is

100 × 99

1 x 2

98

(•91) × ('09)*, &c. The different probabilities are

exhibited for 20 arguments in the following table which has been calculated to 7 figures and contracted to 5 decimal places:

It will be seen from the second column of results that the chance of more than 91 disagreements is 44940, and from the third column that the chance of less than 91 disagreements is 41249 so that (following the analogy of the term 'probable error' in the Theory of Errors) the 'probable number of disagreements' is 91, viz. the number of disagreements is about as likely to exceed as to fall short of 91. It also happens that 91 disagreements in the hundred is the most likely number of disagreements; and it is pretty evident that generally the 'probable number' will not differ much from the most likely number.

The table shows that the odds against more than 20 agreements are about 5000 to 1.

If on the average a per cent. of the calculations performed in a given way are erroneous, then in a piece of work so calculated in duplicate, it follows from elementary considera(100—a)2

tions that the most likely number of disagreements is 100 per cent, and it will not be far from the truth if this be also taken as the 'probable' percentage of disagreements also.

A similar discussion of triplicate calculations is of interest, but the question is complicated by other matters that do not enter into the simpler case just considered.

EXERCISES IN THE INTEGRAL CALCULUS. NO. II.

By Sir James Cockle, F.R.S.

1. AT pp. 249-251 of Vol. III. (Old Series), discussing a unique case in the theory of coresolvents, I have arrived at results which may be expressed as follows: From

Substituting this value of u in (6) and reducing, we have

and, by equating these two values and after expunging the

and when this condition is satisfied, then, referring to (8) and (11), we see that (5) is soluble by means of

3. Equation (12) has an integrating factor, but before proceeding with its general solution I shall notice a special case. Let a = 0, then (12) becomes

Now (see Hymers, 1835, p. 194, Ex. 1) if the modulus

b= CB....

...(16),

.(17).

5. By substitution in (17) we obtain a result of the form

THE COLLECTION OF MODELS OF RULED SURFACES AT SOUTH KENSINGTON.

THIS collection is very remarkable, whether we have regard to the number of different surfaces it includes, to the various means of deforming them, or to the perfect finish of the models themselves. It is believed to be the finest of its kind in existence. Its history is rather curious. Some officers of the Science and Art Department, at South Kensington, had recommended the formation of a collection of models similar to those which had been found so useful in the Conservatoire des Arts et Métiers and in the Ecole Polytechnique in Paris. An order was consequently given, perhaps 12 years ago, to a M. Fabre de Lagrange, who had constructed some of them, to make a set for the Educational Museum at South Kensington. It appears that for some years M. Fabre obtained better employment. At any rate the order was not executed, and was considered as lapsed. During the siege of Paris, however, employment being scarce, M. Fabre set about the execution of the almost forgotten order, and carried it out successfully amidst many risks and privations-trials through which he lost both his wife and his mother. The models themselves narrowly escaped destruction by the bursting of a shell in the next room. the end of the second siege, when Paris was again pretty open, he wrote to South Kensington to ask whether he might consider the order as still outstanding, and under the circumstances it was, very wisely as well as considerately, decided to make the purchase.

At

The preparation of the catalogue was entrusted to Mr. Merrifield, then Principal of the School of Naval Architecture at South Kensington. The catalogue contains a running commentary on the models, as each is described, and is followed by an appendix, containing an account of the application of analysis to the investigation and classification of ruled surfaces. This is chiefly founded on Monge's

* A Catalogue of a Collection of Models of Ruled Surfaces, constructed by M. Fabre de Lagrange, with an Appendix containing an account of the Application of Analysis to their Investigation and Classification. By C. W. Merrifield, F.R.S. London, 1872 (Eyre and Spottiswoode, for the Stationery Office).