proportion is regularly added to or deducted from a known sum of money, or any fixed quantity, at uniform intervals, it can be represented either by an ascending or a descending curve. The rate at which the curve falls depends on what is called the logarithmic decrement. Now when we take such industries as I have indicated above, and graphically record from year to year the relations between capital, revenue, expenditure, and profit, we possess a permanent and continuous diagramatic history of the growth, decay, or conduct of the business. It indicates the changes in time. We have a running view of progress and a clear indication of fall. The signals are as rapid and as certain as those of the barometer in showing changes of weather. They require observant and practical translation. They give us indications of laws. They enable us to forecast and even to prophecy. and expenditure in a very satisfactory way was closed, and after lingering for 12 years more it was wound up. The logarithmic curve became irregular and impossible in the 18th year and showed that reconstruction or winding up ought to have been applied in its 20th year. The diagram shows also that as soon as we can determine the logarithmic decrement (K in the equation) we can forecast the business in any future period if no price-cutting rival enters the field. On the other hand we may have fresh developments introduced which add new sources of income. The curve may become a straight line. It may even ascend. This is occurring with the modern electric light industry. Fresh markets are opening out in electro-chemistry, heat and power. Power and automobiles are "a potentiality of wealth beyond the dreams of avarice," and certainly beyond the scope of the mathematician. The average growth of business per year is the total increase during any period of years divided by the number of years. The growth would be constant and uniform if the business received equal increments in successive years, but the components of most businesses are variable quantities, and the increase per annum itself is also variable. It is, in fact, in most cases, a diminishing quantity due to the approach to saturation in the possible amount of business attainable. We can represent this growth as a diminishing percentage per annum, and this gives us the descending logarithmic curve. Fig. 1 is a curve of an imaginary business. It shows that it took two years to be established. It then began to show profit and it flourished steadily for the next 16 years when some new rival industry made inroads into its business. The capital which had been growing with its revenue I. WATER. Water is the oldest of these industries. The New River Company for supplying London with water for drinking purposes, originated with Sir Hugh Myddelton in the reign of James the First-300 years ago. The supply of pure water which is so necessary to the health and comfort of every human being of the community, is essentially the proper function of the municipality, and in the majority of cases it has been carried out by the local authorities. Many companies, however, exist, especially in serving the metropolis. It is, however, not a speculative or competitive business. It is conducted under strict statutory powers. The amount of water supplied varies directly with the population; its income depends on the rateable value of the district; its growth is steady, and its prosperity fixed. There is, however, one branch of it that is 600,000 GO% 500,000 50% 400,000 40% 300,000 200,000 20% 100,000 10% CAPITAL, GROSS REVENUE 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1893 1899 1900 1901 FIG. 3. 1885 1895 Gas commenced with the 19th century, and was in a flourishing condition of steady progress and steady indolence arising from great prosperity and fat dividends when electricity appeared as a rival in 1878, and gave its proprietors a very serious fright and a very rough shaking. It is, however, remarkable that electricity has proved the friend and not the enemy of gas. It has created an appetite for more light, and has taught gas managers that if they are to keep their businesses together they must look out for "fresh woods and pastures new." Hence, gas for heating, cooking, and power purposes has been much pushed. The supply of gas has not been checked, but, on the contrary, its rate of growth is actually increasing. The advent of the incandescent mantle has had much to do with this. Not so its profits where the business is in the hands of private enterprise, for the diagram III. RAILWAYS. The curves (Fig. 4, p. 13) show that the commercial soundness of the railway interests in the United Kingdom is in anything but a satisfactory condition. The mileage of railways open is 22,078. The expenditure is £67,489,739; ratio to revenue, 63.3 per cent. The profits are £39,069,976; ratio to capital, 3'27 per cent. There is no apparent allocation of this difference to depreciation, reserve, or redemption of capital. While capital is increasing, and the growth of traffic both in passengers and goods is satisfactory, the rate of growth of revenue is stationary, and the difference between expenditure and revenue is seriously diminishing. The consequence is that dividends are also diminishing. application of new regulations. The result of the control of this Department has been most This state of affairs would be very serious if beneficial to the safety of the travelling public. we did not see daylight ahead. What are the causes of the present depres[sion? The first cause is unquestionably Parliament, which legislates for the railway world without the least regard to the science of business or the ordinary requirements of commercial prudence. The enforcement of cheap fares and workmen's trains at the expense of the shareholders is pandering to a sentiment, and savours of a bribe to catch votes. The operation of the Railway and Canal The lives saved annually are untold. We have every reason to be proud of the security of our lines. But the finances of our railway companies have been sadly dislocated by this enforced incessant expenditure, and our managers are much exercised to determine what to charge to capital and what against annual revenue. The local authorities and municipalities also are insatiable in their unscrupulous assessment for local taxation. The growth of this drain on the resources of the companies is Traffic Acts of 1888 and 1892 forbids our railways being worked on commercial lines so far as goods and mineral traffic are concerned. What is to be said of a law which places an impediment in the way of reducing rates by enacting that they may not be restored, if their reduction be found not to have led to the expected result, without liability to an expensive law suit? Or to the refusal of permission to restore the rate to its old figure without such elaborate proof of change of circumstances as shall satisfy the Railway Commissioners that it is right to do so? Other industries could not live if exposed to such conditions, and their effect is most detrimental to our railways. The Board of Trade is ceaseless in its 1895 alarming. The taxation of railways has increased 75 per cent. in the last decade, while that of the whole community has increased only 39 per cent. Trade unionism has generated a serious labour trouble. Shorter hours, greater pay, enlarged staff are very desirable for the men themselves, but these advantages are not to be acquired if they lead to financial deadlock and to the disregard of the dictates of commercial law. The men cannot obey two masters, nor can the first masters submit to the external management of their business by self-constituted second masters. If this were submitted to, chaos and bankruptcy would be the result. The railways are now subject to a very serious competition in the introduction of |