Prof. Ewing, The teaching of Mechanics by experiment. 317 



how to apply these principles to practical questions and how to 

 detect and allow for the causes which produce aberrations from 

 what may be called the theoretical result. He exhibited in 

 illustration of his remarks a number of self-contained pieces of 

 apparatus for experiments in Statics, Dynamics and Elasticity. 



On the calculation of the double integral expressing normal 

 con-elation. By Mr W. F. Sheppard, M.A., LL.M. 



[Read 5 February 1900.] 



[Abstract, v. Transactions, Vol. xix. Pt. I.] 



When the measures of two organs vary about their mean 

 value according to the normal law, and the statistical correlation 

 of the two sets of variations is also normal, the frequency of 

 joint variation within any selected limits is expressed by the 

 integral considered in the paper. The integral involves five 

 parameters, one being the angle which measures the statistical 

 divergence of the one set of variations from the other (this 

 divergence being a right angle when the variations are inde- 

 pendent, and tending towards zero or two right angles for perfect 

 positive or negative correlation) ; but it can be split up into 

 eight terms, each of which is a definite function of two para- 

 meters. It is therefore only necessary to tabulate this function 

 in terms of its two parameters. The table has not been con- 

 structed, but formulae are given by means of which the function 

 can be calculated in any particular case. 



The integral is required in two classes of cases. In the first, 

 the data represent actual measurements, and the object is to 

 determine whether the correlation is normal. For dealing with 

 these cases, in the absence of the desired table, an alternative but 

 less convenient method is given, accompanied by a complete 

 table. In the second class of cases the characteristics observed 

 are not capable of quantitative measurement, but their presence 

 or absence is regarded as dependent on the greater or less de- 

 velopment, according to the normal law, of some physiological 

 factor. (Some of the more important questions of assortative 

 mating and heredity come under this head.) The data then 

 shew the relative frequency of the presence or absence of the one 

 characteristic in connexion with the presence or absence of the 

 other, and the problem is to determine the divergence. This is 

 done by expressing the double integral as a single integral, in 

 which the divergence is the independent variable, and then 

 applying the ordinary methods of quadrature. 



