (e. g. sulphur, phosphorus, &c.) the frost-edge will appear as a receding edge, and the melting-edge as a prominent one. Instead of the prominent cornice that represents the freezing-region on the water model, we shall find on the models for these bodies a terrace-shaped prominence to represent the same region.

* A plaster model of the temperature-surface of water, made by the sculptor Blum, of Aachen, is in the museum of the Aachen Polytechnicum.

If the necessary data were known from experiment for all bodies, we might then exhibit their behaviour in changing state by a series of models-just as certain of their properties are naturally shown by their crystalline forms.

Aachen, June 28, 1877.

XXX. On Unitation.-VIII. Practical Remarks thereon, together with Examples. By W. H. WALENN, Mem. Phys. Soc.

[Continued from vol. iv. p. 379.]

N the general formula for any integer number, given in

29. I art. 28, namely IN

-1

An pn−1 +An-1pm-2 + ... + azNo2 + a2r+a1,

the suffixes to the coefficients which correspond to the digits are so disposed as to show the number of digits at a glance. They also show, by inspection, the place of any one digit in the number, counting the unit's digit as the 1st digit, symbolized by a1, the tens' digit as the 2nd digit, symbolized by a2, the hundreds' digit as the 3rd digit, symbolized by as, and so on. This use of the suffix implies a law; and the law is an extension upon that which has hitherto appeared in relation to suffixes; this extension involves a special interpretation of the symbol a

In this place it must be noted that suffixes have not been used with that attention to perfect congruity which should accompany every mathematical work. In ordinary algebra, these adjuncts to notation have most frequently been used for the series of coefficients in the general formula for an equation, or in an expression in which each term is presumed to have a coefficient, either known or unknown. In these cases, for the most part, the suffixes simply indicate the order in which the coefficients follow one another: sometimes this order is from the right hand to the left, and in opposition to the order of the indices of the powers of the unknown quantity or variable; and sometimes it is in the same direction. A common use of the suffix is to mark the index of the power of the variable to which the coefficient belongs in any particular term. It is used in this way in Hind's 'Algebra (second edition), chap. xi. p. 374, for instance in the formula

amrm

+&c.+a2r2+aïr+aro+a_1r¬1+a-2 r¬2 + &c.

Another use of the suffix is to mark the terms that disappear when a particular operation is performed upon a general ex

sin

(D) X= sin (7.0)A, + sin(.1) A1+ sin (.2)Age2+&c.*

But no full recognition of the function of a suffix appears in any of these uses. When this method of indicating the order of a series of quantities is completely developed, it should be capable of showing, not only the sequence and direction, but also the relative position of each member of the series to a given point, as in the case of the ordinal numbers, 1st, 2nd, 3rd, &c.

The use made of the suffix in unitation, as proposed in art. 28, is in accordance with these views; and, in the general formula for integer numbers above cited, the suffix expresses the order of the digits commencing with the unit's digit, counting the unit's digit as the 1st integer digit from the decimal point, the tens' digit as the 2nd digit, &c. A question then arises which is as important, in relation to ordinal numbers, as the meaning of ao is in the theory of exponents. This question can be answered on a basis as logical as that of the exponential question: it is, "If the series be continued towards the right hand, as

&c. ... a2, α1, α, a-1, a_2, &c.,

ɑn, ɑn-1, ɑn−2, An-3, what does a mean?" It is impossible to think of the next number to the right of the unit's digit (which is in the 1st decimal place) as having any relation to a; for as the unit's digit in the number is the 1st digit to the left of the decimal point, so the next figure to the unit's digit, on the right hand, is the 1st digit to the right of the decimal point. If these suffixes are to be read as ordinal numbers, a must be left out; for the Oth place to the right or left of the decimal point is the decimal point itself. That is, if the above general formula be extended to decimals (or towards the right hand), it must be written

-2

A_pn−1 + An-12n−2+...+Azp2 + a2r11 +a1?1o +α-1?'-1 +a_2r−2 + ... and a has no other meaning than the decimal point itself.

Writing in the top row, which follows, an ordinary number with a finite number of decimal places, the corresponding co

* See Dr. Graves (Bishop of Limerick), 'Law's Mathematical Prize,' 1853, quoted in Carmichael's Calculus of Operations,' p. 100.

Uor M

efficients of the formula are represented in the underneath row as follows:

Thus written, each ordinal suffix has due relation to the position of its corresponding digit, and the whole number, in respect to the suffixes, is read:-5 is the 1st digit (from the decimal point understood), 4 is the 2nd digit, 7 is the 3rd digit. Then (in the opposite direction) 6 is in the first decimal place, 2 is in the 2nd, 5 is in the 3rd.

Normally considered, the operation of unitation always proceeds from right to left; but the negative suffix indicates the possibility of a change from right to left to left to right, under certain circumstances. This view will receive further consideration in the proper place. The use of a, as determining the place from which the direction of the operation is to be reversed, is believed to be new, and may be of use in other departments of mathematical science. Thus the meaning of a, in the series αз, ɑ2, α1, α, α-1, a_2, &c., is satisfactorily made out, according to the principles of the interpretation of symbols, to mean the place from which the order of the suffixes is reckoned, in reference to direction of counting.

30. This interpretation of a, may be well illustrated by a geometrical diagram:-If a vertical line cross a horizontal line, as in the marginal diagram, in the style of Descartes's rectangular coordinates, and if the origin be taken as the point from which the counting is to commence in both horizontal directions, namely backwards and forwards, it is evident that the a which is at the origin is a, and that the a's with the positive suffixes are distant from the origin in the numerical order of their suffixes; also the a's with the negative suffixes are distant from the origin in the numerical order of their suffixes, the negative signs simply indicating that the counting of the suffixes is to proceed in the opposite direction to the counting of the positive suffixes.

An... Az A2 A1 αo A-1 A-2... A—n

31. The substitution of (r-d) for r, in the formula

an-1+an-n-2 + ... + Azə22 +a2r+a1

for a given number N, yields

an(r−8)”−1+an−1(r−8)”—2+...+a3(r−8)2+a2(r−8)+a1. This is a number which has the same remainder to & as N has;

for on expanding the (r-8)" portion of each term by the binomial theorem, it has 8 as a factor in every term of the expansion (of any one term in the latter formula) except the first, which is the same power of as occurs in the corresponding term of the original value of N.

32. In obtaining the remainder to 8 of N, the formula in art. 31 may be extended, by means of negative suffixes, into a ̧ (r−8)”−1+an−1(r−8)n−2+...+a3(r−8)2+a2(r−8)1

+a1(r−8)°+a_(r−8) ̄1+a_2(r−8)2+..., thus making it available for other numbers than whole numbers. In the operation for obtaining the remainder, the number resulting from the first substitution of the digits in the formula is again subjected to the operation; then this last number is again treated in the same way, and so on, each treatment giving a number less than the previous one, and divisible by & with the same remainder that N has. If this treatment be continued until a number less than & is obtained, that number is the unitate of N to the base &. This is according to the definition of a unitate given in the Philosophical Magazine for November 1868, p. 346.

33. This method of obtaining the unitate of N is general, and is therefore valuable. It also affords a means of comparing the properties of UN with those of N in a direct and satisfactory manner.

The repetition of the process of reduction by the formula is peculiar to unitation; and it may be symbolized by U" N, (n) being the number of times the formula is applied to a given determination of UN in order that the ultimate value of UN may be less than 8. This repetition has no analogy in the expression of a number by means of the formula N. The following examples illustrate the repetition of the cess of reduction :

I. If

N=1234567,

U, N=1+2+3+4+5+6+7=28.

U,"N=2+8=10.

U,"N=1+0=1. Here (n)=3.

pro

II. In obtaining U,N, if the formula containing the unreduced powers of 3 be used,

U/ N=36.1+ 33. 2 + 31. 3 + 33. 4 + 32.5 +3.6+7=1636.

U," N=33.1+32.6+3.3+6=96.

7

UN=3.9+6=33.,

UN=3.3+3=12.

U; N=3.1+2=5. Here (n)=5.