PROGRESS OF PURE MATHEMATICS IN 1900. 165 
plicated. The author deserves our gratitude for the effort 
he has made to lighten the labors of the reader. 
Painleve’s latest work was, of course, not accessible to Pro¬ 
fessor Forsyth. Many examples are scattered through the 
text, a welcome feature always present in English books, 
quite a number of which are worked out in great detail, but 
often with the aid of theorems borrowed from the theory of 
the linear equations and iefc undemonstrated. Since the 
author announces his intention to publish a fourth vglume 
on linear equations, one cannot help asking why the order 
was not reversed. 
On the whole, the reviewer desires to congratulate Pro¬ 
fessor Forsyth on his treatise on a subject which is in such 
an unsatisfactory shape for such a work, and to suggest that 
when Painleve’s promised memoirs appear much more de¬ 
finitive matter will be available for a second edition, part of 
which might well take the place of the chapter on sub- 
uniform integrals. 
Conclusion .—Let me conclude with a simile. The ordi¬ 
nary non-linear equations may be likened to a lump of 
rebellious ore unresolved into its elements and resisting all 
known processes. To be sure, the conglomerate mass may 
be studied directly, but the results obtained are affected by 
all the elements and compounds present, and are of little 
value. Consequently every newly discovered reagent is at 
once tried upon it with the hope and breathless expectation 
of isolating some new element. Briot and Bouquet tried 
one reagent on a fragment of the lump, and, although it 
appeared to bite into its very heart, a careful analysis of the 
resulting liquid revealed only elements long known. Fuchs, 
Poincare, Picard, and Painleve in turn used new reagents of 
ever increasing resolvent power, but each time only old ele¬ 
ments resulted. Finally that giant of analysists, Painleve, 
compounded yet a new reagent, and with the audacity of 
youth poured it on the whole lump. The effect was instanta¬ 
neous. The entire mass seemed to melt under its scorching 
fumes. An analysis of only a part of the resulting solution 
disclosed, not one, but three new elements, whose properties 
he has undertaken to discover. 
