MATHEMATICS AND ACTUARIAL SCIENCE. 285 



and is spoken of as the curve of constant fourth differences. From 

 this, by the differential calculus, and that of finite differences, other 

 formula? for interpolation are obtained. 



In quitting, for this present purpose, the branch of life insurance,. 

 I may remark that these mathematical niceties and absolute formulae- 

 added to the mercantile reserves and precautions render a policy 

 on one's life in a good office, spite of the uncertainty of life, one of 

 the most certain things in the world. 



I class Friendly Societies as the next branch of actuarial science 

 where mathematics are applicable. As the question of sickness- 

 incapacitating from work is now involved, as well as that of death,, 

 one is sorry that, though they have more need, they have made less 

 use of the certainties and formulae of science. The principle of 

 the sickness question is that the society receives so much a week 

 from the member during life, or up to a certain age, and undertakes 

 to pay him a certain amount a week when too ill to work. This 

 sick pay is often diminished after the first six months of sickness^ 

 Again the same processes have to be gone through, and first the 

 collection of statistics. Here these are done by the societies, or 

 the larger associations, comprising many societies, such as the 

 Oddfellows, Rechabites, Foresters, &c. Then these are adjusted 

 by the actuar}' by some of the previous methods, and w^hen adjusted, 

 commutation columns, including now additional cokimns for each 

 class of sickness, first six months, second six months, and subse- 

 quently, are constructed, and from them the present value of the 

 sick pay is formed. 



Their assets and liabilities, and their financial position, are also 

 valued by similar methods to those indicated for life companies. 



I hope that these societies will soon pass into the stage of 

 scientific exactitude, as there is no reason why the weekly contribu- 

 tions should not bear a strictly mathematical relation to the sick 

 pay to be received. It will be seen, therefore, that sickness matters 

 are considerably behind life ones in scientific exactitude; for while, 

 as regards both the mathematical theory and the data to which it 

 can be applied, the science of life contingencies may now^ be said to 

 be nearly perfect, the extension of these principles to the work of 

 friendly societies is still in its infancy. 



Insurances against Issue are efi'ected by a good many companies, 

 and to apply the science to these matters we require a combined 

 marriage and mortality table, and to be able to answer such questions 

 as — What is the probability that a bachelor of a given age will (1) 

 marry, or (2) die unmarried in an assigned year from the present 

 time, or (3) be alive and still unmarried after the lapse of a given 

 number of years ; and some progress has been made in the collec- 

 tion of statistics, graduation, &;c. ; also in the preparation of such 

 a table with the value of monetary benefits dependent thereon. 

 Similar to these may be mentioned the qiiestion oi family annuities^ 

 which are spoken of quite recently as a very important and difficult 

 element in some of the State insurance schemes which have lately 



