NA TURE 



[December 23, 1909 



therefore, also be regarded as primitive, thoug-h 

 separated from Decolopoda by many differences. Tlie 

 Pallenidje are closely allied to the Xymphonidse. The 

 Phoxichilidiidae have points of resemblance with the 

 Pallenida?, and the Pycnogonidse are probably allied 

 to them. The ideas here expressed as to the relations 

 of the different families have, however, recently been 

 questioned by Carpenter, whose views have received 

 support from Caiman. .According to these authors, 

 the fifth pair of legs in Decolopoda and in Penta- 

 nymphon may possibly represent a comparativelv new 

 development and not a primitive character. 



UORTAUTY TABLES. 

 The Theory of the Construction of Tables of Mortality 

 and of Similar Statistical Tables in Use by the 

 Actuary. A course of lectures delivered at the 

 Institute of Actuaries, Staple Inn Hall, during the 

 Session 1904-5, by G. F. Hardy. Pp. iii + 141. 

 (London : C. and E. Layton, 1909.) Price ys. 6d. 



THIS course of lectures, which was delivered under 

 the auspices of the Institute of Actuaries, deals 

 "with the construction of mortality and similar tables 

 which, as the author justly observes in his opening 

 •sentence, lie at the very basis of actuarial work. 

 They deal succinctly with familiar methods of gradua- 

 tion, such as the graphic and Woolhouse's difference 

 method, but are for the most part devoted to more 

 TiTiodern theories of curve-fitting, and to the applica- 

 tion of Makeham's hypothesis in dealing with the 

 somewhat intractable curve which arises from the 

 fact that, with assured lives, the rate of mortality' is 

 for several years a function of time that has elapsed 

 since medical examination rather than of age. 



It may be remarked that a subject involving much 

 technical mathematical detail cannot be satisfactorily 

 dealt with in the form of lectures. Mr. Hardy's 

 first two lectures deal with methods of graduation 

 which are familiar to most actuaries, and can be suit- 

 ably presented in this form, but the remaining four 

 lectures contain much original work, which can only 

 be thoroughly understood after careful reading and 

 ■study. Fortunately, Mr. Hardy appears to have 

 realised the limitations of his medium, and in its 

 present form the work is well suited to the actuarial 

 student. 



The publication of this book is of special interest, 

 as a perusal of it shows that modern development in 

 the graduation of tables of mortality has, singularly 

 enough, had its impulse and inspiration from outside 

 the actuarial profession. To Prof. Karl Pearson's 

 original work in the field of biological statistics, 

 actuaries are indebted for the new calculus, which 

 was applied by Mr. Hardy in the graduation of the 

 principal mortality tables compiled from the experi- 

 ence of lives assured in British offices, and published 

 a few years ago. Actuaries, indeed, cannot be said 

 to have taken very readily to the new method, and 

 during the last three years there has been a surpris- 

 ing number of contributions in the Journal of the 

 Institute of Actuaries dealing with the development 

 of those finite diff'erence formulas to which Mr. Hardy 

 .\'0. 2095, VOL. 82] 



devotes only half-a-dozen pages in this book, and 

 which will, we hope, in a few years be considered 

 obsolete. 



The most interesting part of the book is that deal- 

 ing with the Pearsonian frequency curves, and it is 

 suggestive of the exhaustive nature of the work done 

 by Prof. Pearson to find that so original a thinker 

 as Mr. Hardy has practically nothing to add to the 

 information we have already received about this im- 

 portant family of curves. He, in fact, refers the 

 reader to Prof. Pearson's works, and to the treatise 

 of Mr. W. Palin Elderton, for fuller information. The 

 latter work was published only three years ago for 

 the benefit of actuaries, and will, we think, hold the 

 field as the only te.xt-book on the subject for some 

 vears. In these circumstances, it may perhaps be 

 regretted that Mr. Hardy has seen fit to re-number 

 the types of curves, as this may easily confuse anyone 

 who finds occasion for reference to both books, or to 

 Pearson's original work, which Elderton follows. 

 The student will also be in difficulty at the outset 

 owing to Mr. Hardy giving the differential equation 

 from which these curves are derived as 

 I rfr _ Ki - X- 

 y ' lt\~ a - bx - y.\"' 

 instead of the one to which we are accustomed, 

 I dy _ x + a 



y ' dx A|, + b^x + b.^" + . . . 

 and we may mention that a misprint would appear 

 to have been introduced here, as it does not seem 

 possible to derive Pearson's Type I. from the first- 

 mentioned equation. 



It is unfortunate that Mr. Hardy has not illustrated 

 this part of his subject by reference to the chief 

 mortalitv table, in the graduation of which the method 

 of frequency curves has been employed, and 

 was carried out by himself. When it is remembered 

 that this will now be the standard mortality table for 

 many life-insurance purposes, it seems strange that 

 so favourable an opportunity should have been 

 missed. 



In justice, however, to the author, it may be as 

 well to say that, taking into consideration the space 

 at his disposal, he has done wisely in devoting so 

 large a proportion of it to the study of the Makeham 

 curves, in the knowledge and manipulation of which 

 he is so able an exponent. .At the second Inter- 

 national Congress of .Actuaries, one of the most 

 eminent of Continental actuaries stated that, in his 

 opinion, the day had entirely gone past in which 

 Makeham's graduation would be practically applied 

 in the graduation of tables; and it is a singular com- 

 mentary on this statement that the select annuity and 

 assurance tables of the recent experience have been 

 graduated by the application of the formula in ques- 

 tion. In order to prove the importance of this matter 

 to actuaries, it is only necessary to point out that 

 the value of an annuity payable during the joint life 

 of two persons of any age can be found from a table 

 giving the annuity values for two lives of equal age, 

 whereas, in the case of a table graduated on different 

 principles, a large volume would be required. 



Mr. Hardy's illustrations of the application of 



