450 Proceedings of Royal Society of Edinburgh. [sess. 
devotion to this particular branch of astronomical observation. It 
is the desire of the Society to recognise by this award the value and 
importance of Dr Anderson’s work in a field of astronomical re- 
search where results can be obtained only by the most determined 
perseverance and by an unabating enthusiasm and love for science. 
In conclusion, I have to express the extreme regret of the 
Astronomer-Royal for Scotland that illness prevents him from 
being present on this memorable occasion. 
The Keith Prize for 1897-99 was presented to Dr James 
2 ft 2 
Burgess for his paper “ On the Definite Integral — r I e~ dt , 
sJttJ o 
with extended Tables of Values,” printed in vol. xxxix. of the 
Transactions of the Society. 
The Chairman, on presenting the Prize, said : — 
The Keith Prize for the Sessions 1897-8, 1898-9, is awarded to 
James Burgess, Esq., C.I.E., LL.D., for his paper on “The 
2 ft 2 
Definite Integral — — I dt , with Extended Tables of Values,” 
Jttj 0 
published in the Society’s Transactions. This integral is of import- 
ance in various fields of physical science, such as theory of atmospheric 
refraction, conduction of heat, probabilities, errors of observation, 
etc. It is also of fundamental importance in the evaluation of 
many other forms of definite integrals. A closely connected 
integral was tabulated in 1789 by Kramp, and various tables of 
both integrals have been computed or compiled by different 
authors since that date. Dr Burgess’s tables are, however, calcu- 
lated to a greater number of significant figures than in any of these 
earlier tables, being for certain values of the limit computed to 
fifteen decimal places. The logarithms are in these cases given to 
sixteen places, and the table is prepared for all practical purposes 
by being provided with tables of differences as far as the fourth 
order. The arithmetical labour involved in constructing such a 
table must have been enormous, and could have been accomplished 
only by a calculator of rare accuracy and power. In addition to 
the tabulated values, which fill thirty-nine pages of the Society’s 
Transactions , the memoir itself contains a brief history of the 
subject, and a luminous account of the methods adopted in making 
