320 



NATURE 



[Fed. 2, I 



for computing logarithms as well as antilogarithms ; and thus, 

 not only made the operations more convenient, but also caused 

 one set of preparatory tables to he sufficient. 



The principal table in Gray's book above-named consists of the 

 logarithms to twenty-four places of all the possible factors 

 I + (•ooi)»« X r, up to that limit. An auxiliary table contains, 

 also to twenty-four places, the logarithms and their comple- 

 ments of the natural numbers I to 9, these being frequently 

 required to "prepare "the given number. A smaller table to 

 twelve figures only appeared, as already mentioned, in the 

 Journal of the Institute of Actuaries, and was subsequently 

 published separately by Messrs. C. and E. Layton ; but as the 

 twenty-four-figure table can be worked quite easily to any extent 

 up to that limit, there is no particular advantage in the smaller 

 one. 



By means of Gray's tables the work of forming logarithms and 

 antilogarithms is reduced to a minimum, and the process is so 

 simple that any arithmetician can perform it, the more especially 

 as many numerical examples are given in the introduction. 



London, January 23. George King. 



Note on a Problem in Maxima and Minima. 



To find a point such that the sum of the straight lines joining 

 it with the angular points of a given triangle shall be a 

 minimum. 



This problem was proposed by Fermat to Torricelli, who 

 solved it, and sent it to Vincent Viviani, who also solved it, but 

 called it a problem ' ' quod, ut vera fateor, non nisi iteratis 

 oppugnationibus tunc nobis vincere datum fuit." 



The solution is given in Gregory's "Examples of the Differ- 

 ential and Integral Calculus," and in Todhunter's " Differential 

 Calculus," pp. 240-42. 



Yet it can be solved in the most elementary manner. 



Let ABC be the triangle. Describe an equilateral triangle 

 on BC on the side remote from A. Describe a circle round the 

 triangle BCD, Join AD. Then E is the point required. Join 

 BE, CE. 



(i) It follows, from Euc. vi. D, that 



BE -i- EC = ED, 

 . •. BE -1- EC -^ AE = AD, 



and evidently / BEC = BE A - AEC = 120°. 

 (2) Let F be a point on the circumference BC. 



BF -h FC = FD (Euc. vi. D), 

 .-. BF -f FC -t- FA = FD -f FA > AD. 



{3) Let P be a point not on the circumference. Join DP, 



and produce it to the circumference at G. Let fall the perpen- 

 diculars PH and PK, on GB and GC respectively. 



By Euc. i. 26, GH = GK = -iGP. 



Since z GPH = 30° = GPK, 



.-. BH -f KC = PD, 



.-. BP -f PC >PD, 



. . BP -f PC -I- PA > PD + PA > AD. 



(4) It also follows from the above that if z A =: 120", then 

 the point required is A ^ E. 



If / A > 120°, the point A will be within the circle, and A 

 itself will be the point required. R. Chartres. 



Note on the Dimensions and Meaning of J, usually 

 called the Mechanical Equivalent of Heat. 



The title " mechanical equivalent of heat " tends to make 

 one consider that J means the ratio of a quantity of mechanical 

 energy to an equivalent quantity of heat ; but since heat is 

 mechanical energy (in a molecular form) it follows that J on this 

 supposition is equal to tmity, and therefore unnecessary. 



Another way in which J is sometimes regarded is as the ratio 

 between the ordinary units of heat and work ; that is to say, in 

 England, it is the ratio of the British thermal unit to a foot- 

 pound, viz. the number 772. This definition makes it a simple 

 number, the number of work units in a heat unit, a number 

 which depends on the units of heat and work employed, and is 

 different in France and England. 



Now although J generally has one or other of these signifi- 

 cations — that is, must be either unity or some pure number — yet 

 people speak of the dimensions of J as being, not zero, but 



Work 



Mass X Temperature 



It is evident that there must be some confusion here, a con- 

 fusion arising from the fact that most people when talking of 

 quantities mean only so many times the units of those quantities, 

 and so are not always sufficiently careful about the definitions of 

 the various quantities which they introduce. 



Now if we confine our attention to quantities themselves, 

 independently of any systems of measurement, we shall 

 be led to a perfectly consistent mode of regarding J, a way 

 moreover in which it will have the required dimensions 



Work 

 Mass X Temperature 



A British thermal unit is the heat required to raise a pound 

 of water at freezing-point through i° F., and Joule discovered 

 that the mechanical equivalent of that amount of heat was about 

 772 foot-pounds. 



Hence if we wish to consider the work necessary to raise any 

 other mass of water at freezing-point through any small 

 range of temperature, we have only to notice that the 



quantity ^ is constant, and equal 



Mass X Range of Temperature 



772 foot-pounds 



to ^ . 



I pound X I F. 

 This quantity is very fitly denoted by J, and might, if thought 

 convenient, be called a Joule. 



But this quantity is the specific heat of water, according to the 

 definition that specific heat is the heat required to raise a mass 

 through a small range of temperature divided by the mass and 

 the range. So that we have arrived at these conclusions : a 

 quantity of heat is the same thing, whether expressed in British 

 thermal units,or in foot-pounds, or in termsof any other standard ; 

 and the specific heat of water at 0° C. is denoted by the 

 letter J. 



Indeed it may be said that the result of Joule's experiments 

 is the determination of the specific heat of water in absolute 

 measure. Again, if c is the ratio of the specific heat of any 

 substance to that of water, the full expression of its specific heat 

 is fj ; that is, its specific heat is some multiple or fraction of 

 ■X Joule. 



The first law of thermo-dynamics will then be expressed 

 as — 



r/Q - / . dN ~ c]m . dd + m . dl, 

 where ^Q - pdV is the total energy supplied, epndO is the 

 amount of new energy evidenced by increase of temperature, 

 and i?idl is the increment of the latent energy of the body. 



Coopers Hill, Staines, January 19. Alfred Lodge. 



