BOOK NOTES AND BIBLIOGRAPHY. 
BOOK NOTES. 
Interpolationsrechiung, by T. N. Thiele. Leipzig. B. G. Teubner, 1909, M. 10. 
This interesting work which is divided into four chapters begins by evolving the general inter- 
polation formula 
X=A + {.i--a)[b' (ff, 6) + -/>){§"(«,..., e) + (.r-c){S"'(ff, d) + . ..]]-] 
V. , X-A , ^„ , 8' (j.; a) - 8' (a, b) ^ 
where 8 {.v, a)= and 5' (,/•,..., Z))= ^ ' ^ etc. 
This formula does not depend as Newton's formula does on the tabulation of the function for 
even differences in the argument and in its practical application one differences the function in 
the usual way and then divides the diffei-ences so obtained by the corresponding difference in the 
argument. (Cp. De Morgan's Differential and Integral Calculus, Ch. xviii., on Interpolation and 
Summation.) The remainder of the first chapter deals with Newton's Method and with series and 
the problems arising from them. The second chapter is devoted to symbolic work and although 
helpful and interesting in places it is rather overburdened with notation, and the third chapter 
gives auxiliary methods such as the graphic and e.xplains the use of a "qualifizirte Differenz " 
which is based on expressions formed symbolically, {E-a)f{:c) = f{x+l)-af(.v) for the first 
difference and {E''--2cbE+b')f{.v)=f(x + 2)-2cbf{x+l) + b^f(x) for the second difference and 
so on. These differences are applied in connection with exponential and periodic functions. 
The last chapter of the book deals with interpolation when there is more than one independent 
variable, but contains no new suggestions. 
Interpolation is ^lerhaps chiefly serviceable in its arithmetical applications and we can recom- 
mend the jiresent work not only for its examples on this part of the subject but also for its 
interesting theoretical treatment. 
W. P. E. 
The Theory of the Construction of Tables of Mortality. A course of Lectures by 
G. F. Hardy, F.LA., Delivered during the Session 1904—5. Published 
for the Institute of Actuaries by C. and E. Lay ton, 1909. 
These Lectures which were intended mainly for Actuaries deal with graduation and the 
problems arising fi'om it. They recount briefly the older methods of graduation such as the 
Graphic and Woolhouse's Difference Method but a,re chiefly concerned with the fitting of curves 
and with the practical use of Makeham's hypothesis for graduating a mortality Table. 
It is a pity that owing to the omission of those parts of the subject with which his audience 
was familiar and to the fact that Mr Hardy has not had time to enlarge what were notes for 
Biometrika vii 53 
